Dirichlet series
| L(s) = 1 | + 3·3-s − 5·5-s − 6·9-s + 13·11-s − 10·13-s − 15·15-s + 9·17-s + 12·19-s + 5·23-s − 9·25-s − 29·27-s + 10·29-s − 5·31-s + 39·33-s − 4·37-s − 30·39-s − 3·41-s + 27·43-s + 30·45-s + 16·47-s − 32·49-s + 27·51-s + 11·53-s − 65·55-s + 36·57-s + 27·59-s − 7·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 2.23·5-s − 2·9-s + 3.91·11-s − 2.77·13-s − 3.87·15-s + 2.18·17-s + 2.75·19-s + 1.04·23-s − 9/5·25-s − 5.58·27-s + 1.85·29-s − 0.898·31-s + 6.78·33-s − 0.657·37-s − 4.80·39-s − 0.468·41-s + 4.11·43-s + 4.47·45-s + 2.33·47-s − 4.57·49-s + 3.78·51-s + 1.51·53-s − 8.76·55-s + 4.76·57-s + 3.51·59-s − 0.896·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 13^{10} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 13^{10} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{40} \cdot 13^{10} \cdot 29^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.72023\times 10^{16}\) |
| Root analytic conductor: | \(6.94015\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{40} \cdot 13^{10} \cdot 29^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(50.37914844\) |
| \(L(\frac12)\) | \(\approx\) | \(50.37914844\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 13 | \( ( 1 + T )^{10} \) | |
| 29 | \( ( 1 - T )^{10} \) | |
| good | 3 | \( 1 - p T + 5 p T^{2} - 34 T^{3} + 35 p T^{4} - 214 T^{5} + 541 T^{6} - 1036 T^{7} + 2230 T^{8} - 3905 T^{9} + 7372 T^{10} - 3905 p T^{11} + 2230 p^{2} T^{12} - 1036 p^{3} T^{13} + 541 p^{4} T^{14} - 214 p^{5} T^{15} + 35 p^{7} T^{16} - 34 p^{7} T^{17} + 5 p^{9} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 + p T + 34 T^{2} + 128 T^{3} + 106 p T^{4} + 312 p T^{5} + 4981 T^{6} + 12262 T^{7} + 6653 p T^{8} + 72757 T^{9} + 179834 T^{10} + 72757 p T^{11} + 6653 p^{3} T^{12} + 12262 p^{3} T^{13} + 4981 p^{4} T^{14} + 312 p^{6} T^{15} + 106 p^{7} T^{16} + 128 p^{7} T^{17} + 34 p^{8} T^{18} + p^{10} T^{19} + p^{10} T^{20} \) | |
| 7 | \( 1 + 32 T^{2} - T^{3} + 456 T^{4} + 27 p T^{5} + 3893 T^{6} + 5721 T^{7} + 22739 T^{8} + 71315 T^{9} + 129594 T^{10} + 71315 p T^{11} + 22739 p^{2} T^{12} + 5721 p^{3} T^{13} + 3893 p^{4} T^{14} + 27 p^{6} T^{15} + 456 p^{6} T^{16} - p^{7} T^{17} + 32 p^{8} T^{18} + p^{10} T^{20} \) | |
| 11 | \( 1 - 13 T + 12 p T^{2} - 958 T^{3} + 6048 T^{4} - 32554 T^{5} + 159294 T^{6} - 700316 T^{7} + 2848143 T^{8} - 10589211 T^{9} + 36650396 T^{10} - 10589211 p T^{11} + 2848143 p^{2} T^{12} - 700316 p^{3} T^{13} + 159294 p^{4} T^{14} - 32554 p^{5} T^{15} + 6048 p^{6} T^{16} - 958 p^{7} T^{17} + 12 p^{9} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 9 T + 104 T^{2} - 736 T^{3} + 5442 T^{4} - 31954 T^{5} + 185857 T^{6} - 941432 T^{7} + 4651681 T^{8} - 20715021 T^{9} + 89480294 T^{10} - 20715021 p T^{11} + 4651681 p^{2} T^{12} - 941432 p^{3} T^{13} + 185857 p^{4} T^{14} - 31954 p^{5} T^{15} + 5442 p^{6} T^{16} - 736 p^{7} T^{17} + 104 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 12 T + 135 T^{2} - 925 T^{3} + 6474 T^{4} - 33518 T^{5} + 187698 T^{6} - 42756 p T^{7} + 4020901 T^{8} - 15703091 T^{9} + 76173486 T^{10} - 15703091 p T^{11} + 4020901 p^{2} T^{12} - 42756 p^{4} T^{13} + 187698 p^{4} T^{14} - 33518 p^{5} T^{15} + 6474 p^{6} T^{16} - 925 p^{7} T^{17} + 135 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 5 T + 129 T^{2} - 428 T^{3} + 7735 T^{4} - 844 p T^{5} + 320982 T^{6} - 666430 T^{7} + 10188256 T^{8} - 17823281 T^{9} + 257700866 T^{10} - 17823281 p T^{11} + 10188256 p^{2} T^{12} - 666430 p^{3} T^{13} + 320982 p^{4} T^{14} - 844 p^{6} T^{15} + 7735 p^{6} T^{16} - 428 p^{7} T^{17} + 129 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 5 T + 201 T^{2} + 920 T^{3} + 19165 T^{4} + 83000 T^{5} + 1180832 T^{6} + 4903994 T^{7} + 53413138 T^{8} + 207576597 T^{9} + 1869940110 T^{10} + 207576597 p T^{11} + 53413138 p^{2} T^{12} + 4903994 p^{3} T^{13} + 1180832 p^{4} T^{14} + 83000 p^{5} T^{15} + 19165 p^{6} T^{16} + 920 p^{7} T^{17} + 201 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 4 T + 301 T^{2} + 1029 T^{3} + 42399 T^{4} + 124679 T^{5} + 3699099 T^{6} + 9384177 T^{7} + 222137340 T^{8} + 485634215 T^{9} + 9615772984 T^{10} + 485634215 p T^{11} + 222137340 p^{2} T^{12} + 9384177 p^{3} T^{13} + 3699099 p^{4} T^{14} + 124679 p^{5} T^{15} + 42399 p^{6} T^{16} + 1029 p^{7} T^{17} + 301 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 3 T + 297 T^{2} + 793 T^{3} + 42012 T^{4} + 98943 T^{5} + 3756432 T^{6} + 7772583 T^{7} + 237490267 T^{8} + 430723742 T^{9} + 11189177758 T^{10} + 430723742 p T^{11} + 237490267 p^{2} T^{12} + 7772583 p^{3} T^{13} + 3756432 p^{4} T^{14} + 98943 p^{5} T^{15} + 42012 p^{6} T^{16} + 793 p^{7} T^{17} + 297 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 27 T + 445 T^{2} - 5426 T^{3} + 1319 p T^{4} - 542182 T^{5} + 4866833 T^{6} - 40593452 T^{7} + 312088338 T^{8} - 2225744921 T^{9} + 14967849592 T^{10} - 2225744921 p T^{11} + 312088338 p^{2} T^{12} - 40593452 p^{3} T^{13} + 4866833 p^{4} T^{14} - 542182 p^{5} T^{15} + 1319 p^{7} T^{16} - 5426 p^{7} T^{17} + 445 p^{8} T^{18} - 27 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 16 T + 336 T^{2} - 4052 T^{3} + 53538 T^{4} - 517469 T^{5} + 5306435 T^{6} - 43515255 T^{7} + 373458641 T^{8} - 2668443246 T^{9} + 19958078018 T^{10} - 2668443246 p T^{11} + 373458641 p^{2} T^{12} - 43515255 p^{3} T^{13} + 5306435 p^{4} T^{14} - 517469 p^{5} T^{15} + 53538 p^{6} T^{16} - 4052 p^{7} T^{17} + 336 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 - 11 T + 334 T^{2} - 3080 T^{3} + 55686 T^{4} - 445748 T^{5} + 6062538 T^{6} - 42653254 T^{7} + 478604633 T^{8} - 2979608187 T^{9} + 28830947824 T^{10} - 2979608187 p T^{11} + 478604633 p^{2} T^{12} - 42653254 p^{3} T^{13} + 6062538 p^{4} T^{14} - 445748 p^{5} T^{15} + 55686 p^{6} T^{16} - 3080 p^{7} T^{17} + 334 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 27 T + 760 T^{2} - 13170 T^{3} + 218576 T^{4} - 2830944 T^{5} + 34702002 T^{6} - 359980162 T^{7} + 3536742027 T^{8} - 30432920901 T^{9} + 248533437940 T^{10} - 30432920901 p T^{11} + 3536742027 p^{2} T^{12} - 359980162 p^{3} T^{13} + 34702002 p^{4} T^{14} - 2830944 p^{5} T^{15} + 218576 p^{6} T^{16} - 13170 p^{7} T^{17} + 760 p^{8} T^{18} - 27 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 7 T + 313 T^{2} + 2291 T^{3} + 56620 T^{4} + 388131 T^{5} + 6955840 T^{6} + 43736059 T^{7} + 634034651 T^{8} + 3574633568 T^{9} + 43971514174 T^{10} + 3574633568 p T^{11} + 634034651 p^{2} T^{12} + 43736059 p^{3} T^{13} + 6955840 p^{4} T^{14} + 388131 p^{5} T^{15} + 56620 p^{6} T^{16} + 2291 p^{7} T^{17} + 313 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 35 T + 883 T^{2} - 15652 T^{3} + 236453 T^{4} - 3001792 T^{5} + 34629186 T^{6} - 356957574 T^{7} + 3445576466 T^{8} - 30604141187 T^{9} + 259481144894 T^{10} - 30604141187 p T^{11} + 3445576466 p^{2} T^{12} - 356957574 p^{3} T^{13} + 34629186 p^{4} T^{14} - 3001792 p^{5} T^{15} + 236453 p^{6} T^{16} - 15652 p^{7} T^{17} + 883 p^{8} T^{18} - 35 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 21 T + 657 T^{2} - 9344 T^{3} + 166081 T^{4} - 1741348 T^{5} + 22633861 T^{6} - 184670438 T^{7} + 2019559390 T^{8} - 14050966089 T^{9} + 147684926136 T^{10} - 14050966089 p T^{11} + 2019559390 p^{2} T^{12} - 184670438 p^{3} T^{13} + 22633861 p^{4} T^{14} - 1741348 p^{5} T^{15} + 166081 p^{6} T^{16} - 9344 p^{7} T^{17} + 657 p^{8} T^{18} - 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 + 442 T^{2} + 587 T^{3} + 94527 T^{4} + 248436 T^{5} + 13098096 T^{6} + 48034634 T^{7} + 1337231680 T^{8} + 5498558095 T^{9} + 108222369484 T^{10} + 5498558095 p T^{11} + 1337231680 p^{2} T^{12} + 48034634 p^{3} T^{13} + 13098096 p^{4} T^{14} + 248436 p^{5} T^{15} + 94527 p^{6} T^{16} + 587 p^{7} T^{17} + 442 p^{8} T^{18} + p^{10} T^{20} \) | |
| 79 | \( 1 - 12 T + 451 T^{2} - 3969 T^{3} + 88598 T^{4} - 584362 T^{5} + 10627760 T^{6} - 52783640 T^{7} + 948474525 T^{8} - 3797131351 T^{9} + 75650685594 T^{10} - 3797131351 p T^{11} + 948474525 p^{2} T^{12} - 52783640 p^{3} T^{13} + 10627760 p^{4} T^{14} - 584362 p^{5} T^{15} + 88598 p^{6} T^{16} - 3969 p^{7} T^{17} + 451 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 24 T + 621 T^{2} - 8369 T^{3} + 121778 T^{4} - 1084498 T^{5} + 12238084 T^{6} - 85761784 T^{7} + 1031731777 T^{8} - 7201781607 T^{9} + 90859562382 T^{10} - 7201781607 p T^{11} + 1031731777 p^{2} T^{12} - 85761784 p^{3} T^{13} + 12238084 p^{4} T^{14} - 1084498 p^{5} T^{15} + 121778 p^{6} T^{16} - 8369 p^{7} T^{17} + 621 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 23 T + 378 T^{2} + 6021 T^{3} + 83966 T^{4} + 962177 T^{5} + 10132204 T^{6} + 99543185 T^{7} + 967674633 T^{8} + 8653363778 T^{9} + 77786010660 T^{10} + 8653363778 p T^{11} + 967674633 p^{2} T^{12} + 99543185 p^{3} T^{13} + 10132204 p^{4} T^{14} + 962177 p^{5} T^{15} + 83966 p^{6} T^{16} + 6021 p^{7} T^{17} + 378 p^{8} T^{18} + 23 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 2 T + 457 T^{2} - 605 T^{3} + 111502 T^{4} - 150070 T^{5} + 19228484 T^{6} - 28520620 T^{7} + 2551972201 T^{8} - 3863633759 T^{9} + 273348922486 T^{10} - 3863633759 p T^{11} + 2551972201 p^{2} T^{12} - 28520620 p^{3} T^{13} + 19228484 p^{4} T^{14} - 150070 p^{5} T^{15} + 111502 p^{6} T^{16} - 605 p^{7} T^{17} + 457 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.76061122196102400897058737789, −2.72600983947449690413161732093, −2.66568872275587016305533125398, −2.59179382035690188544224684882, −2.58049789896029677737233585562, −2.14670331183888610173501931548, −2.11081401146039261827803354658, −2.07669820378201200055141088732, −2.05410441343237411566038885066, −1.95947320555973759609905244349, −1.89847989026609808838159862783, −1.83030906429261954710226724898, −1.79084966498035865292309363666, −1.56653560292715220160810202950, −1.28429424696546179089836440605, −1.24414981942553243609888832289, −1.01342974182990562106662196619, −1.00181554577237315713068054641, −0.864626626947482912709667097409, −0.76322462555446435012707954690, −0.62396542872566406789994848056, −0.52030966943061468586032957157, −0.50690402575002748553135849601, −0.45259899134108496551831272403, −0.22191258497957777719583215318, 0.22191258497957777719583215318, 0.45259899134108496551831272403, 0.50690402575002748553135849601, 0.52030966943061468586032957157, 0.62396542872566406789994848056, 0.76322462555446435012707954690, 0.864626626947482912709667097409, 1.00181554577237315713068054641, 1.01342974182990562106662196619, 1.24414981942553243609888832289, 1.28429424696546179089836440605, 1.56653560292715220160810202950, 1.79084966498035865292309363666, 1.83030906429261954710226724898, 1.89847989026609808838159862783, 1.95947320555973759609905244349, 2.05410441343237411566038885066, 2.07669820378201200055141088732, 2.11081401146039261827803354658, 2.14670331183888610173501931548, 2.58049789896029677737233585562, 2.59179382035690188544224684882, 2.66568872275587016305533125398, 2.72600983947449690413161732093, 2.76061122196102400897058737789
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.