Properties

Degree 46
Conductor $ 2^{92} \cdot 251^{23} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 8·5-s − 2·7-s − 10·9-s − 8·11-s + 8·13-s − 16·15-s + 19·17-s + 9·19-s + 4·21-s − 21·23-s + 7·25-s + 23·27-s + 10·29-s + 9·31-s + 16·33-s − 16·35-s + 11·37-s − 16·39-s + 35·41-s + 9·43-s − 80·45-s − 37·47-s − 40·49-s − 38·51-s + 38·53-s − 64·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 3.57·5-s − 0.755·7-s − 3.33·9-s − 2.41·11-s + 2.21·13-s − 4.13·15-s + 4.60·17-s + 2.06·19-s + 0.872·21-s − 4.37·23-s + 7/5·25-s + 4.42·27-s + 1.85·29-s + 1.61·31-s + 2.78·33-s − 2.70·35-s + 1.80·37-s − 2.56·39-s + 5.46·41-s + 1.37·43-s − 11.9·45-s − 5.39·47-s − 5.71·49-s − 5.32·51-s + 5.21·53-s − 8.62·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{92} \cdot 251^{23}\right)^{s/2} \, \Gamma_{\C}(s)^{23} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{92} \cdot 251^{23}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{23} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(46\)
\( N \)  =  \(2^{92} \cdot 251^{23}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4016} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(46,\ 2^{92} \cdot 251^{23} ,\ ( \ : [1/2]^{23} ),\ 1 )$
$L(1)$  $\approx$  $82.60123368$
$L(\frac12)$  $\approx$  $82.60123368$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;251\}$,\(F_p(T)\) is a polynomial of degree 46. If $p \in \{2,\;251\}$, then $F_p(T)$ is a polynomial of degree at most 45.
$p$$F_p(T)$
bad2 \( 1 \)
251 \( ( 1 - T )^{23} \)
good3 \( 1 + 2 T + 14 T^{2} + 25 T^{3} + 98 T^{4} + 143 T^{5} + 151 p T^{6} + 533 T^{7} + 182 p^{2} T^{8} + 1624 T^{9} + 1880 p T^{10} + 5201 T^{11} + 20627 T^{12} + 19271 T^{13} + 77380 T^{14} + 69863 T^{15} + 270410 T^{16} + 24182 p^{2} T^{17} + 843031 T^{18} + 578786 T^{19} + 2472794 T^{20} + 1493482 T^{21} + 7253098 T^{22} + 4256272 T^{23} + 7253098 p T^{24} + 1493482 p^{2} T^{25} + 2472794 p^{3} T^{26} + 578786 p^{4} T^{27} + 843031 p^{5} T^{28} + 24182 p^{8} T^{29} + 270410 p^{7} T^{30} + 69863 p^{8} T^{31} + 77380 p^{9} T^{32} + 19271 p^{10} T^{33} + 20627 p^{11} T^{34} + 5201 p^{12} T^{35} + 1880 p^{14} T^{36} + 1624 p^{14} T^{37} + 182 p^{17} T^{38} + 533 p^{16} T^{39} + 151 p^{18} T^{40} + 143 p^{18} T^{41} + 98 p^{19} T^{42} + 25 p^{20} T^{43} + 14 p^{21} T^{44} + 2 p^{22} T^{45} + p^{23} T^{46} \)
5 \( 1 - 8 T + 57 T^{2} - 297 T^{3} + 1407 T^{4} - 5743 T^{5} + 21864 T^{6} - 15152 p T^{7} + 9983 p^{2} T^{8} - 154059 p T^{9} + 2294798 T^{10} - 6529686 T^{11} + 18145939 T^{12} - 48790593 T^{13} + 129113954 T^{14} - 333018454 T^{15} + 848229373 T^{16} - 2112257153 T^{17} + 5197716812 T^{18} - 12515292272 T^{19} + 29769657897 T^{20} - 69303839608 T^{21} + 31865979679 p T^{22} - 358533016454 T^{23} + 31865979679 p^{2} T^{24} - 69303839608 p^{2} T^{25} + 29769657897 p^{3} T^{26} - 12515292272 p^{4} T^{27} + 5197716812 p^{5} T^{28} - 2112257153 p^{6} T^{29} + 848229373 p^{7} T^{30} - 333018454 p^{8} T^{31} + 129113954 p^{9} T^{32} - 48790593 p^{10} T^{33} + 18145939 p^{11} T^{34} - 6529686 p^{12} T^{35} + 2294798 p^{13} T^{36} - 154059 p^{15} T^{37} + 9983 p^{17} T^{38} - 15152 p^{17} T^{39} + 21864 p^{17} T^{40} - 5743 p^{18} T^{41} + 1407 p^{19} T^{42} - 297 p^{20} T^{43} + 57 p^{21} T^{44} - 8 p^{22} T^{45} + p^{23} T^{46} \)
7 \( 1 + 2 T + 44 T^{2} + 81 T^{3} + 1090 T^{4} + 1809 T^{5} + 19918 T^{6} + 30489 T^{7} + 297350 T^{8} + 426283 T^{9} + 3819301 T^{10} + 5168994 T^{11} + 43475307 T^{12} + 56041147 T^{13} + 447335992 T^{14} + 552541757 T^{15} + 4213876171 T^{16} + 5007941488 T^{17} + 36635213763 T^{18} + 42006826668 T^{19} + 295580729477 T^{20} + 327369188873 T^{21} + 2219824920535 T^{22} + 2374813175202 T^{23} + 2219824920535 p T^{24} + 327369188873 p^{2} T^{25} + 295580729477 p^{3} T^{26} + 42006826668 p^{4} T^{27} + 36635213763 p^{5} T^{28} + 5007941488 p^{6} T^{29} + 4213876171 p^{7} T^{30} + 552541757 p^{8} T^{31} + 447335992 p^{9} T^{32} + 56041147 p^{10} T^{33} + 43475307 p^{11} T^{34} + 5168994 p^{12} T^{35} + 3819301 p^{13} T^{36} + 426283 p^{14} T^{37} + 297350 p^{15} T^{38} + 30489 p^{16} T^{39} + 19918 p^{17} T^{40} + 1809 p^{18} T^{41} + 1090 p^{19} T^{42} + 81 p^{20} T^{43} + 44 p^{21} T^{44} + 2 p^{22} T^{45} + p^{23} T^{46} \)
11 \( 1 + 8 T + 85 T^{2} + 531 T^{3} + 3558 T^{4} + 18804 T^{5} + 101980 T^{6} + 488074 T^{7} + 2332588 T^{8} + 10401211 T^{9} + 45601868 T^{10} + 191713545 T^{11} + 787231609 T^{12} + 3148993229 T^{13} + 1116887153 p T^{14} + 47010189836 T^{15} + 176019284214 T^{16} + 647268576335 T^{17} + 2339790722440 T^{18} + 8298239678772 T^{19} + 29076967085774 T^{20} + 99724770760125 T^{21} + 339332306630432 T^{22} + 1129111282465428 T^{23} + 339332306630432 p T^{24} + 99724770760125 p^{2} T^{25} + 29076967085774 p^{3} T^{26} + 8298239678772 p^{4} T^{27} + 2339790722440 p^{5} T^{28} + 647268576335 p^{6} T^{29} + 176019284214 p^{7} T^{30} + 47010189836 p^{8} T^{31} + 1116887153 p^{10} T^{32} + 3148993229 p^{10} T^{33} + 787231609 p^{11} T^{34} + 191713545 p^{12} T^{35} + 45601868 p^{13} T^{36} + 10401211 p^{14} T^{37} + 2332588 p^{15} T^{38} + 488074 p^{16} T^{39} + 101980 p^{17} T^{40} + 18804 p^{18} T^{41} + 3558 p^{19} T^{42} + 531 p^{20} T^{43} + 85 p^{21} T^{44} + 8 p^{22} T^{45} + p^{23} T^{46} \)
13 \( 1 - 8 T + 138 T^{2} - 981 T^{3} + 10034 T^{4} - 63745 T^{5} + 497771 T^{6} - 2860153 T^{7} + 18708064 T^{8} - 98280836 T^{9} + 43388338 p T^{10} - 2734319527 T^{11} + 14151177065 T^{12} - 63829096419 T^{13} + 303468602078 T^{14} - 1283619217139 T^{15} + 5686858402886 T^{16} - 22735955633504 T^{17} + 95000107011099 T^{18} - 361839660829612 T^{19} + 1440847005548176 T^{20} - 5265690284891736 T^{21} + 20142534120479870 T^{22} - 70965453892411560 T^{23} + 20142534120479870 p T^{24} - 5265690284891736 p^{2} T^{25} + 1440847005548176 p^{3} T^{26} - 361839660829612 p^{4} T^{27} + 95000107011099 p^{5} T^{28} - 22735955633504 p^{6} T^{29} + 5686858402886 p^{7} T^{30} - 1283619217139 p^{8} T^{31} + 303468602078 p^{9} T^{32} - 63829096419 p^{10} T^{33} + 14151177065 p^{11} T^{34} - 2734319527 p^{12} T^{35} + 43388338 p^{14} T^{36} - 98280836 p^{14} T^{37} + 18708064 p^{15} T^{38} - 2860153 p^{16} T^{39} + 497771 p^{17} T^{40} - 63745 p^{18} T^{41} + 10034 p^{19} T^{42} - 981 p^{20} T^{43} + 138 p^{21} T^{44} - 8 p^{22} T^{45} + p^{23} T^{46} \)
17 \( 1 - 19 T + 19 p T^{2} - 3772 T^{3} + 40296 T^{4} - 360070 T^{5} + 2998319 T^{6} - 22282581 T^{7} + 156164297 T^{8} - 1006266674 T^{9} + 6163685290 T^{10} - 35275383592 T^{11} + 193008867729 T^{12} - 996400365357 T^{13} + 4941177792716 T^{14} - 23284138720462 T^{15} + 105972744751230 T^{16} - 461715253379157 T^{17} + 1960120821398653 T^{18} - 8057520604913551 T^{19} + 32784277501095762 T^{20} - 131553154214193285 T^{21} + 533842816248554335 T^{22} - 2175949675700281966 T^{23} + 533842816248554335 p T^{24} - 131553154214193285 p^{2} T^{25} + 32784277501095762 p^{3} T^{26} - 8057520604913551 p^{4} T^{27} + 1960120821398653 p^{5} T^{28} - 461715253379157 p^{6} T^{29} + 105972744751230 p^{7} T^{30} - 23284138720462 p^{8} T^{31} + 4941177792716 p^{9} T^{32} - 996400365357 p^{10} T^{33} + 193008867729 p^{11} T^{34} - 35275383592 p^{12} T^{35} + 6163685290 p^{13} T^{36} - 1006266674 p^{14} T^{37} + 156164297 p^{15} T^{38} - 22282581 p^{16} T^{39} + 2998319 p^{17} T^{40} - 360070 p^{18} T^{41} + 40296 p^{19} T^{42} - 3772 p^{20} T^{43} + 19 p^{22} T^{44} - 19 p^{22} T^{45} + p^{23} T^{46} \)
19 \( 1 - 9 T + 206 T^{2} - 1373 T^{3} + 19130 T^{4} - 100157 T^{5} + 58747 p T^{6} - 4629694 T^{7} + 47187785 T^{8} - 151278036 T^{9} + 1584354730 T^{10} - 3702319055 T^{11} + 45493720762 T^{12} - 69862751894 T^{13} + 1181868637605 T^{14} - 984021037348 T^{15} + 28551773518448 T^{16} - 6962655134781 T^{17} + 33845684641780 p T^{18} + 150855929085788 T^{19} + 710759927767408 p T^{20} + 7741610671080157 T^{21} + 267683588371142524 T^{22} + 185517803279571588 T^{23} + 267683588371142524 p T^{24} + 7741610671080157 p^{2} T^{25} + 710759927767408 p^{4} T^{26} + 150855929085788 p^{4} T^{27} + 33845684641780 p^{6} T^{28} - 6962655134781 p^{6} T^{29} + 28551773518448 p^{7} T^{30} - 984021037348 p^{8} T^{31} + 1181868637605 p^{9} T^{32} - 69862751894 p^{10} T^{33} + 45493720762 p^{11} T^{34} - 3702319055 p^{12} T^{35} + 1584354730 p^{13} T^{36} - 151278036 p^{14} T^{37} + 47187785 p^{15} T^{38} - 4629694 p^{16} T^{39} + 58747 p^{18} T^{40} - 100157 p^{18} T^{41} + 19130 p^{19} T^{42} - 1373 p^{20} T^{43} + 206 p^{21} T^{44} - 9 p^{22} T^{45} + p^{23} T^{46} \)
23 \( 1 + 21 T + 417 T^{2} + 5853 T^{3} + 75982 T^{4} + 844016 T^{5} + 8766725 T^{6} + 82718429 T^{7} + 737882514 T^{8} + 6141387958 T^{9} + 48721478508 T^{10} + 365873032696 T^{11} + 2634383801131 T^{12} + 18111227366519 T^{13} + 119894491603658 T^{14} + 762043142752528 T^{15} + 4678009887259510 T^{16} + 27673055539650698 T^{17} + 158444074540437625 T^{18} + 876278870819477713 T^{19} + 4697210607535983566 T^{20} + 24356289942769944423 T^{21} + \)\(12\!\cdots\!86\)\( T^{22} + \)\(59\!\cdots\!04\)\( T^{23} + \)\(12\!\cdots\!86\)\( p T^{24} + 24356289942769944423 p^{2} T^{25} + 4697210607535983566 p^{3} T^{26} + 876278870819477713 p^{4} T^{27} + 158444074540437625 p^{5} T^{28} + 27673055539650698 p^{6} T^{29} + 4678009887259510 p^{7} T^{30} + 762043142752528 p^{8} T^{31} + 119894491603658 p^{9} T^{32} + 18111227366519 p^{10} T^{33} + 2634383801131 p^{11} T^{34} + 365873032696 p^{12} T^{35} + 48721478508 p^{13} T^{36} + 6141387958 p^{14} T^{37} + 737882514 p^{15} T^{38} + 82718429 p^{16} T^{39} + 8766725 p^{17} T^{40} + 844016 p^{18} T^{41} + 75982 p^{19} T^{42} + 5853 p^{20} T^{43} + 417 p^{21} T^{44} + 21 p^{22} T^{45} + p^{23} T^{46} \)
29 \( 1 - 10 T + 320 T^{2} - 2538 T^{3} + 47690 T^{4} - 311575 T^{5} + 4527718 T^{6} - 24699797 T^{7} + 312655631 T^{8} - 1414234202 T^{9} + 16904648527 T^{10} - 61416175064 T^{11} + 750507358267 T^{12} - 2033418011431 T^{13} + 28334279819168 T^{14} - 48047262575832 T^{15} + 938184822568139 T^{16} - 533814246820859 T^{17} + 28184547152967933 T^{18} + 17200116366678882 T^{19} + 801251194072980351 T^{20} + 1295216860208595141 T^{21} + 22605699782976821411 T^{22} + 46540103422923029594 T^{23} + 22605699782976821411 p T^{24} + 1295216860208595141 p^{2} T^{25} + 801251194072980351 p^{3} T^{26} + 17200116366678882 p^{4} T^{27} + 28184547152967933 p^{5} T^{28} - 533814246820859 p^{6} T^{29} + 938184822568139 p^{7} T^{30} - 48047262575832 p^{8} T^{31} + 28334279819168 p^{9} T^{32} - 2033418011431 p^{10} T^{33} + 750507358267 p^{11} T^{34} - 61416175064 p^{12} T^{35} + 16904648527 p^{13} T^{36} - 1414234202 p^{14} T^{37} + 312655631 p^{15} T^{38} - 24699797 p^{16} T^{39} + 4527718 p^{17} T^{40} - 311575 p^{18} T^{41} + 47690 p^{19} T^{42} - 2538 p^{20} T^{43} + 320 p^{21} T^{44} - 10 p^{22} T^{45} + p^{23} T^{46} \)
31 \( 1 - 9 T + 309 T^{2} - 2414 T^{3} + 48644 T^{4} - 346895 T^{5} + 5270183 T^{6} - 35014374 T^{7} + 14245310 p T^{8} - 2762966597 T^{9} + 30438022579 T^{10} - 180492415843 T^{11} + 1791146742118 T^{12} - 10109769801040 T^{13} + 92204603692939 T^{14} - 496858634832037 T^{15} + 4221562270137499 T^{16} - 21759558232041320 T^{17} + 173863053158385388 T^{18} - 857924125682156229 T^{19} + 6489362982535015867 T^{20} - 30650901652033516267 T^{21} + \)\(22\!\cdots\!81\)\( T^{22} - \)\(99\!\cdots\!06\)\( T^{23} + \)\(22\!\cdots\!81\)\( p T^{24} - 30650901652033516267 p^{2} T^{25} + 6489362982535015867 p^{3} T^{26} - 857924125682156229 p^{4} T^{27} + 173863053158385388 p^{5} T^{28} - 21759558232041320 p^{6} T^{29} + 4221562270137499 p^{7} T^{30} - 496858634832037 p^{8} T^{31} + 92204603692939 p^{9} T^{32} - 10109769801040 p^{10} T^{33} + 1791146742118 p^{11} T^{34} - 180492415843 p^{12} T^{35} + 30438022579 p^{13} T^{36} - 2762966597 p^{14} T^{37} + 14245310 p^{16} T^{38} - 35014374 p^{16} T^{39} + 5270183 p^{17} T^{40} - 346895 p^{18} T^{41} + 48644 p^{19} T^{42} - 2414 p^{20} T^{43} + 309 p^{21} T^{44} - 9 p^{22} T^{45} + p^{23} T^{46} \)
37 \( 1 - 11 T + 373 T^{2} - 2898 T^{3} + 62189 T^{4} - 340255 T^{5} + 6407122 T^{6} - 21439028 T^{7} + 470473417 T^{8} - 458240398 T^{9} + 27190740068 T^{10} + 51368286940 T^{11} + 1375165545142 T^{12} + 6839141608544 T^{13} + 68563295285466 T^{14} + 482857120682700 T^{15} + 3589991390841137 T^{16} + 25401111342779487 T^{17} + 188761236520700025 T^{18} + 1113423522178502742 T^{19} + 9188535062932007651 T^{20} + 43834954624335089097 T^{21} + \)\(39\!\cdots\!41\)\( T^{22} + \)\(16\!\cdots\!04\)\( T^{23} + \)\(39\!\cdots\!41\)\( p T^{24} + 43834954624335089097 p^{2} T^{25} + 9188535062932007651 p^{3} T^{26} + 1113423522178502742 p^{4} T^{27} + 188761236520700025 p^{5} T^{28} + 25401111342779487 p^{6} T^{29} + 3589991390841137 p^{7} T^{30} + 482857120682700 p^{8} T^{31} + 68563295285466 p^{9} T^{32} + 6839141608544 p^{10} T^{33} + 1375165545142 p^{11} T^{34} + 51368286940 p^{12} T^{35} + 27190740068 p^{13} T^{36} - 458240398 p^{14} T^{37} + 470473417 p^{15} T^{38} - 21439028 p^{16} T^{39} + 6407122 p^{17} T^{40} - 340255 p^{18} T^{41} + 62189 p^{19} T^{42} - 2898 p^{20} T^{43} + 373 p^{21} T^{44} - 11 p^{22} T^{45} + p^{23} T^{46} \)
41 \( 1 - 35 T + 1050 T^{2} - 21841 T^{3} + 409171 T^{4} - 6420370 T^{5} + 93334111 T^{6} - 1211422658 T^{7} + 14815823586 T^{8} - 166810589781 T^{9} + 1789303618792 T^{10} - 17980584177536 T^{11} + 173462449063525 T^{12} - 1585151657798839 T^{13} + 13983378954685896 T^{14} - 117718870832440460 T^{15} + 960538973342513643 T^{16} - 7518053774474798218 T^{17} + 57202128736598770022 T^{18} - \)\(41\!\cdots\!21\)\( T^{19} + \)\(29\!\cdots\!86\)\( T^{20} - \)\(20\!\cdots\!52\)\( T^{21} + \)\(13\!\cdots\!80\)\( T^{22} - \)\(89\!\cdots\!24\)\( T^{23} + \)\(13\!\cdots\!80\)\( p T^{24} - \)\(20\!\cdots\!52\)\( p^{2} T^{25} + \)\(29\!\cdots\!86\)\( p^{3} T^{26} - \)\(41\!\cdots\!21\)\( p^{4} T^{27} + 57202128736598770022 p^{5} T^{28} - 7518053774474798218 p^{6} T^{29} + 960538973342513643 p^{7} T^{30} - 117718870832440460 p^{8} T^{31} + 13983378954685896 p^{9} T^{32} - 1585151657798839 p^{10} T^{33} + 173462449063525 p^{11} T^{34} - 17980584177536 p^{12} T^{35} + 1789303618792 p^{13} T^{36} - 166810589781 p^{14} T^{37} + 14815823586 p^{15} T^{38} - 1211422658 p^{16} T^{39} + 93334111 p^{17} T^{40} - 6420370 p^{18} T^{41} + 409171 p^{19} T^{42} - 21841 p^{20} T^{43} + 1050 p^{21} T^{44} - 35 p^{22} T^{45} + p^{23} T^{46} \)
43 \( 1 - 9 T + 335 T^{2} - 2232 T^{3} + 55845 T^{4} - 323267 T^{5} + 6649770 T^{6} - 35792786 T^{7} + 625248409 T^{8} - 3233344596 T^{9} + 48975580316 T^{10} - 250124106010 T^{11} + 3322750909866 T^{12} - 395004268180 p T^{13} + 200199943485698 T^{14} - 1028836173484940 T^{15} + 10953154582122989 T^{16} - 56513713331682463 T^{17} + 554362952319017479 T^{18} - 2853051709727224174 T^{19} + 26328837525879438635 T^{20} - \)\(13\!\cdots\!61\)\( T^{21} + \)\(11\!\cdots\!39\)\( T^{22} - \)\(59\!\cdots\!80\)\( T^{23} + \)\(11\!\cdots\!39\)\( p T^{24} - \)\(13\!\cdots\!61\)\( p^{2} T^{25} + 26328837525879438635 p^{3} T^{26} - 2853051709727224174 p^{4} T^{27} + 554362952319017479 p^{5} T^{28} - 56513713331682463 p^{6} T^{29} + 10953154582122989 p^{7} T^{30} - 1028836173484940 p^{8} T^{31} + 200199943485698 p^{9} T^{32} - 395004268180 p^{11} T^{33} + 3322750909866 p^{11} T^{34} - 250124106010 p^{12} T^{35} + 48975580316 p^{13} T^{36} - 3233344596 p^{14} T^{37} + 625248409 p^{15} T^{38} - 35792786 p^{16} T^{39} + 6649770 p^{17} T^{40} - 323267 p^{18} T^{41} + 55845 p^{19} T^{42} - 2232 p^{20} T^{43} + 335 p^{21} T^{44} - 9 p^{22} T^{45} + p^{23} T^{46} \)
47 \( 1 + 37 T + 1098 T^{2} + 22911 T^{3} + 417850 T^{4} + 6395999 T^{5} + 89107490 T^{6} + 1107403502 T^{7} + 12831893594 T^{8} + 136805063059 T^{9} + 1382986285654 T^{10} + 13116822545379 T^{11} + 119526244166134 T^{12} + 1036320893101505 T^{13} + 8727207400266279 T^{14} + 70677314794869192 T^{15} + 560816075586943214 T^{16} + 4312317848064259658 T^{17} + 32688248335609413612 T^{18} + \)\(24\!\cdots\!78\)\( T^{19} + \)\(17\!\cdots\!00\)\( T^{20} + \)\(12\!\cdots\!46\)\( T^{21} + \)\(88\!\cdots\!00\)\( T^{22} + \)\(60\!\cdots\!24\)\( T^{23} + \)\(88\!\cdots\!00\)\( p T^{24} + \)\(12\!\cdots\!46\)\( p^{2} T^{25} + \)\(17\!\cdots\!00\)\( p^{3} T^{26} + \)\(24\!\cdots\!78\)\( p^{4} T^{27} + 32688248335609413612 p^{5} T^{28} + 4312317848064259658 p^{6} T^{29} + 560816075586943214 p^{7} T^{30} + 70677314794869192 p^{8} T^{31} + 8727207400266279 p^{9} T^{32} + 1036320893101505 p^{10} T^{33} + 119526244166134 p^{11} T^{34} + 13116822545379 p^{12} T^{35} + 1382986285654 p^{13} T^{36} + 136805063059 p^{14} T^{37} + 12831893594 p^{15} T^{38} + 1107403502 p^{16} T^{39} + 89107490 p^{17} T^{40} + 6395999 p^{18} T^{41} + 417850 p^{19} T^{42} + 22911 p^{20} T^{43} + 1098 p^{21} T^{44} + 37 p^{22} T^{45} + p^{23} T^{46} \)
53 \( 1 - 38 T + 1218 T^{2} - 28423 T^{3} + 588560 T^{4} - 10515345 T^{5} + 172232981 T^{6} - 2567966681 T^{7} + 35791392077 T^{8} - 465466363781 T^{9} + 5728626087208 T^{10} - 66698942760443 T^{11} + 740939811806234 T^{12} - 7853386121661342 T^{13} + 79864846176294401 T^{14} - 779274868607227884 T^{15} + 7323792988196914474 T^{16} - 66289040250186455044 T^{17} + \)\(57\!\cdots\!76\)\( T^{18} - \)\(48\!\cdots\!02\)\( T^{19} + \)\(39\!\cdots\!40\)\( T^{20} - \)\(31\!\cdots\!74\)\( T^{21} + \)\(24\!\cdots\!46\)\( T^{22} - \)\(17\!\cdots\!18\)\( T^{23} + \)\(24\!\cdots\!46\)\( p T^{24} - \)\(31\!\cdots\!74\)\( p^{2} T^{25} + \)\(39\!\cdots\!40\)\( p^{3} T^{26} - \)\(48\!\cdots\!02\)\( p^{4} T^{27} + \)\(57\!\cdots\!76\)\( p^{5} T^{28} - 66289040250186455044 p^{6} T^{29} + 7323792988196914474 p^{7} T^{30} - 779274868607227884 p^{8} T^{31} + 79864846176294401 p^{9} T^{32} - 7853386121661342 p^{10} T^{33} + 740939811806234 p^{11} T^{34} - 66698942760443 p^{12} T^{35} + 5728626087208 p^{13} T^{36} - 465466363781 p^{14} T^{37} + 35791392077 p^{15} T^{38} - 2567966681 p^{16} T^{39} + 172232981 p^{17} T^{40} - 10515345 p^{18} T^{41} + 588560 p^{19} T^{42} - 28423 p^{20} T^{43} + 1218 p^{21} T^{44} - 38 p^{22} T^{45} + p^{23} T^{46} \)
59 \( 1 + 17 T + 722 T^{2} + 9934 T^{3} + 249227 T^{4} + 2963907 T^{5} + 56851930 T^{6} + 606984133 T^{7} + 9803560458 T^{8} + 96176259523 T^{9} + 1371750091441 T^{10} + 12543615849142 T^{11} + 162273631173394 T^{12} + 1395372971841281 T^{13} + 16645862035831771 T^{14} + 135348092491710876 T^{15} + 1505302429427212158 T^{16} + 11614077053132958998 T^{17} + \)\(12\!\cdots\!16\)\( T^{18} + \)\(89\!\cdots\!44\)\( T^{19} + \)\(87\!\cdots\!54\)\( T^{20} + \)\(10\!\cdots\!94\)\( p T^{21} + \)\(57\!\cdots\!52\)\( T^{22} + \)\(38\!\cdots\!26\)\( T^{23} + \)\(57\!\cdots\!52\)\( p T^{24} + \)\(10\!\cdots\!94\)\( p^{3} T^{25} + \)\(87\!\cdots\!54\)\( p^{3} T^{26} + \)\(89\!\cdots\!44\)\( p^{4} T^{27} + \)\(12\!\cdots\!16\)\( p^{5} T^{28} + 11614077053132958998 p^{6} T^{29} + 1505302429427212158 p^{7} T^{30} + 135348092491710876 p^{8} T^{31} + 16645862035831771 p^{9} T^{32} + 1395372971841281 p^{10} T^{33} + 162273631173394 p^{11} T^{34} + 12543615849142 p^{12} T^{35} + 1371750091441 p^{13} T^{36} + 96176259523 p^{14} T^{37} + 9803560458 p^{15} T^{38} + 606984133 p^{16} T^{39} + 56851930 p^{17} T^{40} + 2963907 p^{18} T^{41} + 249227 p^{19} T^{42} + 9934 p^{20} T^{43} + 722 p^{21} T^{44} + 17 p^{22} T^{45} + p^{23} T^{46} \)
61 \( 1 + 944 T^{2} + 977 T^{3} + 438397 T^{4} + 838839 T^{5} + 134009158 T^{6} + 351981049 T^{7} + 30368540535 T^{8} + 96360832652 T^{9} + 5436774072556 T^{10} + 19361893614779 T^{11} + 798900133908077 T^{12} + 3041318466474603 T^{13} + 1618829362005756 p T^{14} + 387962702470766776 T^{15} + 10436316879676269463 T^{16} + 41181691633352931897 T^{17} + \)\(95\!\cdots\!39\)\( T^{18} + \)\(36\!\cdots\!64\)\( T^{19} + \)\(75\!\cdots\!15\)\( T^{20} + \)\(28\!\cdots\!97\)\( T^{21} + \)\(52\!\cdots\!79\)\( T^{22} + \)\(18\!\cdots\!86\)\( T^{23} + \)\(52\!\cdots\!79\)\( p T^{24} + \)\(28\!\cdots\!97\)\( p^{2} T^{25} + \)\(75\!\cdots\!15\)\( p^{3} T^{26} + \)\(36\!\cdots\!64\)\( p^{4} T^{27} + \)\(95\!\cdots\!39\)\( p^{5} T^{28} + 41181691633352931897 p^{6} T^{29} + 10436316879676269463 p^{7} T^{30} + 387962702470766776 p^{8} T^{31} + 1618829362005756 p^{10} T^{32} + 3041318466474603 p^{10} T^{33} + 798900133908077 p^{11} T^{34} + 19361893614779 p^{12} T^{35} + 5436774072556 p^{13} T^{36} + 96360832652 p^{14} T^{37} + 30368540535 p^{15} T^{38} + 351981049 p^{16} T^{39} + 134009158 p^{17} T^{40} + 838839 p^{18} T^{41} + 438397 p^{19} T^{42} + 977 p^{20} T^{43} + 944 p^{21} T^{44} + p^{23} T^{46} \)
67 \( 1 + 9 T + 338 T^{2} + 4409 T^{3} + 91590 T^{4} + 1089204 T^{5} + 18321447 T^{6} + 209284640 T^{7} + 44226423 p T^{8} + 32069156506 T^{9} + 407826386037 T^{10} + 4177174974859 T^{11} + 48723700962151 T^{12} + 475630323979803 T^{13} + 5169860595107327 T^{14} + 48204784532456289 T^{15} + 494023359956885898 T^{16} + 4409521719913069357 T^{17} + 42890934538655247686 T^{18} + \)\(36\!\cdots\!67\)\( T^{19} + \)\(34\!\cdots\!70\)\( T^{20} + \)\(27\!\cdots\!57\)\( T^{21} + \)\(24\!\cdots\!36\)\( T^{22} + \)\(19\!\cdots\!48\)\( T^{23} + \)\(24\!\cdots\!36\)\( p T^{24} + \)\(27\!\cdots\!57\)\( p^{2} T^{25} + \)\(34\!\cdots\!70\)\( p^{3} T^{26} + \)\(36\!\cdots\!67\)\( p^{4} T^{27} + 42890934538655247686 p^{5} T^{28} + 4409521719913069357 p^{6} T^{29} + 494023359956885898 p^{7} T^{30} + 48204784532456289 p^{8} T^{31} + 5169860595107327 p^{9} T^{32} + 475630323979803 p^{10} T^{33} + 48723700962151 p^{11} T^{34} + 4177174974859 p^{12} T^{35} + 407826386037 p^{13} T^{36} + 32069156506 p^{14} T^{37} + 44226423 p^{16} T^{38} + 209284640 p^{16} T^{39} + 18321447 p^{17} T^{40} + 1089204 p^{18} T^{41} + 91590 p^{19} T^{42} + 4409 p^{20} T^{43} + 338 p^{21} T^{44} + 9 p^{22} T^{45} + p^{23} T^{46} \)
71 \( 1 + 13 T + 712 T^{2} + 7391 T^{3} + 237060 T^{4} + 1971353 T^{5} + 49677970 T^{6} + 323633734 T^{7} + 7494118606 T^{8} + 36192233845 T^{9} + 893730990856 T^{10} + 2881598959571 T^{11} + 91560374865700 T^{12} + 163254647790265 T^{13} + 8615429224325635 T^{14} + 5477527730873288 T^{15} + 766685546138686130 T^{16} - 163700460858992470 T^{17} + 64171026255753194764 T^{18} - 56370266703015090450 T^{19} + \)\(50\!\cdots\!96\)\( T^{20} - \)\(68\!\cdots\!38\)\( T^{21} + \)\(37\!\cdots\!44\)\( T^{22} - \)\(57\!\cdots\!16\)\( T^{23} + \)\(37\!\cdots\!44\)\( p T^{24} - \)\(68\!\cdots\!38\)\( p^{2} T^{25} + \)\(50\!\cdots\!96\)\( p^{3} T^{26} - 56370266703015090450 p^{4} T^{27} + 64171026255753194764 p^{5} T^{28} - 163700460858992470 p^{6} T^{29} + 766685546138686130 p^{7} T^{30} + 5477527730873288 p^{8} T^{31} + 8615429224325635 p^{9} T^{32} + 163254647790265 p^{10} T^{33} + 91560374865700 p^{11} T^{34} + 2881598959571 p^{12} T^{35} + 893730990856 p^{13} T^{36} + 36192233845 p^{14} T^{37} + 7494118606 p^{15} T^{38} + 323633734 p^{16} T^{39} + 49677970 p^{17} T^{40} + 1971353 p^{18} T^{41} + 237060 p^{19} T^{42} + 7391 p^{20} T^{43} + 712 p^{21} T^{44} + 13 p^{22} T^{45} + p^{23} T^{46} \)
73 \( 1 - 41 T + 1251 T^{2} - 26764 T^{3} + 494198 T^{4} - 7681121 T^{5} + 109934141 T^{6} - 1418861932 T^{7} + 17427585242 T^{8} - 199294209509 T^{9} + 2210424389913 T^{10} - 23293958123313 T^{11} + 241811242748874 T^{12} - 2419347955864134 T^{13} + 24072802537822873 T^{14} - 232475444975438077 T^{15} + 2241163411236580221 T^{16} - 21015927437953950626 T^{17} + \)\(19\!\cdots\!16\)\( T^{18} - \)\(17\!\cdots\!61\)\( T^{19} + \)\(16\!\cdots\!53\)\( T^{20} - \)\(14\!\cdots\!19\)\( T^{21} + \)\(12\!\cdots\!69\)\( T^{22} - \)\(10\!\cdots\!90\)\( T^{23} + \)\(12\!\cdots\!69\)\( p T^{24} - \)\(14\!\cdots\!19\)\( p^{2} T^{25} + \)\(16\!\cdots\!53\)\( p^{3} T^{26} - \)\(17\!\cdots\!61\)\( p^{4} T^{27} + \)\(19\!\cdots\!16\)\( p^{5} T^{28} - 21015927437953950626 p^{6} T^{29} + 2241163411236580221 p^{7} T^{30} - 232475444975438077 p^{8} T^{31} + 24072802537822873 p^{9} T^{32} - 2419347955864134 p^{10} T^{33} + 241811242748874 p^{11} T^{34} - 23293958123313 p^{12} T^{35} + 2210424389913 p^{13} T^{36} - 199294209509 p^{14} T^{37} + 17427585242 p^{15} T^{38} - 1418861932 p^{16} T^{39} + 109934141 p^{17} T^{40} - 7681121 p^{18} T^{41} + 494198 p^{19} T^{42} - 26764 p^{20} T^{43} + 1251 p^{21} T^{44} - 41 p^{22} T^{45} + p^{23} T^{46} \)
79 \( 1 - 36 T + 1623 T^{2} - 41683 T^{3} + 1137591 T^{4} - 23423970 T^{5} + 489353877 T^{6} - 8561173354 T^{7} + 149264087308 T^{8} - 2295553578020 T^{9} + 34899072488353 T^{10} - 482346361120176 T^{11} + 6563993022934221 T^{12} - 82791693614308013 T^{13} + 1026109896478182184 T^{14} - 11943343286789567983 T^{15} + \)\(13\!\cdots\!60\)\( T^{16} - \)\(14\!\cdots\!22\)\( T^{17} + \)\(15\!\cdots\!42\)\( T^{18} - \)\(15\!\cdots\!85\)\( T^{19} + \)\(15\!\cdots\!60\)\( T^{20} - \)\(15\!\cdots\!42\)\( T^{21} + \)\(14\!\cdots\!13\)\( T^{22} - \)\(12\!\cdots\!40\)\( T^{23} + \)\(14\!\cdots\!13\)\( p T^{24} - \)\(15\!\cdots\!42\)\( p^{2} T^{25} + \)\(15\!\cdots\!60\)\( p^{3} T^{26} - \)\(15\!\cdots\!85\)\( p^{4} T^{27} + \)\(15\!\cdots\!42\)\( p^{5} T^{28} - \)\(14\!\cdots\!22\)\( p^{6} T^{29} + \)\(13\!\cdots\!60\)\( p^{7} T^{30} - 11943343286789567983 p^{8} T^{31} + 1026109896478182184 p^{9} T^{32} - 82791693614308013 p^{10} T^{33} + 6563993022934221 p^{11} T^{34} - 482346361120176 p^{12} T^{35} + 34899072488353 p^{13} T^{36} - 2295553578020 p^{14} T^{37} + 149264087308 p^{15} T^{38} - 8561173354 p^{16} T^{39} + 489353877 p^{17} T^{40} - 23423970 p^{18} T^{41} + 1137591 p^{19} T^{42} - 41683 p^{20} T^{43} + 1623 p^{21} T^{44} - 36 p^{22} T^{45} + p^{23} T^{46} \)
83 \( 1 + 29 T + 1352 T^{2} + 28480 T^{3} + 781975 T^{4} + 13057143 T^{5} + 269425631 T^{6} + 3711601325 T^{7} + 63149775351 T^{8} + 727997489105 T^{9} + 10749276783083 T^{10} + 102599617412977 T^{11} + 1366670093289802 T^{12} + 10197963677386724 T^{13} + 128552939235704731 T^{14} + 595054256678589592 T^{15} + 8129032255719035330 T^{16} - 10857709809535120491 T^{17} + \)\(16\!\cdots\!46\)\( T^{18} - \)\(78\!\cdots\!33\)\( T^{19} - \)\(36\!\cdots\!12\)\( T^{20} - \)\(11\!\cdots\!70\)\( T^{21} - \)\(57\!\cdots\!28\)\( T^{22} - \)\(11\!\cdots\!82\)\( T^{23} - \)\(57\!\cdots\!28\)\( p T^{24} - \)\(11\!\cdots\!70\)\( p^{2} T^{25} - \)\(36\!\cdots\!12\)\( p^{3} T^{26} - \)\(78\!\cdots\!33\)\( p^{4} T^{27} + \)\(16\!\cdots\!46\)\( p^{5} T^{28} - 10857709809535120491 p^{6} T^{29} + 8129032255719035330 p^{7} T^{30} + 595054256678589592 p^{8} T^{31} + 128552939235704731 p^{9} T^{32} + 10197963677386724 p^{10} T^{33} + 1366670093289802 p^{11} T^{34} + 102599617412977 p^{12} T^{35} + 10749276783083 p^{13} T^{36} + 727997489105 p^{14} T^{37} + 63149775351 p^{15} T^{38} + 3711601325 p^{16} T^{39} + 269425631 p^{17} T^{40} + 13057143 p^{18} T^{41} + 781975 p^{19} T^{42} + 28480 p^{20} T^{43} + 1352 p^{21} T^{44} + 29 p^{22} T^{45} + p^{23} T^{46} \)
89 \( 1 - 36 T + 1678 T^{2} - 44267 T^{3} + 1234528 T^{4} - 26136545 T^{5} + 554192530 T^{6} - 9914857148 T^{7} + 174665509236 T^{8} - 2730767165819 T^{9} + 41852302089396 T^{10} - 585698790092128 T^{11} + 8038492033976386 T^{12} - 102566505307067273 T^{13} + 1286593102989216090 T^{14} - 15189203540702708715 T^{15} + \)\(17\!\cdots\!46\)\( T^{16} - \)\(19\!\cdots\!50\)\( T^{17} + \)\(21\!\cdots\!57\)\( T^{18} - \)\(22\!\cdots\!18\)\( T^{19} + \)\(23\!\cdots\!04\)\( T^{20} - \)\(22\!\cdots\!84\)\( T^{21} + \)\(22\!\cdots\!48\)\( T^{22} - \)\(21\!\cdots\!14\)\( T^{23} + \)\(22\!\cdots\!48\)\( p T^{24} - \)\(22\!\cdots\!84\)\( p^{2} T^{25} + \)\(23\!\cdots\!04\)\( p^{3} T^{26} - \)\(22\!\cdots\!18\)\( p^{4} T^{27} + \)\(21\!\cdots\!57\)\( p^{5} T^{28} - \)\(19\!\cdots\!50\)\( p^{6} T^{29} + \)\(17\!\cdots\!46\)\( p^{7} T^{30} - 15189203540702708715 p^{8} T^{31} + 1286593102989216090 p^{9} T^{32} - 102566505307067273 p^{10} T^{33} + 8038492033976386 p^{11} T^{34} - 585698790092128 p^{12} T^{35} + 41852302089396 p^{13} T^{36} - 2730767165819 p^{14} T^{37} + 174665509236 p^{15} T^{38} - 9914857148 p^{16} T^{39} + 554192530 p^{17} T^{40} - 26136545 p^{18} T^{41} + 1234528 p^{19} T^{42} - 44267 p^{20} T^{43} + 1678 p^{21} T^{44} - 36 p^{22} T^{45} + p^{23} T^{46} \)
97 \( 1 - 40 T + 1870 T^{2} - 51422 T^{3} + 1445831 T^{4} - 30978980 T^{5} + 657785135 T^{6} - 11646636354 T^{7} + 202648516787 T^{8} - 3062411832836 T^{9} + 45461042129711 T^{10} - 597761706107766 T^{11} + 7752991379060562 T^{12} - 89782140170744920 T^{13} + 1034851827419184749 T^{14} - 10645753941620982712 T^{15} + \)\(11\!\cdots\!42\)\( T^{16} - \)\(10\!\cdots\!36\)\( T^{17} + \)\(98\!\cdots\!40\)\( T^{18} - \)\(83\!\cdots\!92\)\( T^{19} + \)\(78\!\cdots\!62\)\( T^{20} - \)\(63\!\cdots\!52\)\( T^{21} + \)\(63\!\cdots\!46\)\( T^{22} - \)\(54\!\cdots\!56\)\( T^{23} + \)\(63\!\cdots\!46\)\( p T^{24} - \)\(63\!\cdots\!52\)\( p^{2} T^{25} + \)\(78\!\cdots\!62\)\( p^{3} T^{26} - \)\(83\!\cdots\!92\)\( p^{4} T^{27} + \)\(98\!\cdots\!40\)\( p^{5} T^{28} - \)\(10\!\cdots\!36\)\( p^{6} T^{29} + \)\(11\!\cdots\!42\)\( p^{7} T^{30} - 10645753941620982712 p^{8} T^{31} + 1034851827419184749 p^{9} T^{32} - 89782140170744920 p^{10} T^{33} + 7752991379060562 p^{11} T^{34} - 597761706107766 p^{12} T^{35} + 45461042129711 p^{13} T^{36} - 3062411832836 p^{14} T^{37} + 202648516787 p^{15} T^{38} - 11646636354 p^{16} T^{39} + 657785135 p^{17} T^{40} - 30978980 p^{18} T^{41} + 1445831 p^{19} T^{42} - 51422 p^{20} T^{43} + 1870 p^{21} T^{44} - 40 p^{22} T^{45} + p^{23} T^{46} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{46} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.56618666361384379902507205548, −1.50182027629641387545992719374, −1.45476707413079181268051807104, −1.45459059593748622659370423384, −1.41730891806122888634773669812, −1.20888607211190496904610847376, −1.16666374012568492408831229437, −1.11378111298221052593387432073, −1.09262056208496493735861799200, −1.01873155319082119995949406872, −0.795605398493410347806565803314, −0.77774178253262237122664530317, −0.77049089389587631588947243599, −0.75089331920601669324232082907, −0.71233223546209213725545763149, −0.65705871577796517167448261819, −0.64152781743006954211110076330, −0.56228732428409182259452808687, −0.54058863156484521598132696001, −0.51711490602191842557264367422, −0.50312947777833023468855912453, −0.44690094835506484034641559343, −0.25761067189979796360831373952, −0.19543783440016964273481579782, −0.06240768949309930971462466713, 0.06240768949309930971462466713, 0.19543783440016964273481579782, 0.25761067189979796360831373952, 0.44690094835506484034641559343, 0.50312947777833023468855912453, 0.51711490602191842557264367422, 0.54058863156484521598132696001, 0.56228732428409182259452808687, 0.64152781743006954211110076330, 0.65705871577796517167448261819, 0.71233223546209213725545763149, 0.75089331920601669324232082907, 0.77049089389587631588947243599, 0.77774178253262237122664530317, 0.795605398493410347806565803314, 1.01873155319082119995949406872, 1.09262056208496493735861799200, 1.11378111298221052593387432073, 1.16666374012568492408831229437, 1.20888607211190496904610847376, 1.41730891806122888634773669812, 1.45459059593748622659370423384, 1.45476707413079181268051807104, 1.50182027629641387545992719374, 1.56618666361384379902507205548

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.