Dirichlet series
| L(s) = 1 | + 2-s − 3·3-s + 5·5-s − 3·6-s + 7-s + 14·9-s + 5·10-s − 11·11-s + 10·13-s + 14-s − 15·15-s + 17-s + 14·18-s + 3·19-s − 3·21-s − 11·22-s + 12·23-s + 16·25-s + 10·26-s − 33·27-s + 2·29-s − 15·30-s − 5·31-s + 33·33-s + 34-s + 5·35-s + 27·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 2.23·5-s − 1.22·6-s + 0.377·7-s + 14/3·9-s + 1.58·10-s − 3.31·11-s + 2.77·13-s + 0.267·14-s − 3.87·15-s + 0.242·17-s + 3.29·18-s + 0.688·19-s − 0.654·21-s − 2.34·22-s + 2.50·23-s + 16/5·25-s + 1.96·26-s − 6.35·27-s + 0.371·29-s − 2.73·30-s − 0.898·31-s + 5.74·33-s + 0.171·34-s + 0.845·35-s + 4.43·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 7^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 7^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 7^{10} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(12627.8\) |
| Root analytic conductor: | \(1.60349\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 7^{10} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(7.021178285\) |
| \(L(\frac12)\) | \(\approx\) | \(7.021178285\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 7 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) | |
| 23 | \( 1 - 12 T + 100 T^{2} - 705 T^{3} + 4236 T^{4} - 22595 T^{5} + 4236 p T^{6} - 705 p^{2} T^{7} + 100 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 3 | \( 1 + p T - 5 T^{2} - 8 p T^{3} + p^{2} T^{4} + 11 p^{2} T^{5} + 28 T^{6} - 235 T^{7} - 206 T^{8} + 241 T^{9} + 703 T^{10} + 241 p T^{11} - 206 p^{2} T^{12} - 235 p^{3} T^{13} + 28 p^{4} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} - 8 p^{8} T^{17} - 5 p^{8} T^{18} + p^{10} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 - p T + 9 T^{2} + 13 T^{3} - 11 p T^{4} + 9 p T^{5} + 6 T^{6} + p^{4} T^{7} - 1208 T^{8} - 22 p^{3} T^{9} + 15951 T^{10} - 22 p^{4} T^{11} - 1208 p^{2} T^{12} + p^{7} T^{13} + 6 p^{4} T^{14} + 9 p^{6} T^{15} - 11 p^{7} T^{16} + 13 p^{7} T^{17} + 9 p^{8} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 + p T + 6 p T^{2} + 3 p^{2} T^{3} + 174 p T^{4} + 805 p T^{5} + 317 p^{2} T^{6} + 1292 p^{2} T^{7} + 4833 p^{2} T^{8} + 17429 p^{2} T^{9} + 60259 p^{2} T^{10} + 17429 p^{3} T^{11} + 4833 p^{4} T^{12} + 1292 p^{5} T^{13} + 317 p^{6} T^{14} + 805 p^{6} T^{15} + 174 p^{7} T^{16} + 3 p^{9} T^{17} + 6 p^{9} T^{18} + p^{10} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 10 T + 54 T^{2} - 190 T^{3} + 736 T^{4} - 2536 T^{5} + 8433 T^{6} - 21706 T^{7} + 59350 T^{8} + 1936 p T^{9} - 380423 T^{10} + 1936 p^{2} T^{11} + 59350 p^{2} T^{12} - 21706 p^{3} T^{13} + 8433 p^{4} T^{14} - 2536 p^{5} T^{15} + 736 p^{6} T^{16} - 190 p^{7} T^{17} + 54 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - T - 16 T^{2} + 77 T^{3} - 47 T^{4} - 82 p T^{5} + 1995 T^{6} + 31911 T^{7} - 100278 T^{8} - 441241 T^{9} + 2686859 T^{10} - 441241 p T^{11} - 100278 p^{2} T^{12} + 31911 p^{3} T^{13} + 1995 p^{4} T^{14} - 82 p^{6} T^{15} - 47 p^{6} T^{16} + 77 p^{7} T^{17} - 16 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 3 T + T^{2} - 100 T^{3} + 545 T^{4} - 153 T^{5} - 3868 T^{6} + 2312 T^{7} - 61935 T^{8} + 769724 T^{9} - 4325729 T^{10} + 769724 p T^{11} - 61935 p^{2} T^{12} + 2312 p^{3} T^{13} - 3868 p^{4} T^{14} - 153 p^{5} T^{15} + 545 p^{6} T^{16} - 100 p^{7} T^{17} + p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 2 T - 36 T^{2} - 156 T^{3} + 1125 T^{4} + 12273 T^{5} - 3007 T^{6} - 388953 T^{7} - 1543187 T^{8} + 2823524 T^{9} + 85742669 T^{10} + 2823524 p T^{11} - 1543187 p^{2} T^{12} - 388953 p^{3} T^{13} - 3007 p^{4} T^{14} + 12273 p^{5} T^{15} + 1125 p^{6} T^{16} - 156 p^{7} T^{17} - 36 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 5 T - 6 T^{2} + 79 T^{3} + 537 T^{4} + 2370 T^{5} + 25321 T^{6} - 60341 T^{7} - 162920 T^{8} + 836477 T^{9} - 18475127 T^{10} + 836477 p T^{11} - 162920 p^{2} T^{12} - 60341 p^{3} T^{13} + 25321 p^{4} T^{14} + 2370 p^{5} T^{15} + 537 p^{6} T^{16} + 79 p^{7} T^{17} - 6 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 27 T + 252 T^{2} + 102 T^{3} - 21318 T^{4} + 147069 T^{5} + 374016 T^{6} - 10860717 T^{7} + 47893359 T^{8} + 210331385 T^{9} - 3057029271 T^{10} + 210331385 p T^{11} + 47893359 p^{2} T^{12} - 10860717 p^{3} T^{13} + 374016 p^{4} T^{14} + 147069 p^{5} T^{15} - 21318 p^{6} T^{16} + 102 p^{7} T^{17} + 252 p^{8} T^{18} - 27 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 12 T + 37 T^{2} - 708 T^{3} - 9122 T^{4} - 25590 T^{5} + 343748 T^{6} + 4306332 T^{7} + 14125850 T^{8} - 103841294 T^{9} - 1304636323 T^{10} - 103841294 p T^{11} + 14125850 p^{2} T^{12} + 4306332 p^{3} T^{13} + 343748 p^{4} T^{14} - 25590 p^{5} T^{15} - 9122 p^{6} T^{16} - 708 p^{7} T^{17} + 37 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 38 T + 653 T^{2} - 6383 T^{3} + 35109 T^{4} - 65438 T^{5} - 493567 T^{6} + 5220267 T^{7} - 46819018 T^{8} + 501984786 T^{9} - 4070658373 T^{10} + 501984786 p T^{11} - 46819018 p^{2} T^{12} + 5220267 p^{3} T^{13} - 493567 p^{4} T^{14} - 65438 p^{5} T^{15} + 35109 p^{6} T^{16} - 6383 p^{7} T^{17} + 653 p^{8} T^{18} - 38 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 - T + 176 T^{2} - 317 T^{3} + 14093 T^{4} - 24523 T^{5} + 14093 p T^{6} - 317 p^{2} T^{7} + 176 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( 1 + 16 T + 181 T^{2} + 1740 T^{3} + 14331 T^{4} + 145172 T^{5} + 1340899 T^{6} + 11935126 T^{7} + 93881239 T^{8} + 647445110 T^{9} + 4753677159 T^{10} + 647445110 p T^{11} + 93881239 p^{2} T^{12} + 11935126 p^{3} T^{13} + 1340899 p^{4} T^{14} + 145172 p^{5} T^{15} + 14331 p^{6} T^{16} + 1740 p^{7} T^{17} + 181 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 5 T - 34 T^{2} + 723 T^{3} + 5621 T^{4} - 7556 T^{5} + 159989 T^{6} + 1263349 T^{7} + 14795932 T^{8} + 106069293 T^{9} - 187755171 T^{10} + 106069293 p T^{11} + 14795932 p^{2} T^{12} + 1263349 p^{3} T^{13} + 159989 p^{4} T^{14} - 7556 p^{5} T^{15} + 5621 p^{6} T^{16} + 723 p^{7} T^{17} - 34 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 26 T + 329 T^{2} + 3382 T^{3} + 32839 T^{4} + 235408 T^{5} + 1058461 T^{6} + 1265094 T^{7} - 51923775 T^{8} - 871618330 T^{9} - 8308477209 T^{10} - 871618330 p T^{11} - 51923775 p^{2} T^{12} + 1265094 p^{3} T^{13} + 1058461 p^{4} T^{14} + 235408 p^{5} T^{15} + 32839 p^{6} T^{16} + 3382 p^{7} T^{17} + 329 p^{8} T^{18} + 26 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 4 T - 139 T^{2} - 1726 T^{3} + 77 p T^{4} + 3530 p T^{5} + 943763 T^{6} - 16385472 T^{7} - 161977047 T^{8} + 482204800 T^{9} + 13395241685 T^{10} + 482204800 p T^{11} - 161977047 p^{2} T^{12} - 16385472 p^{3} T^{13} + 943763 p^{4} T^{14} + 3530 p^{6} T^{15} + 77 p^{7} T^{16} - 1726 p^{7} T^{17} - 139 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - T + 7 T^{2} + 636 T^{3} + 1023 T^{4} - 24179 T^{5} + 538594 T^{6} + 4029920 T^{7} - 3860789 T^{8} + 106859764 T^{9} + 2791009827 T^{10} + 106859764 p T^{11} - 3860789 p^{2} T^{12} + 4029920 p^{3} T^{13} + 538594 p^{4} T^{14} - 24179 p^{5} T^{15} + 1023 p^{6} T^{16} + 636 p^{7} T^{17} + 7 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 9 T + 85 T^{2} - 548 T^{3} + 2093 T^{4} - 13593 T^{5} + 167768 T^{6} - 1201955 T^{7} - 6373056 T^{8} + 176806729 T^{9} - 3539214029 T^{10} + 176806729 p T^{11} - 6373056 p^{2} T^{12} - 1201955 p^{3} T^{13} + 167768 p^{4} T^{14} - 13593 p^{5} T^{15} + 2093 p^{6} T^{16} - 548 p^{7} T^{17} + 85 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 50 T + 1090 T^{2} + 13282 T^{3} + 82737 T^{4} - 184466 T^{5} - 9679570 T^{6} - 86682345 T^{7} + 23833664 T^{8} + 9432744857 T^{9} + 120917307829 T^{10} + 9432744857 p T^{11} + 23833664 p^{2} T^{12} - 86682345 p^{3} T^{13} - 9679570 p^{4} T^{14} - 184466 p^{5} T^{15} + 82737 p^{6} T^{16} + 13282 p^{7} T^{17} + 1090 p^{8} T^{18} + 50 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 29 T + 120 T^{2} - 54 p T^{3} - 55832 T^{4} - 15441 T^{5} + 4287126 T^{6} + 54397447 T^{7} + 326720051 T^{8} - 3095878093 T^{9} - 66420068901 T^{10} - 3095878093 p T^{11} + 326720051 p^{2} T^{12} + 54397447 p^{3} T^{13} + 4287126 p^{4} T^{14} - 15441 p^{5} T^{15} - 55832 p^{6} T^{16} - 54 p^{8} T^{17} + 120 p^{8} T^{18} + 29 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 7 T + 290 T^{2} - 1792 T^{3} + 34738 T^{4} - 226304 T^{5} + 3050076 T^{6} - 28676586 T^{7} + 290064213 T^{8} - 3577705087 T^{9} + 27068154356 T^{10} - 3577705087 p T^{11} + 290064213 p^{2} T^{12} - 28676586 p^{3} T^{13} + 3050076 p^{4} T^{14} - 226304 p^{5} T^{15} + 34738 p^{6} T^{16} - 1792 p^{7} T^{17} + 290 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 12 T + 355 T^{2} - 2282 T^{3} + 46981 T^{4} - 43790 T^{5} + 2518977 T^{6} + 37933888 T^{7} - 98109797 T^{8} + 6971574654 T^{9} - 24466755037 T^{10} + 6971574654 p T^{11} - 98109797 p^{2} T^{12} + 37933888 p^{3} T^{13} + 2518977 p^{4} T^{14} - 43790 p^{5} T^{15} + 46981 p^{6} T^{16} - 2282 p^{7} T^{17} + 355 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−4.52863552559732473841603787545, −4.37515715153226147513654646851, −4.36645935709069772426800191112, −4.12811418453049879307475415980, −4.03141908952439540138606686791, −3.96675370035217843666552036939, −3.79337135962468353441874757434, −3.71858788560951833483373546004, −3.65060117860343486731037412180, −3.06719242400443771758479582934, −2.99861070567276851514239704354, −2.97441609031470376837844972085, −2.77763154819111230320944882629, −2.65544924166654849828067141114, −2.63367395155565043166283256663, −2.61325891922718761942911950574, −2.41064167082261804741826448711, −1.90421415421086556926452566721, −1.67738823533735312535781932857, −1.44645852654285609987524557241, −1.44495181859295011733365134060, −1.32112310680595665606696606757, −1.10726228578672800443094734959, −1.10709605141670411766060464158, −0.44948701794037164273630303664, 0.44948701794037164273630303664, 1.10709605141670411766060464158, 1.10726228578672800443094734959, 1.32112310680595665606696606757, 1.44495181859295011733365134060, 1.44645852654285609987524557241, 1.67738823533735312535781932857, 1.90421415421086556926452566721, 2.41064167082261804741826448711, 2.61325891922718761942911950574, 2.63367395155565043166283256663, 2.65544924166654849828067141114, 2.77763154819111230320944882629, 2.97441609031470376837844972085, 2.99861070567276851514239704354, 3.06719242400443771758479582934, 3.65060117860343486731037412180, 3.71858788560951833483373546004, 3.79337135962468353441874757434, 3.96675370035217843666552036939, 4.03141908952439540138606686791, 4.12811418453049879307475415980, 4.36645935709069772426800191112, 4.37515715153226147513654646851, 4.52863552559732473841603787545
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.