Properties

Label 40-73e20-1.1-c1e20-0-0
Degree $40$
Conductor $1.847\times 10^{37}$
Sign $1$
Analytic cond. $2.05115\times 10^{-5}$
Root an. cond. $0.763484$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 14·4-s − 4·5-s − 2·7-s − 36·8-s + 14·9-s + 16·10-s − 6·11-s − 16·13-s + 8·14-s + 85·16-s + 8·17-s − 56·18-s − 12·19-s − 56·20-s + 24·22-s − 6·23-s − 10·25-s + 64·26-s − 28·28-s − 6·29-s + 20·31-s − 174·32-s − 32·34-s + 8·35-s + 196·36-s − 8·37-s + ⋯
L(s)  = 1  − 2.82·2-s + 7·4-s − 1.78·5-s − 0.755·7-s − 12.7·8-s + 14/3·9-s + 5.05·10-s − 1.80·11-s − 4.43·13-s + 2.13·14-s + 85/4·16-s + 1.94·17-s − 13.1·18-s − 2.75·19-s − 12.5·20-s + 5.11·22-s − 1.25·23-s − 2·25-s + 12.5·26-s − 5.29·28-s − 1.11·29-s + 3.59·31-s − 30.7·32-s − 5.48·34-s + 1.35·35-s + 98/3·36-s − 1.31·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(73^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(73^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(73^{20}\)
Sign: $1$
Analytic conductor: \(2.05115\times 10^{-5}\)
Root analytic conductor: \(0.763484\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{73} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 73^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.06180815473\)
\(L(\frac12)\) \(\approx\) \(0.06180815473\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 + 16 T + 94 T^{2} + 1404 T^{3} + 28350 T^{4} + 271546 T^{5} + 1421348 T^{6} + 19173090 T^{7} + 279379458 T^{8} + 1820859772 T^{9} + 9378350075 T^{10} + 1820859772 p T^{11} + 279379458 p^{2} T^{12} + 19173090 p^{3} T^{13} + 1421348 p^{4} T^{14} + 271546 p^{5} T^{15} + 28350 p^{6} T^{16} + 1404 p^{7} T^{17} + 94 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \)
good2 \( 1 + p^{2} T + p T^{2} - 3 p^{2} T^{3} - 17 T^{4} + 3 p T^{5} + 17 p T^{6} + 13 p^{2} T^{7} + 3 p T^{8} - 85 p T^{9} - 231 T^{10} + 35 p T^{11} + 347 T^{12} + 63 p^{2} T^{13} + 11 p^{3} T^{14} - 39 p^{2} T^{15} - 1005 T^{16} - 667 p T^{17} + 733 T^{18} + 1337 p T^{19} + 2941 T^{20} + 1337 p^{2} T^{21} + 733 p^{2} T^{22} - 667 p^{4} T^{23} - 1005 p^{4} T^{24} - 39 p^{7} T^{25} + 11 p^{9} T^{26} + 63 p^{9} T^{27} + 347 p^{8} T^{28} + 35 p^{10} T^{29} - 231 p^{10} T^{30} - 85 p^{12} T^{31} + 3 p^{13} T^{32} + 13 p^{15} T^{33} + 17 p^{15} T^{34} + 3 p^{16} T^{35} - 17 p^{16} T^{36} - 3 p^{19} T^{37} + p^{19} T^{38} + p^{21} T^{39} + p^{20} T^{40} \)
3 \( 1 - 14 T^{2} + 107 T^{4} - 662 T^{6} + 3607 T^{8} - 17246 T^{10} + 2740 p^{3} T^{12} - 10790 p^{3} T^{14} + 117665 p^{2} T^{16} - 3549952 T^{18} + 11032168 T^{20} - 3549952 p^{2} T^{22} + 117665 p^{6} T^{24} - 10790 p^{9} T^{26} + 2740 p^{11} T^{28} - 17246 p^{10} T^{30} + 3607 p^{12} T^{32} - 662 p^{14} T^{34} + 107 p^{16} T^{36} - 14 p^{18} T^{38} + p^{20} T^{40} \)
5 \( 1 + 4 T + 26 T^{2} + 66 T^{3} + 249 T^{4} + 304 T^{5} + 772 T^{6} - 332 p T^{7} - 656 p T^{8} - 25588 T^{9} - 25659 T^{10} - 20892 p T^{11} + 102778 T^{12} + 45982 T^{13} + 1899538 T^{14} + 1580934 T^{15} + 9269849 T^{16} - 736684 p T^{17} + 491081 p^{2} T^{18} - 795526 p^{3} T^{19} - 1763069 p^{2} T^{20} - 795526 p^{4} T^{21} + 491081 p^{4} T^{22} - 736684 p^{4} T^{23} + 9269849 p^{4} T^{24} + 1580934 p^{5} T^{25} + 1899538 p^{6} T^{26} + 45982 p^{7} T^{27} + 102778 p^{8} T^{28} - 20892 p^{10} T^{29} - 25659 p^{10} T^{30} - 25588 p^{11} T^{31} - 656 p^{13} T^{32} - 332 p^{14} T^{33} + 772 p^{14} T^{34} + 304 p^{15} T^{35} + 249 p^{16} T^{36} + 66 p^{17} T^{37} + 26 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \)
7 \( 1 + 2 T + 2 T^{2} + 20 T^{3} + p T^{4} - 106 T^{5} - 26 T^{6} - 10 p T^{7} - 1577 T^{8} - 3522 T^{9} + 510 T^{10} - 7912 T^{11} - 11476 p T^{12} - 106350 T^{13} + 120934 T^{14} - 906076 T^{15} - 1433631 T^{16} + 10616386 T^{17} + 21876796 T^{18} + 70204312 T^{19} + 350103420 T^{20} + 70204312 p T^{21} + 21876796 p^{2} T^{22} + 10616386 p^{3} T^{23} - 1433631 p^{4} T^{24} - 906076 p^{5} T^{25} + 120934 p^{6} T^{26} - 106350 p^{7} T^{27} - 11476 p^{9} T^{28} - 7912 p^{9} T^{29} + 510 p^{10} T^{30} - 3522 p^{11} T^{31} - 1577 p^{12} T^{32} - 10 p^{14} T^{33} - 26 p^{14} T^{34} - 106 p^{15} T^{35} + p^{17} T^{36} + 20 p^{17} T^{37} + 2 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \)
11 \( 1 + 6 T + 84 T^{2} + 380 T^{3} + 290 p T^{4} + 11824 T^{5} + 77672 T^{6} + 254508 T^{7} + 1458646 T^{8} + 4610832 T^{9} + 23879628 T^{10} + 77491790 T^{11} + 358159820 T^{12} + 109909964 p T^{13} + 4927504160 T^{14} + 17156019920 T^{15} + 62418557046 T^{16} + 222004739112 T^{17} + 741250642424 T^{18} + 2645647306094 T^{19} + 8357262055767 T^{20} + 2645647306094 p T^{21} + 741250642424 p^{2} T^{22} + 222004739112 p^{3} T^{23} + 62418557046 p^{4} T^{24} + 17156019920 p^{5} T^{25} + 4927504160 p^{6} T^{26} + 109909964 p^{8} T^{27} + 358159820 p^{8} T^{28} + 77491790 p^{9} T^{29} + 23879628 p^{10} T^{30} + 4610832 p^{11} T^{31} + 1458646 p^{12} T^{32} + 254508 p^{13} T^{33} + 77672 p^{14} T^{34} + 11824 p^{15} T^{35} + 290 p^{17} T^{36} + 380 p^{17} T^{37} + 84 p^{18} T^{38} + 6 p^{19} T^{39} + p^{20} T^{40} \)
13 \( 1 + 16 T + 110 T^{2} + 252 T^{3} - 1509 T^{4} - 12820 T^{5} - 17926 T^{6} + 221900 T^{7} + 1343535 T^{8} + 1456454 T^{9} - 16355537 T^{10} - 77256844 T^{11} - 4817412 T^{12} + 1144114588 T^{13} + 4165204024 T^{14} + 532613298 T^{15} - 43843841913 T^{16} - 146945333810 T^{17} - 42142856274 T^{18} + 1456256372062 T^{19} + 7298599373270 T^{20} + 1456256372062 p T^{21} - 42142856274 p^{2} T^{22} - 146945333810 p^{3} T^{23} - 43843841913 p^{4} T^{24} + 532613298 p^{5} T^{25} + 4165204024 p^{6} T^{26} + 1144114588 p^{7} T^{27} - 4817412 p^{8} T^{28} - 77256844 p^{9} T^{29} - 16355537 p^{10} T^{30} + 1456454 p^{11} T^{31} + 1343535 p^{12} T^{32} + 221900 p^{13} T^{33} - 17926 p^{14} T^{34} - 12820 p^{15} T^{35} - 1509 p^{16} T^{36} + 252 p^{17} T^{37} + 110 p^{18} T^{38} + 16 p^{19} T^{39} + p^{20} T^{40} \)
17 \( 1 - 8 T + 32 T^{2} - 96 T^{3} + 210 T^{4} + 994 T^{5} - 592 p T^{6} + 67378 T^{7} - 393135 T^{8} + 1595082 T^{9} - 4041166 T^{10} + 2872406 T^{11} + 80694176 T^{12} - 582466458 T^{13} + 2549310226 T^{14} - 9702606816 T^{15} + 26734150038 T^{16} - 703716232 T^{17} - 343676967052 T^{18} + 3018435808210 T^{19} - 15962697482619 T^{20} + 3018435808210 p T^{21} - 343676967052 p^{2} T^{22} - 703716232 p^{3} T^{23} + 26734150038 p^{4} T^{24} - 9702606816 p^{5} T^{25} + 2549310226 p^{6} T^{26} - 582466458 p^{7} T^{27} + 80694176 p^{8} T^{28} + 2872406 p^{9} T^{29} - 4041166 p^{10} T^{30} + 1595082 p^{11} T^{31} - 393135 p^{12} T^{32} + 67378 p^{13} T^{33} - 592 p^{15} T^{34} + 994 p^{15} T^{35} + 210 p^{16} T^{36} - 96 p^{17} T^{37} + 32 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \)
19 \( 1 + 12 T + 157 T^{2} + 1308 T^{3} + 10370 T^{4} + 68580 T^{5} + 22153 p T^{6} + 2345688 T^{7} + 12103492 T^{8} + 57972000 T^{9} + 255405546 T^{10} + 1017211260 T^{11} + 3527077491 T^{12} + 8988552204 T^{13} + 5060697073 T^{14} - 155327901372 T^{15} - 1430391188017 T^{16} - 9322238506584 T^{17} - 51148591406891 T^{18} - 251806093215072 T^{19} - 1145335823233201 T^{20} - 251806093215072 p T^{21} - 51148591406891 p^{2} T^{22} - 9322238506584 p^{3} T^{23} - 1430391188017 p^{4} T^{24} - 155327901372 p^{5} T^{25} + 5060697073 p^{6} T^{26} + 8988552204 p^{7} T^{27} + 3527077491 p^{8} T^{28} + 1017211260 p^{9} T^{29} + 255405546 p^{10} T^{30} + 57972000 p^{11} T^{31} + 12103492 p^{12} T^{32} + 2345688 p^{13} T^{33} + 22153 p^{15} T^{34} + 68580 p^{15} T^{35} + 10370 p^{16} T^{36} + 1308 p^{17} T^{37} + 157 p^{18} T^{38} + 12 p^{19} T^{39} + p^{20} T^{40} \)
23 \( 1 + 6 T + 136 T^{2} + 744 T^{3} + 9803 T^{4} + 1968 p T^{5} + 452600 T^{6} + 1680642 T^{7} + 14410139 T^{8} + 36993348 T^{9} + 315957200 T^{10} + 269861808 T^{11} + 4666996734 T^{12} - 11196800184 T^{13} + 59578447616 T^{14} - 395435770860 T^{15} + 1888223084293 T^{16} - 3759582189294 T^{17} + 81949421412648 T^{18} + 75067158984768 T^{19} + 2367060652986617 T^{20} + 75067158984768 p T^{21} + 81949421412648 p^{2} T^{22} - 3759582189294 p^{3} T^{23} + 1888223084293 p^{4} T^{24} - 395435770860 p^{5} T^{25} + 59578447616 p^{6} T^{26} - 11196800184 p^{7} T^{27} + 4666996734 p^{8} T^{28} + 269861808 p^{9} T^{29} + 315957200 p^{10} T^{30} + 36993348 p^{11} T^{31} + 14410139 p^{12} T^{32} + 1680642 p^{13} T^{33} + 452600 p^{14} T^{34} + 1968 p^{16} T^{35} + 9803 p^{16} T^{36} + 744 p^{17} T^{37} + 136 p^{18} T^{38} + 6 p^{19} T^{39} + p^{20} T^{40} \)
29 \( 1 + 6 T - 66 T^{2} - 328 T^{3} + 2758 T^{4} - 2906 T^{5} - 151216 T^{6} + 530034 T^{7} + 4705108 T^{8} - 24118944 T^{9} - 10974495 T^{10} + 1044253784 T^{11} - 2347945498 T^{12} - 21561000536 T^{13} + 86922536798 T^{14} - 151465055602 T^{15} - 2988859973280 T^{16} + 7293799222590 T^{17} + 10218959497544 T^{18} - 1232099600518 T^{19} + 2020549132604388 T^{20} - 1232099600518 p T^{21} + 10218959497544 p^{2} T^{22} + 7293799222590 p^{3} T^{23} - 2988859973280 p^{4} T^{24} - 151465055602 p^{5} T^{25} + 86922536798 p^{6} T^{26} - 21561000536 p^{7} T^{27} - 2347945498 p^{8} T^{28} + 1044253784 p^{9} T^{29} - 10974495 p^{10} T^{30} - 24118944 p^{11} T^{31} + 4705108 p^{12} T^{32} + 530034 p^{13} T^{33} - 151216 p^{14} T^{34} - 2906 p^{15} T^{35} + 2758 p^{16} T^{36} - 328 p^{17} T^{37} - 66 p^{18} T^{38} + 6 p^{19} T^{39} + p^{20} T^{40} \)
31 \( 1 - 20 T + 347 T^{2} - 3940 T^{3} + 38893 T^{4} - 297588 T^{5} + 1949790 T^{6} - 9656040 T^{7} + 33572601 T^{8} + 8691074 T^{9} - 1201951264 T^{10} + 12812179898 T^{11} - 92115884823 T^{12} + 569999589484 T^{13} - 2921548883523 T^{14} + 12756849942312 T^{15} - 32177555560566 T^{16} - 79665424251942 T^{17} + 67159138752830 p T^{18} - 17973883451409986 T^{19} + 116271993993733653 T^{20} - 17973883451409986 p T^{21} + 67159138752830 p^{3} T^{22} - 79665424251942 p^{3} T^{23} - 32177555560566 p^{4} T^{24} + 12756849942312 p^{5} T^{25} - 2921548883523 p^{6} T^{26} + 569999589484 p^{7} T^{27} - 92115884823 p^{8} T^{28} + 12812179898 p^{9} T^{29} - 1201951264 p^{10} T^{30} + 8691074 p^{11} T^{31} + 33572601 p^{12} T^{32} - 9656040 p^{13} T^{33} + 1949790 p^{14} T^{34} - 297588 p^{15} T^{35} + 38893 p^{16} T^{36} - 3940 p^{17} T^{37} + 347 p^{18} T^{38} - 20 p^{19} T^{39} + p^{20} T^{40} \)
37 \( 1 + 8 T - 117 T^{2} - 2112 T^{3} - 1952 T^{4} + 173078 T^{5} + 1366445 T^{6} - 2841158 T^{7} - 102787826 T^{8} - 558721752 T^{9} + 1851840962 T^{10} + 43008617802 T^{11} + 210511300706 T^{12} - 801892292758 T^{13} - 15681623824975 T^{14} - 63800305621570 T^{15} + 312893262858410 T^{16} + 4556647051884312 T^{17} + 15017093917032718 T^{18} - 88353921654613370 T^{19} - 1085039591406399452 T^{20} - 88353921654613370 p T^{21} + 15017093917032718 p^{2} T^{22} + 4556647051884312 p^{3} T^{23} + 312893262858410 p^{4} T^{24} - 63800305621570 p^{5} T^{25} - 15681623824975 p^{6} T^{26} - 801892292758 p^{7} T^{27} + 210511300706 p^{8} T^{28} + 43008617802 p^{9} T^{29} + 1851840962 p^{10} T^{30} - 558721752 p^{11} T^{31} - 102787826 p^{12} T^{32} - 2841158 p^{13} T^{33} + 1366445 p^{14} T^{34} + 173078 p^{15} T^{35} - 1952 p^{16} T^{36} - 2112 p^{17} T^{37} - 117 p^{18} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \)
41 \( 1 - 10 T - 199 T^{2} + 1670 T^{3} + 24479 T^{4} - 135314 T^{5} - 2344136 T^{6} + 7462788 T^{7} + 178414901 T^{8} - 332634234 T^{9} - 11255063297 T^{10} + 13634275652 T^{11} + 629361362988 T^{12} - 558823756146 T^{13} - 32852465167703 T^{14} + 22037333676844 T^{15} + 1622840947580711 T^{16} - 720195009918822 T^{17} - 74768012603624012 T^{18} + 11787417960062140 T^{19} + 3185951996094648794 T^{20} + 11787417960062140 p T^{21} - 74768012603624012 p^{2} T^{22} - 720195009918822 p^{3} T^{23} + 1622840947580711 p^{4} T^{24} + 22037333676844 p^{5} T^{25} - 32852465167703 p^{6} T^{26} - 558823756146 p^{7} T^{27} + 629361362988 p^{8} T^{28} + 13634275652 p^{9} T^{29} - 11255063297 p^{10} T^{30} - 332634234 p^{11} T^{31} + 178414901 p^{12} T^{32} + 7462788 p^{13} T^{33} - 2344136 p^{14} T^{34} - 135314 p^{15} T^{35} + 24479 p^{16} T^{36} + 1670 p^{17} T^{37} - 199 p^{18} T^{38} - 10 p^{19} T^{39} + p^{20} T^{40} \)
43 \( 1 - 12 T + 72 T^{2} - 612 T^{3} + 12092 T^{4} - 110700 T^{5} + 645048 T^{6} - 5404908 T^{7} + 66762560 T^{8} - 501358428 T^{9} + 2844525096 T^{10} - 546900444 p T^{11} + 239452554644 T^{12} - 1582227969300 T^{13} + 8784223741656 T^{14} - 71567184815748 T^{15} + 644952306138376 T^{16} - 3862871094240300 T^{17} + 21070477869774408 T^{18} - 168892010968941540 T^{19} + 1352370018902865858 T^{20} - 168892010968941540 p T^{21} + 21070477869774408 p^{2} T^{22} - 3862871094240300 p^{3} T^{23} + 644952306138376 p^{4} T^{24} - 71567184815748 p^{5} T^{25} + 8784223741656 p^{6} T^{26} - 1582227969300 p^{7} T^{27} + 239452554644 p^{8} T^{28} - 546900444 p^{10} T^{29} + 2844525096 p^{10} T^{30} - 501358428 p^{11} T^{31} + 66762560 p^{12} T^{32} - 5404908 p^{13} T^{33} + 645048 p^{14} T^{34} - 110700 p^{15} T^{35} + 12092 p^{16} T^{36} - 612 p^{17} T^{37} + 72 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \)
47 \( 1 + 20 T + 215 T^{2} + 2104 T^{3} + 17008 T^{4} + 114536 T^{5} + 784123 T^{6} + 4186992 T^{7} + 14196202 T^{8} + 28844168 T^{9} - 255161376 T^{10} - 3416780888 T^{11} - 16442564839 T^{12} - 94156289068 T^{13} - 897958316975 T^{14} - 6711379010956 T^{15} - 75994117243839 T^{16} - 940065002208924 T^{17} - 8326213270843909 T^{18} - 1519020490345876 p T^{19} - 12037235533071675 p T^{20} - 1519020490345876 p^{2} T^{21} - 8326213270843909 p^{2} T^{22} - 940065002208924 p^{3} T^{23} - 75994117243839 p^{4} T^{24} - 6711379010956 p^{5} T^{25} - 897958316975 p^{6} T^{26} - 94156289068 p^{7} T^{27} - 16442564839 p^{8} T^{28} - 3416780888 p^{9} T^{29} - 255161376 p^{10} T^{30} + 28844168 p^{11} T^{31} + 14196202 p^{12} T^{32} + 4186992 p^{13} T^{33} + 784123 p^{14} T^{34} + 114536 p^{15} T^{35} + 17008 p^{16} T^{36} + 2104 p^{17} T^{37} + 215 p^{18} T^{38} + 20 p^{19} T^{39} + p^{20} T^{40} \)
53 \( 1 - 24 T + 255 T^{2} - 828 T^{3} - 13150 T^{4} + 190690 T^{5} - 15075 p T^{6} - 6616876 T^{7} + 109753268 T^{8} - 413449082 T^{9} - 4762979614 T^{10} + 72976274930 T^{11} - 332657778669 T^{12} - 1497436874928 T^{13} + 29530165371731 T^{14} - 130242159258034 T^{15} - 684097991386205 T^{16} + 11892094794870328 T^{17} - 42733030310327261 T^{18} - 401537873203180152 T^{19} + 5583528691148103591 T^{20} - 401537873203180152 p T^{21} - 42733030310327261 p^{2} T^{22} + 11892094794870328 p^{3} T^{23} - 684097991386205 p^{4} T^{24} - 130242159258034 p^{5} T^{25} + 29530165371731 p^{6} T^{26} - 1497436874928 p^{7} T^{27} - 332657778669 p^{8} T^{28} + 72976274930 p^{9} T^{29} - 4762979614 p^{10} T^{30} - 413449082 p^{11} T^{31} + 109753268 p^{12} T^{32} - 6616876 p^{13} T^{33} - 15075 p^{15} T^{34} + 190690 p^{15} T^{35} - 13150 p^{16} T^{36} - 828 p^{17} T^{37} + 255 p^{18} T^{38} - 24 p^{19} T^{39} + p^{20} T^{40} \)
59 \( 1 + 18 T + 6 T^{2} - 1632 T^{3} - 6336 T^{4} + 53728 T^{5} + 204624 T^{6} - 1070922 T^{7} + 1535940 T^{8} - 67676220 T^{9} - 1494937360 T^{10} - 581393586 T^{11} + 127144933632 T^{12} + 701939673234 T^{13} - 7181564836398 T^{14} - 80349067247456 T^{15} + 193586514571368 T^{16} + 3868562311882062 T^{17} - 4652913662886504 T^{18} - 38106036749302038 T^{19} + 857206140193551031 T^{20} - 38106036749302038 p T^{21} - 4652913662886504 p^{2} T^{22} + 3868562311882062 p^{3} T^{23} + 193586514571368 p^{4} T^{24} - 80349067247456 p^{5} T^{25} - 7181564836398 p^{6} T^{26} + 701939673234 p^{7} T^{27} + 127144933632 p^{8} T^{28} - 581393586 p^{9} T^{29} - 1494937360 p^{10} T^{30} - 67676220 p^{11} T^{31} + 1535940 p^{12} T^{32} - 1070922 p^{13} T^{33} + 204624 p^{14} T^{34} + 53728 p^{15} T^{35} - 6336 p^{16} T^{36} - 1632 p^{17} T^{37} + 6 p^{18} T^{38} + 18 p^{19} T^{39} + p^{20} T^{40} \)
61 \( 1 - 42 T + 1141 T^{2} - 23226 T^{3} + 382430 T^{4} - 5319324 T^{5} + 64451601 T^{6} - 11407392 p T^{7} + 6901817174 T^{8} - 64918609074 T^{9} + 600003237168 T^{10} - 5589034310754 T^{11} + 52643216832046 T^{12} - 493147158164910 T^{13} + 4508261242003245 T^{14} - 650184010010136 p T^{15} + 335224137900232502 T^{16} - 2737987339835676870 T^{17} + 21830038826706025218 T^{18} - \)\(17\!\cdots\!00\)\( T^{19} + \)\(13\!\cdots\!08\)\( T^{20} - \)\(17\!\cdots\!00\)\( p T^{21} + 21830038826706025218 p^{2} T^{22} - 2737987339835676870 p^{3} T^{23} + 335224137900232502 p^{4} T^{24} - 650184010010136 p^{6} T^{25} + 4508261242003245 p^{6} T^{26} - 493147158164910 p^{7} T^{27} + 52643216832046 p^{8} T^{28} - 5589034310754 p^{9} T^{29} + 600003237168 p^{10} T^{30} - 64918609074 p^{11} T^{31} + 6901817174 p^{12} T^{32} - 11407392 p^{14} T^{33} + 64451601 p^{14} T^{34} - 5319324 p^{15} T^{35} + 382430 p^{16} T^{36} - 23226 p^{17} T^{37} + 1141 p^{18} T^{38} - 42 p^{19} T^{39} + p^{20} T^{40} \)
67 \( 1 + 42 T + 1266 T^{2} + 28476 T^{3} + 539866 T^{4} + 8813304 T^{5} + 128485040 T^{6} + 1694281344 T^{7} + 20582712270 T^{8} + 232598675748 T^{9} + 2478213269038 T^{10} + 374935829430 p T^{11} + 245183869559860 T^{12} + 2322692237452164 T^{13} + 21547932514410238 T^{14} + 196466930936659968 T^{15} + 1765512364313054658 T^{16} + 15603490425252979896 T^{17} + \)\(13\!\cdots\!64\)\( T^{18} + \)\(11\!\cdots\!38\)\( T^{19} + \)\(95\!\cdots\!87\)\( T^{20} + \)\(11\!\cdots\!38\)\( p T^{21} + \)\(13\!\cdots\!64\)\( p^{2} T^{22} + 15603490425252979896 p^{3} T^{23} + 1765512364313054658 p^{4} T^{24} + 196466930936659968 p^{5} T^{25} + 21547932514410238 p^{6} T^{26} + 2322692237452164 p^{7} T^{27} + 245183869559860 p^{8} T^{28} + 374935829430 p^{10} T^{29} + 2478213269038 p^{10} T^{30} + 232598675748 p^{11} T^{31} + 20582712270 p^{12} T^{32} + 1694281344 p^{13} T^{33} + 128485040 p^{14} T^{34} + 8813304 p^{15} T^{35} + 539866 p^{16} T^{36} + 28476 p^{17} T^{37} + 1266 p^{18} T^{38} + 42 p^{19} T^{39} + p^{20} T^{40} \)
71 \( 1 - 4 T - 440 T^{2} - 208 T^{3} + 108948 T^{4} + 403900 T^{5} - 17295832 T^{6} - 116737680 T^{7} + 1898504152 T^{8} + 19323528856 T^{9} - 142366245458 T^{10} - 2220154862316 T^{11} + 6289223332192 T^{12} + 189329576524064 T^{13} + 23205747831220 T^{14} - 12264843187280816 T^{15} - 32574841280736980 T^{16} + 584191270856250168 T^{17} + 3583789250746317792 T^{18} - 14290365540285381684 T^{19} - \)\(28\!\cdots\!89\)\( T^{20} - 14290365540285381684 p T^{21} + 3583789250746317792 p^{2} T^{22} + 584191270856250168 p^{3} T^{23} - 32574841280736980 p^{4} T^{24} - 12264843187280816 p^{5} T^{25} + 23205747831220 p^{6} T^{26} + 189329576524064 p^{7} T^{27} + 6289223332192 p^{8} T^{28} - 2220154862316 p^{9} T^{29} - 142366245458 p^{10} T^{30} + 19323528856 p^{11} T^{31} + 1898504152 p^{12} T^{32} - 116737680 p^{13} T^{33} - 17295832 p^{14} T^{34} + 403900 p^{15} T^{35} + 108948 p^{16} T^{36} - 208 p^{17} T^{37} - 440 p^{18} T^{38} - 4 p^{19} T^{39} + p^{20} T^{40} \)
79 \( 1 - 54 T + 1964 T^{2} - 53568 T^{3} + 1217388 T^{4} - 23757936 T^{5} + 412062020 T^{6} - 6446018694 T^{7} + 92575338476 T^{8} - 1232843629176 T^{9} + 15411450935318 T^{10} - 182299937304006 T^{11} + 2060368146101196 T^{12} - 22389577940717202 T^{13} + 235562724353813348 T^{14} - 2407729436099707776 T^{15} + 23990691322818097888 T^{16} - \)\(23\!\cdots\!98\)\( T^{17} + \)\(22\!\cdots\!24\)\( T^{18} - \)\(20\!\cdots\!66\)\( T^{19} + \)\(18\!\cdots\!55\)\( T^{20} - \)\(20\!\cdots\!66\)\( p T^{21} + \)\(22\!\cdots\!24\)\( p^{2} T^{22} - \)\(23\!\cdots\!98\)\( p^{3} T^{23} + 23990691322818097888 p^{4} T^{24} - 2407729436099707776 p^{5} T^{25} + 235562724353813348 p^{6} T^{26} - 22389577940717202 p^{7} T^{27} + 2060368146101196 p^{8} T^{28} - 182299937304006 p^{9} T^{29} + 15411450935318 p^{10} T^{30} - 1232843629176 p^{11} T^{31} + 92575338476 p^{12} T^{32} - 6446018694 p^{13} T^{33} + 412062020 p^{14} T^{34} - 23757936 p^{15} T^{35} + 1217388 p^{16} T^{36} - 53568 p^{17} T^{37} + 1964 p^{18} T^{38} - 54 p^{19} T^{39} + p^{20} T^{40} \)
83 \( 1 + 30 T + 450 T^{2} + 5944 T^{3} + 95605 T^{4} + 1410842 T^{5} + 16968578 T^{6} + 203343912 T^{7} + 2695553170 T^{8} + 32750241390 T^{9} + 351141202920 T^{10} + 3869482640368 T^{11} + 44789785336313 T^{12} + 475907380629938 T^{13} + 4691253245328074 T^{14} + 48289800523534186 T^{15} + 504762607906444230 T^{16} + 4832525965176358350 T^{17} + 44376178221440697962 T^{18} + \)\(42\!\cdots\!16\)\( T^{19} + \)\(40\!\cdots\!10\)\( T^{20} + \)\(42\!\cdots\!16\)\( p T^{21} + 44376178221440697962 p^{2} T^{22} + 4832525965176358350 p^{3} T^{23} + 504762607906444230 p^{4} T^{24} + 48289800523534186 p^{5} T^{25} + 4691253245328074 p^{6} T^{26} + 475907380629938 p^{7} T^{27} + 44789785336313 p^{8} T^{28} + 3869482640368 p^{9} T^{29} + 351141202920 p^{10} T^{30} + 32750241390 p^{11} T^{31} + 2695553170 p^{12} T^{32} + 203343912 p^{13} T^{33} + 16968578 p^{14} T^{34} + 1410842 p^{15} T^{35} + 95605 p^{16} T^{36} + 5944 p^{17} T^{37} + 450 p^{18} T^{38} + 30 p^{19} T^{39} + p^{20} T^{40} \)
89 \( 1 + 22 T - 364 T^{2} - 9964 T^{3} + 87718 T^{4} + 2546654 T^{5} - 18024552 T^{6} - 469950672 T^{7} + 3313545092 T^{8} + 67919708608 T^{9} - 553790812631 T^{10} - 8055575558214 T^{11} + 83165848851212 T^{12} + 797255343548938 T^{13} - 11164424605657168 T^{14} - 65246898990694390 T^{15} + 1333827226506449618 T^{16} + 4097643940166812774 T^{17} - \)\(14\!\cdots\!56\)\( T^{18} - \)\(13\!\cdots\!64\)\( T^{19} + \)\(13\!\cdots\!60\)\( T^{20} - \)\(13\!\cdots\!64\)\( p T^{21} - \)\(14\!\cdots\!56\)\( p^{2} T^{22} + 4097643940166812774 p^{3} T^{23} + 1333827226506449618 p^{4} T^{24} - 65246898990694390 p^{5} T^{25} - 11164424605657168 p^{6} T^{26} + 797255343548938 p^{7} T^{27} + 83165848851212 p^{8} T^{28} - 8055575558214 p^{9} T^{29} - 553790812631 p^{10} T^{30} + 67919708608 p^{11} T^{31} + 3313545092 p^{12} T^{32} - 469950672 p^{13} T^{33} - 18024552 p^{14} T^{34} + 2546654 p^{15} T^{35} + 87718 p^{16} T^{36} - 9964 p^{17} T^{37} - 364 p^{18} T^{38} + 22 p^{19} T^{39} + p^{20} T^{40} \)
97 \( 1 - 1506 T^{2} + 1107655 T^{4} - 528798826 T^{6} + 183715231671 T^{8} - 49359802576028 T^{10} + 10639228252040530 T^{12} - 1883570555977455668 T^{14} + \)\(27\!\cdots\!81\)\( T^{16} - \)\(34\!\cdots\!38\)\( T^{18} + \)\(36\!\cdots\!57\)\( T^{20} - \)\(34\!\cdots\!38\)\( p^{2} T^{22} + \)\(27\!\cdots\!81\)\( p^{4} T^{24} - 1883570555977455668 p^{6} T^{26} + 10639228252040530 p^{8} T^{28} - 49359802576028 p^{10} T^{30} + 183715231671 p^{12} T^{32} - 528798826 p^{14} T^{34} + 1107655 p^{16} T^{36} - 1506 p^{18} T^{38} + p^{20} T^{40} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.08585798811990095270120900264, −4.06292218582996448000299119271, −4.02709759175460366787509795824, −3.95037004690564142640566508105, −3.90981186437798635349402819085, −3.66792436091964613997455196992, −3.59438477809704294661121643248, −3.50665647079416143782249614317, −3.34030175917645157039853410201, −3.28622707858785150657172855753, −2.91965745836372899176250392663, −2.89957091277877891348150661013, −2.86374836602940847752107260282, −2.85938124396687415927740767820, −2.56288927648249866632184685158, −2.55598225386579804629686244088, −2.55441312245444866236410383617, −2.31399372535348298443778950223, −2.11347276926257703104396827397, −2.06442456773439672978256101160, −1.88371965047175996034022647625, −1.80730583519354466387419607056, −1.54927322932926526713938602307, −1.53583126528139858507137394849, −1.10414595671770536804622163242, 1.10414595671770536804622163242, 1.53583126528139858507137394849, 1.54927322932926526713938602307, 1.80730583519354466387419607056, 1.88371965047175996034022647625, 2.06442456773439672978256101160, 2.11347276926257703104396827397, 2.31399372535348298443778950223, 2.55441312245444866236410383617, 2.55598225386579804629686244088, 2.56288927648249866632184685158, 2.85938124396687415927740767820, 2.86374836602940847752107260282, 2.89957091277877891348150661013, 2.91965745836372899176250392663, 3.28622707858785150657172855753, 3.34030175917645157039853410201, 3.50665647079416143782249614317, 3.59438477809704294661121643248, 3.66792436091964613997455196992, 3.90981186437798635349402819085, 3.95037004690564142640566508105, 4.02709759175460366787509795824, 4.06292218582996448000299119271, 4.08585798811990095270120900264

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

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