Properties

Label 40-483e20-1.1-c1e20-0-1
Degree $40$
Conductor $4.775\times 10^{53}$
Sign $1$
Analytic cond. $5.30254\times 10^{11}$
Root an. cond. $1.96386$
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  − 3·2-s + 10·3-s + 7·4-s + 5·5-s − 30·6-s − 10·8-s + 45·9-s − 15·10-s − 8·11-s + 70·12-s + 50·15-s + 10·16-s + 11·17-s − 135·18-s − 19-s + 35·20-s + 24·22-s − 10·23-s − 100·24-s + 27·25-s + 110·27-s + 44·29-s − 150·30-s + 3·31-s − 2·32-s − 80·33-s − 33·34-s + ⋯
L(s)  = 1  − 2.12·2-s + 5.77·3-s + 7/2·4-s + 2.23·5-s − 12.2·6-s − 3.53·8-s + 15·9-s − 4.74·10-s − 2.41·11-s + 20.2·12-s + 12.9·15-s + 5/2·16-s + 2.66·17-s − 31.8·18-s − 0.229·19-s + 7.82·20-s + 5.11·22-s − 2.08·23-s − 20.4·24-s + 27/5·25-s + 21.1·27-s + 8.17·29-s − 27.3·30-s + 0.538·31-s − 0.353·32-s − 13.9·33-s − 5.65·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 7^{20} \cdot 23^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 7^{20} \cdot 23^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(3^{20} \cdot 7^{20} \cdot 23^{20}\)
Sign: $1$
Analytic conductor: \(5.30254\times 10^{11}\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{483} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 3^{20} \cdot 7^{20} \cdot 23^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(266.6694548\)
\(L(\frac12)\) \(\approx\) \(266.6694548\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T + T^{2} )^{10} \)
7 \( 1 - 11 T^{2} + 38 T^{3} + 13 p T^{4} - 426 T^{5} - 26 T^{6} + 438 p T^{7} - 6093 T^{8} - 1784 p T^{9} + 7267 p T^{10} - 1784 p^{2} T^{11} - 6093 p^{2} T^{12} + 438 p^{4} T^{13} - 26 p^{4} T^{14} - 426 p^{5} T^{15} + 13 p^{7} T^{16} + 38 p^{7} T^{17} - 11 p^{8} T^{18} + p^{10} T^{20} \)
23 \( ( 1 + T + T^{2} )^{10} \)
good2 \( 1 + 3 T + p T^{2} - 5 T^{3} - 9 T^{4} + 13 T^{6} + 9 T^{7} - 17 T^{8} - 9 p^{2} T^{9} + 13 T^{10} + 141 T^{11} + 103 p T^{12} + 45 T^{13} - 309 T^{14} - 243 p T^{15} + p^{3} T^{16} + 111 p^{3} T^{17} + 15 p^{6} T^{18} - 91 p^{2} T^{19} - 433 p^{2} T^{20} - 91 p^{3} T^{21} + 15 p^{8} T^{22} + 111 p^{6} T^{23} + p^{7} T^{24} - 243 p^{6} T^{25} - 309 p^{6} T^{26} + 45 p^{7} T^{27} + 103 p^{9} T^{28} + 141 p^{9} T^{29} + 13 p^{10} T^{30} - 9 p^{13} T^{31} - 17 p^{12} T^{32} + 9 p^{13} T^{33} + 13 p^{14} T^{34} - 9 p^{16} T^{36} - 5 p^{17} T^{37} + p^{19} T^{38} + 3 p^{19} T^{39} + p^{20} T^{40} \)
5 \( 1 - p T - 2 T^{2} + 13 T^{3} + 133 T^{4} - 144 T^{5} - 734 T^{6} - 548 T^{7} + 2926 T^{8} + 10238 T^{9} - 614 p^{2} T^{10} - 11328 T^{11} - 53461 T^{12} + 206373 T^{13} - 248478 T^{14} - 385429 T^{15} + 495299 T^{16} + 97692 p^{2} T^{17} + 2266148 p T^{18} - 28977094 T^{19} - 4706544 T^{20} - 28977094 p T^{21} + 2266148 p^{3} T^{22} + 97692 p^{5} T^{23} + 495299 p^{4} T^{24} - 385429 p^{5} T^{25} - 248478 p^{6} T^{26} + 206373 p^{7} T^{27} - 53461 p^{8} T^{28} - 11328 p^{9} T^{29} - 614 p^{12} T^{30} + 10238 p^{11} T^{31} + 2926 p^{12} T^{32} - 548 p^{13} T^{33} - 734 p^{14} T^{34} - 144 p^{15} T^{35} + 133 p^{16} T^{36} + 13 p^{17} T^{37} - 2 p^{18} T^{38} - p^{20} T^{39} + p^{20} T^{40} \)
11 \( 1 + 8 T - 2 p T^{2} - 384 T^{3} - 345 T^{4} + 8504 T^{5} + 23282 T^{6} - 115760 T^{7} - 585921 T^{8} + 786584 T^{9} + 9821436 T^{10} + 7726800 T^{11} - 109913878 T^{12} - 29161960 p T^{13} + 623438052 T^{14} + 5184380672 T^{15} + 4334630045 T^{16} - 51644609240 T^{17} - 166884104994 T^{18} + 20976027208 p T^{19} + 217029933683 p T^{20} + 20976027208 p^{2} T^{21} - 166884104994 p^{2} T^{22} - 51644609240 p^{3} T^{23} + 4334630045 p^{4} T^{24} + 5184380672 p^{5} T^{25} + 623438052 p^{6} T^{26} - 29161960 p^{8} T^{27} - 109913878 p^{8} T^{28} + 7726800 p^{9} T^{29} + 9821436 p^{10} T^{30} + 786584 p^{11} T^{31} - 585921 p^{12} T^{32} - 115760 p^{13} T^{33} + 23282 p^{14} T^{34} + 8504 p^{15} T^{35} - 345 p^{16} T^{36} - 384 p^{17} T^{37} - 2 p^{19} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \)
13 \( ( 1 + 73 T^{2} + 32 T^{3} + 2827 T^{4} + 1656 T^{5} + 74194 T^{6} + 47428 T^{7} + 1433473 T^{8} + 67856 p T^{9} + 1632147 p T^{10} + 67856 p^{2} T^{11} + 1433473 p^{2} T^{12} + 47428 p^{3} T^{13} + 74194 p^{4} T^{14} + 1656 p^{5} T^{15} + 2827 p^{6} T^{16} + 32 p^{7} T^{17} + 73 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
17 \( 1 - 11 T - 8 T^{2} + 373 T^{3} + 147 T^{4} - 7048 T^{5} - 14108 T^{6} + 132148 T^{7} + 54756 T^{8} - 1012184 T^{9} + 786686 T^{10} + 13231444 T^{11} + 40280859 T^{12} - 605631641 T^{13} - 1112965516 T^{14} + 5234276687 T^{15} + 35711180633 T^{16} + 1308191452 T^{17} - 280177314162 T^{18} + 10443164468 p T^{19} - 258314765500 p T^{20} + 10443164468 p^{2} T^{21} - 280177314162 p^{2} T^{22} + 1308191452 p^{3} T^{23} + 35711180633 p^{4} T^{24} + 5234276687 p^{5} T^{25} - 1112965516 p^{6} T^{26} - 605631641 p^{7} T^{27} + 40280859 p^{8} T^{28} + 13231444 p^{9} T^{29} + 786686 p^{10} T^{30} - 1012184 p^{11} T^{31} + 54756 p^{12} T^{32} + 132148 p^{13} T^{33} - 14108 p^{14} T^{34} - 7048 p^{15} T^{35} + 147 p^{16} T^{36} + 373 p^{17} T^{37} - 8 p^{18} T^{38} - 11 p^{19} T^{39} + p^{20} T^{40} \)
19 \( 1 + T - 86 T^{2} + 271 T^{3} + 3722 T^{4} - 23399 T^{5} - 40148 T^{6} + 827771 T^{7} - 2085791 T^{8} - 7452976 T^{9} + 76069354 T^{10} - 16718958 p T^{11} + 313649444 T^{12} + 9043215258 T^{13} - 62464232012 T^{14} + 65698743972 T^{15} + 1183999056725 T^{16} - 6070817725523 T^{17} + 5273786368634 T^{18} + 70634113072993 T^{19} - 435944176464498 T^{20} + 70634113072993 p T^{21} + 5273786368634 p^{2} T^{22} - 6070817725523 p^{3} T^{23} + 1183999056725 p^{4} T^{24} + 65698743972 p^{5} T^{25} - 62464232012 p^{6} T^{26} + 9043215258 p^{7} T^{27} + 313649444 p^{8} T^{28} - 16718958 p^{10} T^{29} + 76069354 p^{10} T^{30} - 7452976 p^{11} T^{31} - 2085791 p^{12} T^{32} + 827771 p^{13} T^{33} - 40148 p^{14} T^{34} - 23399 p^{15} T^{35} + 3722 p^{16} T^{36} + 271 p^{17} T^{37} - 86 p^{18} T^{38} + p^{19} T^{39} + p^{20} T^{40} \)
29 \( ( 1 - 22 T + 396 T^{2} - 5028 T^{3} + 56213 T^{4} - 523954 T^{5} + 4431344 T^{6} - 32918302 T^{7} + 225039538 T^{8} - 47551366 p T^{9} + 7821784840 T^{10} - 47551366 p^{2} T^{11} + 225039538 p^{2} T^{12} - 32918302 p^{3} T^{13} + 4431344 p^{4} T^{14} - 523954 p^{5} T^{15} + 56213 p^{6} T^{16} - 5028 p^{7} T^{17} + 396 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
31 \( 1 - 3 T - 142 T^{2} + 543 T^{3} + 9840 T^{4} - 51405 T^{5} - 465748 T^{6} + 3463807 T^{7} + 16017597 T^{8} - 186228212 T^{9} - 299147976 T^{10} + 8270624692 T^{11} - 6810509818 T^{12} - 301060195322 T^{13} + 935763305122 T^{14} + 8866205720168 T^{15} - 53101034625055 T^{16} - 199917641182515 T^{17} + 2236405962648596 T^{18} + 2295711688620145 T^{19} - 76362381126087634 T^{20} + 2295711688620145 p T^{21} + 2236405962648596 p^{2} T^{22} - 199917641182515 p^{3} T^{23} - 53101034625055 p^{4} T^{24} + 8866205720168 p^{5} T^{25} + 935763305122 p^{6} T^{26} - 301060195322 p^{7} T^{27} - 6810509818 p^{8} T^{28} + 8270624692 p^{9} T^{29} - 299147976 p^{10} T^{30} - 186228212 p^{11} T^{31} + 16017597 p^{12} T^{32} + 3463807 p^{13} T^{33} - 465748 p^{14} T^{34} - 51405 p^{15} T^{35} + 9840 p^{16} T^{36} + 543 p^{17} T^{37} - 142 p^{18} T^{38} - 3 p^{19} T^{39} + p^{20} T^{40} \)
37 \( 1 - 3 T - 152 T^{2} + 73 T^{3} + 10602 T^{4} + 21583 T^{5} - 404436 T^{6} - 1403847 T^{7} + 8203143 T^{8} + 7615460 T^{9} - 203578600 T^{10} + 1495207608 T^{11} + 16737126820 T^{12} - 25126879222 T^{13} - 824573972516 T^{14} - 660230132196 T^{15} + 26849037432453 T^{16} - 1887023420423 T^{17} - 1265688494252188 T^{18} + 595644446788739 T^{19} + 59390719446702534 T^{20} + 595644446788739 p T^{21} - 1265688494252188 p^{2} T^{22} - 1887023420423 p^{3} T^{23} + 26849037432453 p^{4} T^{24} - 660230132196 p^{5} T^{25} - 824573972516 p^{6} T^{26} - 25126879222 p^{7} T^{27} + 16737126820 p^{8} T^{28} + 1495207608 p^{9} T^{29} - 203578600 p^{10} T^{30} + 7615460 p^{11} T^{31} + 8203143 p^{12} T^{32} - 1403847 p^{13} T^{33} - 404436 p^{14} T^{34} + 21583 p^{15} T^{35} + 10602 p^{16} T^{36} + 73 p^{17} T^{37} - 152 p^{18} T^{38} - 3 p^{19} T^{39} + p^{20} T^{40} \)
41 \( ( 1 + 26 T + 512 T^{2} + 6720 T^{3} + 76013 T^{4} + 685282 T^{5} + 5667904 T^{6} + 40070790 T^{7} + 276322178 T^{8} + 1731175150 T^{9} + 11388208416 T^{10} + 1731175150 p T^{11} + 276322178 p^{2} T^{12} + 40070790 p^{3} T^{13} + 5667904 p^{4} T^{14} + 685282 p^{5} T^{15} + 76013 p^{6} T^{16} + 6720 p^{7} T^{17} + 512 p^{8} T^{18} + 26 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
43 \( ( 1 - 27 T + 581 T^{2} - 9052 T^{3} + 120253 T^{4} - 1359081 T^{5} + 13645946 T^{6} - 122446777 T^{7} + 999237334 T^{8} - 7437857957 T^{9} + 50915042966 T^{10} - 7437857957 p T^{11} + 999237334 p^{2} T^{12} - 122446777 p^{3} T^{13} + 13645946 p^{4} T^{14} - 1359081 p^{5} T^{15} + 120253 p^{6} T^{16} - 9052 p^{7} T^{17} + 581 p^{8} T^{18} - 27 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
47 \( 1 + 11 T - 218 T^{2} - 4287 T^{3} + 12817 T^{4} + 727080 T^{5} + 2440682 T^{6} - 70466718 T^{7} - 605875012 T^{8} + 3411433894 T^{9} + 67174425662 T^{10} + 61306576416 T^{11} - 4439317655813 T^{12} - 25268598966687 T^{13} + 153660195227514 T^{14} + 2209135350363947 T^{15} + 2213984222876307 T^{16} - 106818386003826504 T^{17} - 637650928452138808 T^{18} + 2143925301143494372 T^{19} + 41393633785898236120 T^{20} + 2143925301143494372 p T^{21} - 637650928452138808 p^{2} T^{22} - 106818386003826504 p^{3} T^{23} + 2213984222876307 p^{4} T^{24} + 2209135350363947 p^{5} T^{25} + 153660195227514 p^{6} T^{26} - 25268598966687 p^{7} T^{27} - 4439317655813 p^{8} T^{28} + 61306576416 p^{9} T^{29} + 67174425662 p^{10} T^{30} + 3411433894 p^{11} T^{31} - 605875012 p^{12} T^{32} - 70466718 p^{13} T^{33} + 2440682 p^{14} T^{34} + 727080 p^{15} T^{35} + 12817 p^{16} T^{36} - 4287 p^{17} T^{37} - 218 p^{18} T^{38} + 11 p^{19} T^{39} + p^{20} T^{40} \)
53 \( 1 + 5 T - 202 T^{2} + 643 T^{3} + 25537 T^{4} - 209568 T^{5} - 890906 T^{6} + 23364822 T^{7} - 90276320 T^{8} - 753220526 T^{9} + 11190035400 T^{10} - 51193008768 T^{11} - 126139524575 T^{12} + 4817271033341 T^{13} - 41557880234226 T^{14} + 78006561041187 T^{15} + 2246148723059633 T^{16} - 23746165304122152 T^{17} + 68838202077781114 T^{18} + 789320148308548768 T^{19} - 9900049371565258396 T^{20} + 789320148308548768 p T^{21} + 68838202077781114 p^{2} T^{22} - 23746165304122152 p^{3} T^{23} + 2246148723059633 p^{4} T^{24} + 78006561041187 p^{5} T^{25} - 41557880234226 p^{6} T^{26} + 4817271033341 p^{7} T^{27} - 126139524575 p^{8} T^{28} - 51193008768 p^{9} T^{29} + 11190035400 p^{10} T^{30} - 753220526 p^{11} T^{31} - 90276320 p^{12} T^{32} + 23364822 p^{13} T^{33} - 890906 p^{14} T^{34} - 209568 p^{15} T^{35} + 25537 p^{16} T^{36} + 643 p^{17} T^{37} - 202 p^{18} T^{38} + 5 p^{19} T^{39} + p^{20} T^{40} \)
59 \( 1 - 10 T - 324 T^{2} + 2128 T^{3} + 67883 T^{4} - 212606 T^{5} - 9781068 T^{6} + 1944910 T^{7} + 1053548511 T^{8} + 2282863278 T^{9} - 86400240528 T^{10} - 364882825222 T^{11} + 5506642572786 T^{12} + 33670144853992 T^{13} - 276312172351880 T^{14} - 2164006327168314 T^{15} + 11190036831652957 T^{16} + 97156632637566928 T^{17} - 408639650599979452 T^{18} - 2106317415309109224 T^{19} + 18398089700247113725 T^{20} - 2106317415309109224 p T^{21} - 408639650599979452 p^{2} T^{22} + 97156632637566928 p^{3} T^{23} + 11190036831652957 p^{4} T^{24} - 2164006327168314 p^{5} T^{25} - 276312172351880 p^{6} T^{26} + 33670144853992 p^{7} T^{27} + 5506642572786 p^{8} T^{28} - 364882825222 p^{9} T^{29} - 86400240528 p^{10} T^{30} + 2282863278 p^{11} T^{31} + 1053548511 p^{12} T^{32} + 1944910 p^{13} T^{33} - 9781068 p^{14} T^{34} - 212606 p^{15} T^{35} + 67883 p^{16} T^{36} + 2128 p^{17} T^{37} - 324 p^{18} T^{38} - 10 p^{19} T^{39} + p^{20} T^{40} \)
61 \( 1 + 22 T - 230 T^{2} - 6896 T^{3} + 50619 T^{4} + 1411800 T^{5} - 9219598 T^{6} - 207788858 T^{7} + 1483260631 T^{8} + 24107204764 T^{9} - 201944289460 T^{10} - 2261399824080 T^{11} + 23372961891018 T^{12} + 173565665640168 T^{13} - 2312698573708380 T^{14} - 10700736008746332 T^{15} + 197729167260190541 T^{16} + 490867797420293850 T^{17} - 14718177519393519026 T^{18} - 11149314468883636840 T^{19} + \)\(95\!\cdots\!13\)\( T^{20} - 11149314468883636840 p T^{21} - 14718177519393519026 p^{2} T^{22} + 490867797420293850 p^{3} T^{23} + 197729167260190541 p^{4} T^{24} - 10700736008746332 p^{5} T^{25} - 2312698573708380 p^{6} T^{26} + 173565665640168 p^{7} T^{27} + 23372961891018 p^{8} T^{28} - 2261399824080 p^{9} T^{29} - 201944289460 p^{10} T^{30} + 24107204764 p^{11} T^{31} + 1483260631 p^{12} T^{32} - 207788858 p^{13} T^{33} - 9219598 p^{14} T^{34} + 1411800 p^{15} T^{35} + 50619 p^{16} T^{36} - 6896 p^{17} T^{37} - 230 p^{18} T^{38} + 22 p^{19} T^{39} + p^{20} T^{40} \)
67 \( 1 - 2 T - 479 T^{2} + 758 T^{3} + 120434 T^{4} - 139536 T^{5} - 20909157 T^{6} + 13779610 T^{7} + 2803065148 T^{8} - 331669520 T^{9} - 307776243255 T^{10} - 123219507786 T^{11} + 28789002668436 T^{12} + 23032410422464 T^{13} - 2369065448508819 T^{14} - 2290455590365882 T^{15} + 177096296785377007 T^{16} + 142077440869011942 T^{17} - 12428185358008167690 T^{18} - 3898538890143358840 T^{19} + \)\(84\!\cdots\!28\)\( T^{20} - 3898538890143358840 p T^{21} - 12428185358008167690 p^{2} T^{22} + 142077440869011942 p^{3} T^{23} + 177096296785377007 p^{4} T^{24} - 2290455590365882 p^{5} T^{25} - 2369065448508819 p^{6} T^{26} + 23032410422464 p^{7} T^{27} + 28789002668436 p^{8} T^{28} - 123219507786 p^{9} T^{29} - 307776243255 p^{10} T^{30} - 331669520 p^{11} T^{31} + 2803065148 p^{12} T^{32} + 13779610 p^{13} T^{33} - 20909157 p^{14} T^{34} - 139536 p^{15} T^{35} + 120434 p^{16} T^{36} + 758 p^{17} T^{37} - 479 p^{18} T^{38} - 2 p^{19} T^{39} + p^{20} T^{40} \)
71 \( ( 1 - 27 T + 637 T^{2} - 11032 T^{3} + 170394 T^{4} - 2275812 T^{5} + 27918611 T^{6} - 308350531 T^{7} + 3166296871 T^{8} - 29776947596 T^{9} + 261349838452 T^{10} - 29776947596 p T^{11} + 3166296871 p^{2} T^{12} - 308350531 p^{3} T^{13} + 27918611 p^{4} T^{14} - 2275812 p^{5} T^{15} + 170394 p^{6} T^{16} - 11032 p^{7} T^{17} + 637 p^{8} T^{18} - 27 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
73 \( 1 - 8 T - 281 T^{2} + 1840 T^{3} + 46110 T^{4} - 252236 T^{5} - 3880167 T^{6} + 17861160 T^{7} + 62578114 T^{8} - 476496660 T^{9} + 33071141945 T^{10} - 75241699260 T^{11} - 4692008242868 T^{12} + 11308965420780 T^{13} + 309549415860921 T^{14} - 1011903404085044 T^{15} - 956766932860251 T^{16} + 56013369233273128 T^{17} - 1847555476244342418 T^{18} - 1700166335974389812 T^{19} + \)\(20\!\cdots\!88\)\( T^{20} - 1700166335974389812 p T^{21} - 1847555476244342418 p^{2} T^{22} + 56013369233273128 p^{3} T^{23} - 956766932860251 p^{4} T^{24} - 1011903404085044 p^{5} T^{25} + 309549415860921 p^{6} T^{26} + 11308965420780 p^{7} T^{27} - 4692008242868 p^{8} T^{28} - 75241699260 p^{9} T^{29} + 33071141945 p^{10} T^{30} - 476496660 p^{11} T^{31} + 62578114 p^{12} T^{32} + 17861160 p^{13} T^{33} - 3880167 p^{14} T^{34} - 252236 p^{15} T^{35} + 46110 p^{16} T^{36} + 1840 p^{17} T^{37} - 281 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \)
79 \( 1 + 21 T - 84 T^{2} - 5227 T^{3} - 23664 T^{4} + 480397 T^{5} + 4940152 T^{6} - 12009521 T^{7} - 419706793 T^{8} - 1592732046 T^{9} + 22860682898 T^{10} + 278131174516 T^{11} - 865784819238 T^{12} - 33513883694742 T^{13} - 59163531827354 T^{14} + 2796738093133870 T^{15} + 15669039465989393 T^{16} - 135907353312502121 T^{17} - 1463338803919673058 T^{18} + 2981461750353013295 T^{19} + \)\(10\!\cdots\!90\)\( T^{20} + 2981461750353013295 p T^{21} - 1463338803919673058 p^{2} T^{22} - 135907353312502121 p^{3} T^{23} + 15669039465989393 p^{4} T^{24} + 2796738093133870 p^{5} T^{25} - 59163531827354 p^{6} T^{26} - 33513883694742 p^{7} T^{27} - 865784819238 p^{8} T^{28} + 278131174516 p^{9} T^{29} + 22860682898 p^{10} T^{30} - 1592732046 p^{11} T^{31} - 419706793 p^{12} T^{32} - 12009521 p^{13} T^{33} + 4940152 p^{14} T^{34} + 480397 p^{15} T^{35} - 23664 p^{16} T^{36} - 5227 p^{17} T^{37} - 84 p^{18} T^{38} + 21 p^{19} T^{39} + p^{20} T^{40} \)
83 \( ( 1 + 12 T + 362 T^{2} + 2682 T^{3} + 66305 T^{4} + 380966 T^{5} + 8925952 T^{6} + 41824294 T^{7} + 974731974 T^{8} + 3873777678 T^{9} + 87218795372 T^{10} + 3873777678 p T^{11} + 974731974 p^{2} T^{12} + 41824294 p^{3} T^{13} + 8925952 p^{4} T^{14} + 380966 p^{5} T^{15} + 66305 p^{6} T^{16} + 2682 p^{7} T^{17} + 362 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
89 \( 1 + 6 T - 554 T^{2} - 4768 T^{3} + 152583 T^{4} + 1677656 T^{5} - 26338026 T^{6} - 366855138 T^{7} + 3083234355 T^{8} + 54988463660 T^{9} - 252789071788 T^{10} - 5796534328744 T^{11} + 17751617675198 T^{12} + 409920883467724 T^{13} - 2078810110368252 T^{14} - 15671347828827980 T^{15} + 370106826357218253 T^{16} - 84761338554430962 T^{17} - 53976132385861118462 T^{18} + 23435828150237127388 T^{19} + \)\(56\!\cdots\!41\)\( T^{20} + 23435828150237127388 p T^{21} - 53976132385861118462 p^{2} T^{22} - 84761338554430962 p^{3} T^{23} + 370106826357218253 p^{4} T^{24} - 15671347828827980 p^{5} T^{25} - 2078810110368252 p^{6} T^{26} + 409920883467724 p^{7} T^{27} + 17751617675198 p^{8} T^{28} - 5796534328744 p^{9} T^{29} - 252789071788 p^{10} T^{30} + 54988463660 p^{11} T^{31} + 3083234355 p^{12} T^{32} - 366855138 p^{13} T^{33} - 26338026 p^{14} T^{34} + 1677656 p^{15} T^{35} + 152583 p^{16} T^{36} - 4768 p^{17} T^{37} - 554 p^{18} T^{38} + 6 p^{19} T^{39} + p^{20} T^{40} \)
97 \( ( 1 - 6 T + 286 T^{2} - 2290 T^{3} + 42685 T^{4} - 304468 T^{5} + 6024104 T^{6} - 33297868 T^{7} + 750467106 T^{8} - 5053993384 T^{9} + 76658721332 T^{10} - 5053993384 p T^{11} + 750467106 p^{2} T^{12} - 33297868 p^{3} T^{13} + 6024104 p^{4} T^{14} - 304468 p^{5} T^{15} + 42685 p^{6} T^{16} - 2290 p^{7} T^{17} + 286 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
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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

−2.61202311070631603020310297695, −2.56749659325077015462191412457, −2.50259776961288147640890900244, −2.46635395335930269644316932787, −2.39041908772093576831996885402, −2.26752791881068973263659267807, −2.20919835795570190689007096155, −2.17366536885554949041516962114, −2.09150977352406282340023226080, −1.98407105637847481472499540306, −1.77611766155768946216789031613, −1.71426181594447319013837080181, −1.66535305996224421131580637362, −1.57930549792747379789326483748, −1.55169961043904939226911510325, −1.41461360840507551873904449931, −1.35615717072271873587125412889, −1.31212976995492606460724433925, −1.20640667234527928484342250152, −1.07914828938560232273166651709, −0.78013281836934378093277937930, −0.72448161224095031829397617916, −0.69379355540087947715909347217, −0.47379162985919674995987383752, −0.45556921967375248499126572233, 0.45556921967375248499126572233, 0.47379162985919674995987383752, 0.69379355540087947715909347217, 0.72448161224095031829397617916, 0.78013281836934378093277937930, 1.07914828938560232273166651709, 1.20640667234527928484342250152, 1.31212976995492606460724433925, 1.35615717072271873587125412889, 1.41461360840507551873904449931, 1.55169961043904939226911510325, 1.57930549792747379789326483748, 1.66535305996224421131580637362, 1.71426181594447319013837080181, 1.77611766155768946216789031613, 1.98407105637847481472499540306, 2.09150977352406282340023226080, 2.17366536885554949041516962114, 2.20919835795570190689007096155, 2.26752791881068973263659267807, 2.39041908772093576831996885402, 2.46635395335930269644316932787, 2.50259776961288147640890900244, 2.56749659325077015462191412457, 2.61202311070631603020310297695

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

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