Dirichlet series
| L(s) = 1 | + 4·5-s − 4·7-s + 8·11-s − 5·16-s − 4·17-s − 44·23-s + 10·25-s − 16·35-s + 52·43-s − 4·47-s + 8·49-s + 32·55-s + 32·61-s − 20·73-s − 32·77-s − 20·80-s + 116·83-s − 16·85-s − 16·101-s + 20·112-s − 176·115-s + 16·119-s − 52·121-s + 36·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.51·7-s + 2.41·11-s − 5/4·16-s − 0.970·17-s − 9.17·23-s + 2·25-s − 2.70·35-s + 7.92·43-s − 0.583·47-s + 8/7·49-s + 4.31·55-s + 4.09·61-s − 2.34·73-s − 3.64·77-s − 2.23·80-s + 12.7·83-s − 1.73·85-s − 1.59·101-s + 1.88·112-s − 16.4·115-s + 1.46·119-s − 4.72·121-s + 3.21·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{40} \cdot 5^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{40} \cdot 5^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{20} \cdot 3^{40} \cdot 5^{20} \cdot 19^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.07533\times 10^{22}\) |
| Root analytic conductor: | \(3.69518\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{20} \cdot 3^{40} \cdot 5^{20} \cdot 19^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(57.52193949\) |
| \(L(\frac12)\) | \(\approx\) | \(57.52193949\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{4} )^{5} \) |
| 3 | \( 1 \) | |
| 5 | \( ( 1 - 2 T + T^{2} - 8 T^{3} + 22 T^{4} - 52 T^{5} + 22 p T^{6} - 8 p^{2} T^{7} + p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 19 | \( 1 + 54 T^{2} + 1909 T^{4} + 43720 T^{6} + 834034 T^{8} + 14825156 T^{10} + 834034 p^{2} T^{12} + 43720 p^{4} T^{14} + 1909 p^{6} T^{16} + 54 p^{8} T^{18} + p^{10} T^{20} \) | |
| good | 7 | \( ( 1 + 2 T + 2 T^{2} + 22 T^{3} + 5 T^{4} - 32 T^{5} + 24 p T^{6} + 752 T^{7} - 710 T^{8} - 4428 T^{9} + 8780 T^{10} - 4428 p T^{11} - 710 p^{2} T^{12} + 752 p^{3} T^{13} + 24 p^{5} T^{14} - 32 p^{5} T^{15} + 5 p^{6} T^{16} + 22 p^{7} T^{17} + 2 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) |
| 11 | \( ( 1 - 2 T + 23 T^{2} - 80 T^{3} + 38 p T^{4} - 1004 T^{5} + 38 p^{2} T^{6} - 80 p^{2} T^{7} + 23 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{4} \) | |
| 13 | \( 1 + 394 T^{4} + 49245 T^{8} - 310216 T^{12} - 475711438 T^{16} - 36658781444 T^{20} - 475711438 p^{4} T^{24} - 310216 p^{8} T^{28} + 49245 p^{12} T^{32} + 394 p^{16} T^{36} + p^{20} T^{40} \) | |
| 17 | \( ( 1 + 2 T + 2 T^{2} + 18 T^{3} + 37 T^{4} - 264 T^{5} - 440 T^{6} - 4008 T^{7} + 50186 T^{8} + 92636 T^{9} + 63884 T^{10} + 92636 p T^{11} + 50186 p^{2} T^{12} - 4008 p^{3} T^{13} - 440 p^{4} T^{14} - 264 p^{5} T^{15} + 37 p^{6} T^{16} + 18 p^{7} T^{17} + 2 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 23 | \( ( 1 + 22 T + 242 T^{2} + 1970 T^{3} + 14021 T^{4} + 90816 T^{5} + 545320 T^{6} + 3130352 T^{7} + 17263066 T^{8} + 89682716 T^{9} + 439992428 T^{10} + 89682716 p T^{11} + 17263066 p^{2} T^{12} + 3130352 p^{3} T^{13} + 545320 p^{4} T^{14} + 90816 p^{5} T^{15} + 14021 p^{6} T^{16} + 1970 p^{7} T^{17} + 242 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 29 | \( ( 1 + 154 T^{2} + 13381 T^{4} + 781592 T^{6} + 33862338 T^{8} + 1116062556 T^{10} + 33862338 p^{2} T^{12} + 781592 p^{4} T^{14} + 13381 p^{6} T^{16} + 154 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 31 | \( ( 1 - 174 T^{2} + 16605 T^{4} - 1057928 T^{6} + 49482530 T^{8} - 1750617876 T^{10} + 49482530 p^{2} T^{12} - 1057928 p^{4} T^{14} + 16605 p^{6} T^{16} - 174 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 37 | \( 1 - 6870 T^{4} + 19838781 T^{8} - 28798551368 T^{12} + 17864303461554 T^{16} - 4119255381666372 T^{20} + 17864303461554 p^{4} T^{24} - 28798551368 p^{8} T^{28} + 19838781 p^{12} T^{32} - 6870 p^{16} T^{36} + p^{20} T^{40} \) | |
| 41 | \( ( 1 - 194 T^{2} + 18661 T^{4} - 1177768 T^{6} + 56689978 T^{8} - 2395807948 T^{10} + 56689978 p^{2} T^{12} - 1177768 p^{4} T^{14} + 18661 p^{6} T^{16} - 194 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 43 | \( ( 1 - 26 T + 338 T^{2} - 3246 T^{3} + 26693 T^{4} - 190376 T^{5} + 1195800 T^{6} - 6457112 T^{7} + 27650866 T^{8} - 104552492 T^{9} + 534178924 T^{10} - 104552492 p T^{11} + 27650866 p^{2} T^{12} - 6457112 p^{3} T^{13} + 1195800 p^{4} T^{14} - 190376 p^{5} T^{15} + 26693 p^{6} T^{16} - 3246 p^{7} T^{17} + 338 p^{8} T^{18} - 26 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 47 | \( ( 1 + 2 T + 2 T^{2} + 1062 T^{3} + 6165 T^{4} - 5056 T^{5} + 541480 T^{6} + 5016464 T^{7} + 4811418 T^{8} + 120756084 T^{9} + 2233931468 T^{10} + 120756084 p T^{11} + 4811418 p^{2} T^{12} + 5016464 p^{3} T^{13} + 541480 p^{4} T^{14} - 5056 p^{5} T^{15} + 6165 p^{6} T^{16} + 1062 p^{7} T^{17} + 2 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 53 | \( 1 - 3718 T^{4} + 21469341 T^{8} - 41973520328 T^{12} + 221674170955666 T^{16} - 405771832358172132 T^{20} + 221674170955666 p^{4} T^{24} - 41973520328 p^{8} T^{28} + 21469341 p^{12} T^{32} - 3718 p^{16} T^{36} + p^{20} T^{40} \) | |
| 59 | \( ( 1 + 446 T^{2} + 94685 T^{4} + 12630840 T^{6} + 1177962250 T^{8} + 80575224628 T^{10} + 1177962250 p^{2} T^{12} + 12630840 p^{4} T^{14} + 94685 p^{6} T^{16} + 446 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 61 | \( ( 1 - 8 T + 289 T^{2} - 1840 T^{3} + 34362 T^{4} - 165264 T^{5} + 34362 p T^{6} - 1840 p^{2} T^{7} + 289 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} )^{4} \) | |
| 67 | \( 1 - 5990 T^{4} - 25704643 T^{8} + 271551620280 T^{12} + 202002443610130 T^{16} - 6356983071451297828 T^{20} + 202002443610130 p^{4} T^{24} + 271551620280 p^{8} T^{28} - 25704643 p^{12} T^{32} - 5990 p^{16} T^{36} + p^{20} T^{40} \) | |
| 71 | \( ( 1 - 478 T^{2} + 114525 T^{4} - 17788776 T^{6} + 1967920210 T^{8} - 161370485236 T^{10} + 1967920210 p^{2} T^{12} - 17788776 p^{4} T^{14} + 114525 p^{6} T^{16} - 478 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 73 | \( ( 1 + 10 T + 50 T^{2} + 1434 T^{3} + 15373 T^{4} - 13928 T^{5} + 120248 T^{6} + 2848472 T^{7} - 108763454 T^{8} - 1043040772 T^{9} - 2659104084 T^{10} - 1043040772 p T^{11} - 108763454 p^{2} T^{12} + 2848472 p^{3} T^{13} + 120248 p^{4} T^{14} - 13928 p^{5} T^{15} + 15373 p^{6} T^{16} + 1434 p^{7} T^{17} + 50 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 79 | \( ( 1 + 606 T^{2} + 174525 T^{4} + 31601096 T^{6} + 3989973218 T^{8} + 367012757556 T^{10} + 3989973218 p^{2} T^{12} + 31601096 p^{4} T^{14} + 174525 p^{6} T^{16} + 606 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 83 | \( ( 1 - 58 T + 1682 T^{2} - 34590 T^{3} + 586861 T^{4} - 8668952 T^{5} + 113933064 T^{6} - 1361047912 T^{7} + 14963593546 T^{8} - 151896077932 T^{9} + 1432347968812 T^{10} - 151896077932 p T^{11} + 14963593546 p^{2} T^{12} - 1361047912 p^{3} T^{13} + 113933064 p^{4} T^{14} - 8668952 p^{5} T^{15} + 586861 p^{6} T^{16} - 34590 p^{7} T^{17} + 1682 p^{8} T^{18} - 58 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 89 | \( ( 1 + 418 T^{2} + 83941 T^{4} + 10954856 T^{6} + 1087940634 T^{8} + 97111095180 T^{10} + 1087940634 p^{2} T^{12} + 10954856 p^{4} T^{14} + 83941 p^{6} T^{16} + 418 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 97 | \( 1 - 30614 T^{4} + 404194349 T^{8} - 1846939408392 T^{12} - 21181414394442158 T^{16} + \)\(39\!\cdots\!16\)\( T^{20} - 21181414394442158 p^{4} T^{24} - 1846939408392 p^{8} T^{28} + 404194349 p^{12} T^{32} - 30614 p^{16} T^{36} + p^{20} T^{40} \) | |
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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
−2.00534270912952353013363307360, −1.98247788975163507533074513862, −1.97946122212877075278472264978, −1.93832251319381497314924193079, −1.92013379303700057424232316655, −1.90960684210304095978044664807, −1.67517762396396035595493943511, −1.50298921547898647130248749420, −1.43225788053910691845692817376, −1.38273675799466936992792203747, −1.24371159933570713410706137037, −1.21416654337134159742497677905, −1.18891436115489559784152402604, −1.15412793610730817328087623295, −1.02115089396666829268939922760, −1.01489966329999986634513805880, −0.869123571132206243527211192525, −0.71333945446280298905810409105, −0.70277546661185463656287733277, −0.58360206796172013868757043488, −0.48221483699648247528921992508, −0.34093878768750934948300644557, −0.30486318574693969738578383909, −0.27089840288741839166509887570, −0.25867940434603369271400333870, 0.25867940434603369271400333870, 0.27089840288741839166509887570, 0.30486318574693969738578383909, 0.34093878768750934948300644557, 0.48221483699648247528921992508, 0.58360206796172013868757043488, 0.70277546661185463656287733277, 0.71333945446280298905810409105, 0.869123571132206243527211192525, 1.01489966329999986634513805880, 1.02115089396666829268939922760, 1.15412793610730817328087623295, 1.18891436115489559784152402604, 1.21416654337134159742497677905, 1.24371159933570713410706137037, 1.38273675799466936992792203747, 1.43225788053910691845692817376, 1.50298921547898647130248749420, 1.67517762396396035595493943511, 1.90960684210304095978044664807, 1.92013379303700057424232316655, 1.93832251319381497314924193079, 1.97946122212877075278472264978, 1.98247788975163507533074513862, 2.00534270912952353013363307360
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.