Dirichlet series
| L(s) = 1 | + 2-s + 3·3-s + 4·4-s − 5-s + 3·6-s − 10·7-s + 3·8-s + 12·9-s − 10-s + 2·11-s + 12·12-s − 10·14-s − 3·15-s + 11·16-s − 7·17-s + 12·18-s − 7·19-s − 4·20-s − 30·21-s + 2·22-s − 2·23-s + 9·24-s + 12·25-s + 25·27-s − 40·28-s − 6·29-s − 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 2·4-s − 0.447·5-s + 1.22·6-s − 3.77·7-s + 1.06·8-s + 4·9-s − 0.316·10-s + 0.603·11-s + 3.46·12-s − 2.67·14-s − 0.774·15-s + 11/4·16-s − 1.69·17-s + 2.82·18-s − 1.60·19-s − 0.894·20-s − 6.54·21-s + 0.426·22-s − 0.417·23-s + 1.83·24-s + 12/5·25-s + 4.81·27-s − 7.55·28-s − 1.11·29-s − 0.547·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 19^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 19^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(7^{10} \cdot 19^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.82510\) |
| Root analytic conductor: | \(1.03053\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 7^{10} \cdot 19^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(3.579243169\) |
| \(L(\frac12)\) | \(\approx\) | \(3.579243169\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( ( 1 + T )^{10} \) |
| 19 | \( 1 + 7 T + 30 T^{2} + 77 T^{3} - 13 p T^{4} - 2112 T^{5} - 13 p^{2} T^{6} + 77 p^{2} T^{7} + 30 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 2 | \( 1 - T - 3 T^{2} + p^{2} T^{3} - T^{5} + 7 T^{6} - p^{4} T^{7} + 3 p^{2} T^{8} + 5 p^{2} T^{9} - 63 T^{10} + 5 p^{3} T^{11} + 3 p^{4} T^{12} - p^{7} T^{13} + 7 p^{4} T^{14} - p^{5} T^{15} + p^{9} T^{17} - 3 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) |
| 3 | \( 1 - p T - p T^{2} + 20 T^{3} - 5 T^{4} - 61 T^{5} + 49 T^{6} + 103 T^{7} - 49 p T^{8} - 65 T^{9} + 298 T^{10} - 65 p T^{11} - 49 p^{3} T^{12} + 103 p^{3} T^{13} + 49 p^{4} T^{14} - 61 p^{5} T^{15} - 5 p^{6} T^{16} + 20 p^{7} T^{17} - p^{9} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) | |
| 5 | \( 1 + T - 11 T^{2} + 8 T^{3} + 13 p T^{4} - 137 T^{5} - 143 T^{6} + 757 T^{7} - 77 T^{8} - 1229 T^{9} + 2554 T^{10} - 1229 p T^{11} - 77 p^{2} T^{12} + 757 p^{3} T^{13} - 143 p^{4} T^{14} - 137 p^{5} T^{15} + 13 p^{7} T^{16} + 8 p^{7} T^{17} - 11 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( ( 1 - T + 43 T^{2} - 25 T^{3} + 822 T^{4} - 324 T^{5} + 822 p T^{6} - 25 p^{2} T^{7} + 43 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 13 | \( 1 - 45 T^{2} - 22 T^{3} + 6 p^{2} T^{4} + 672 T^{5} - 19088 T^{6} - 2950 T^{7} + 329874 T^{8} - 26544 T^{9} - 4750136 T^{10} - 26544 p T^{11} + 329874 p^{2} T^{12} - 2950 p^{3} T^{13} - 19088 p^{4} T^{14} + 672 p^{5} T^{15} + 6 p^{8} T^{16} - 22 p^{7} T^{17} - 45 p^{8} T^{18} + p^{10} T^{20} \) | |
| 17 | \( 1 + 7 T - T^{2} - 90 T^{3} + 23 T^{4} + 1703 T^{5} + 6661 T^{6} - 877 T^{7} - 6743 p T^{8} + 300773 T^{9} + 4739582 T^{10} + 300773 p T^{11} - 6743 p^{3} T^{12} - 877 p^{3} T^{13} + 6661 p^{4} T^{14} + 1703 p^{5} T^{15} + 23 p^{6} T^{16} - 90 p^{7} T^{17} - p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 + 2 T - 11 T^{2} + 144 T^{3} - 406 T^{4} - 2570 T^{5} + 23814 T^{6} - 76298 T^{7} + 66988 T^{8} + 73442 p T^{9} - 650636 p T^{10} + 73442 p^{2} T^{11} + 66988 p^{2} T^{12} - 76298 p^{3} T^{13} + 23814 p^{4} T^{14} - 2570 p^{5} T^{15} - 406 p^{6} T^{16} + 144 p^{7} T^{17} - 11 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 6 T - 25 T^{2} - 88 T^{3} - 404 T^{4} - 7350 T^{5} - 28514 T^{6} - 160128 T^{7} + 834884 T^{8} + 8749388 T^{9} + 13820564 T^{10} + 8749388 p T^{11} + 834884 p^{2} T^{12} - 160128 p^{3} T^{13} - 28514 p^{4} T^{14} - 7350 p^{5} T^{15} - 404 p^{6} T^{16} - 88 p^{7} T^{17} - 25 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( ( 1 + 3 T + 75 T^{2} + 445 T^{3} + 3066 T^{4} + 20544 T^{5} + 3066 p T^{6} + 445 p^{2} T^{7} + 75 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 37 | \( ( 1 + 5 T + 95 T^{2} + 565 T^{3} + 5888 T^{4} + 25196 T^{5} + 5888 p T^{6} + 565 p^{2} T^{7} + 95 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 41 | \( 1 - 19 T + 75 T^{2} + 274 T^{3} + 2253 T^{4} - 25969 T^{5} - 283001 T^{6} + 1478777 T^{7} + 16445319 T^{8} - 63083491 T^{9} - 327001494 T^{10} - 63083491 p T^{11} + 16445319 p^{2} T^{12} + 1478777 p^{3} T^{13} - 283001 p^{4} T^{14} - 25969 p^{5} T^{15} + 2253 p^{6} T^{16} + 274 p^{7} T^{17} + 75 p^{8} T^{18} - 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 16 T + 83 T^{2} + 26 T^{3} - 1692 T^{4} - 13146 T^{5} + 83590 T^{6} + 1857908 T^{7} + 7417866 T^{8} - 15555664 T^{9} - 220827096 T^{10} - 15555664 p T^{11} + 7417866 p^{2} T^{12} + 1857908 p^{3} T^{13} + 83590 p^{4} T^{14} - 13146 p^{5} T^{15} - 1692 p^{6} T^{16} + 26 p^{7} T^{17} + 83 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 10 T - 86 T^{2} + 1240 T^{3} + 3191 T^{4} - 73648 T^{5} - 95240 T^{6} + 2930606 T^{7} + 2751037 T^{8} - 60353800 T^{9} - 28770062 T^{10} - 60353800 p T^{11} + 2751037 p^{2} T^{12} + 2930606 p^{3} T^{13} - 95240 p^{4} T^{14} - 73648 p^{5} T^{15} + 3191 p^{6} T^{16} + 1240 p^{7} T^{17} - 86 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 - 7 T - 51 T^{2} - 308 T^{3} + 6051 T^{4} + 30005 T^{5} + 148567 T^{6} - 4717123 T^{7} - 11545683 T^{8} + 39595265 T^{9} + 1895810886 T^{10} + 39595265 p T^{11} - 11545683 p^{2} T^{12} - 4717123 p^{3} T^{13} + 148567 p^{4} T^{14} + 30005 p^{5} T^{15} + 6051 p^{6} T^{16} - 308 p^{7} T^{17} - 51 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 16 T - 67 T^{2} + 1638 T^{3} + 10892 T^{4} - 143456 T^{5} - 1083296 T^{6} + 7258984 T^{7} + 96403502 T^{8} - 225915218 T^{9} - 5736741976 T^{10} - 225915218 p T^{11} + 96403502 p^{2} T^{12} + 7258984 p^{3} T^{13} - 1083296 p^{4} T^{14} - 143456 p^{5} T^{15} + 10892 p^{6} T^{16} + 1638 p^{7} T^{17} - 67 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 27 T + 254 T^{2} + 757 T^{3} - 210 T^{4} + 4485 T^{5} - 268576 T^{6} - 3342359 T^{7} + 6811593 T^{8} + 226339546 T^{9} + 1675447764 T^{10} + 226339546 p T^{11} + 6811593 p^{2} T^{12} - 3342359 p^{3} T^{13} - 268576 p^{4} T^{14} + 4485 p^{5} T^{15} - 210 p^{6} T^{16} + 757 p^{7} T^{17} + 254 p^{8} T^{18} + 27 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 11 T - 163 T^{2} - 1692 T^{3} + 19895 T^{4} + 155513 T^{5} - 1824603 T^{6} - 9569399 T^{7} + 131526481 T^{8} + 240822533 T^{9} - 9085247422 T^{10} + 240822533 p T^{11} + 131526481 p^{2} T^{12} - 9569399 p^{3} T^{13} - 1824603 p^{4} T^{14} + 155513 p^{5} T^{15} + 19895 p^{6} T^{16} - 1692 p^{7} T^{17} - 163 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 313 T^{2} - 94 T^{3} + 56200 T^{4} + 21054 T^{5} - 7140290 T^{6} - 1770648 T^{7} + 704092610 T^{8} + 60332612 T^{9} - 55516870384 T^{10} + 60332612 p T^{11} + 704092610 p^{2} T^{12} - 1770648 p^{3} T^{13} - 7140290 p^{4} T^{14} + 21054 p^{5} T^{15} + 56200 p^{6} T^{16} - 94 p^{7} T^{17} - 313 p^{8} T^{18} + p^{10} T^{20} \) | |
| 73 | \( 1 - 5 T - 74 T^{2} + 69 T^{3} - 142 T^{4} + 43349 T^{5} + 5620 p T^{6} - 4239155 T^{7} - 16306571 T^{8} + 173965070 T^{9} - 698814308 T^{10} + 173965070 p T^{11} - 16306571 p^{2} T^{12} - 4239155 p^{3} T^{13} + 5620 p^{5} T^{14} + 43349 p^{5} T^{15} - 142 p^{6} T^{16} + 69 p^{7} T^{17} - 74 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 21 T + 108 T^{2} - 1973 T^{3} + 41634 T^{4} - 204207 T^{5} + 2076118 T^{6} - 42505193 T^{7} + 173431773 T^{8} - 1513669770 T^{9} + 32149725436 T^{10} - 1513669770 p T^{11} + 173431773 p^{2} T^{12} - 42505193 p^{3} T^{13} + 2076118 p^{4} T^{14} - 204207 p^{5} T^{15} + 41634 p^{6} T^{16} - 1973 p^{7} T^{17} + 108 p^{8} T^{18} - 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( ( 1 + 8 T + 386 T^{2} + 2489 T^{3} + 61741 T^{4} + 303094 T^{5} + 61741 p T^{6} + 2489 p^{2} T^{7} + 386 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 89 | \( 1 - 22 T + 34 T^{2} + 348 T^{3} + 26759 T^{4} - 77264 T^{5} - 2852952 T^{6} + 2787130 T^{7} + 127927981 T^{8} + 708276172 T^{9} - 15819320134 T^{10} + 708276172 p T^{11} + 127927981 p^{2} T^{12} + 2787130 p^{3} T^{13} - 2852952 p^{4} T^{14} - 77264 p^{5} T^{15} + 26759 p^{6} T^{16} + 348 p^{7} T^{17} + 34 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 28 T + p T^{2} + 1626 T^{3} + 34986 T^{4} - 533370 T^{5} - 4147698 T^{6} + 33630078 T^{7} + 748402710 T^{8} - 1610693350 T^{9} - 79430831192 T^{10} - 1610693350 p T^{11} + 748402710 p^{2} T^{12} + 33630078 p^{3} T^{13} - 4147698 p^{4} T^{14} - 533370 p^{5} T^{15} + 34986 p^{6} T^{16} + 1626 p^{7} T^{17} + p^{9} T^{18} - 28 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−5.46669776056911658560392533122, −5.22306832739015393192495614973, −5.05732954754412385700537259599, −4.91045457006401459423237603488, −4.73152043767526555333419725119, −4.47288886337877619366259473440, −4.37046518162420013167574775919, −4.33315547137194065391323818435, −3.99493413519057037820380198810, −3.88608789058171469228001304955, −3.79935253998410267954753936056, −3.75188859484122449131796252369, −3.59814890751358469420442356770, −3.52667304151399150096628941073, −3.18447081067282310515922974970, −3.00198075767861424392948924317, −2.95951016861133912375016555799, −2.66244058755131552638518345659, −2.57786110626891290694170286861, −2.52798249580046786061669564454, −2.00418263880680965135969477687, −1.90556098207000685319424761016, −1.81629851325147717252862842135, −1.50409758835562327275052501071, −0.806178251081531900361128864959, 0.806178251081531900361128864959, 1.50409758835562327275052501071, 1.81629851325147717252862842135, 1.90556098207000685319424761016, 2.00418263880680965135969477687, 2.52798249580046786061669564454, 2.57786110626891290694170286861, 2.66244058755131552638518345659, 2.95951016861133912375016555799, 3.00198075767861424392948924317, 3.18447081067282310515922974970, 3.52667304151399150096628941073, 3.59814890751358469420442356770, 3.75188859484122449131796252369, 3.79935253998410267954753936056, 3.88608789058171469228001304955, 3.99493413519057037820380198810, 4.33315547137194065391323818435, 4.37046518162420013167574775919, 4.47288886337877619366259473440, 4.73152043767526555333419725119, 4.91045457006401459423237603488, 5.05732954754412385700537259599, 5.22306832739015393192495614973, 5.46669776056911658560392533122
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.