Dirichlet series
| L(s) = 1 | + 3·2-s + 3-s + 11·5-s + 3·6-s − 10·8-s − 9·9-s + 33·10-s + 2·11-s + 10·13-s + 11·15-s − 13·16-s + 13·17-s − 27·18-s − 5·19-s + 6·22-s + 2·23-s − 10·24-s + 45·25-s + 30·26-s − 14·27-s + 7·29-s + 33·30-s − 11·31-s + 2·33-s + 39·34-s + 3·37-s − 15·38-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 0.577·3-s + 4.91·5-s + 1.22·6-s − 3.53·8-s − 3·9-s + 10.4·10-s + 0.603·11-s + 2.77·13-s + 2.84·15-s − 3.25·16-s + 3.15·17-s − 6.36·18-s − 1.14·19-s + 1.27·22-s + 0.417·23-s − 2.04·24-s + 9·25-s + 5.88·26-s − 2.69·27-s + 1.29·29-s + 6.02·30-s − 1.97·31-s + 0.348·33-s + 6.68·34-s + 0.493·37-s − 2.43·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(163^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(163^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(163^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.32780\) |
| Root analytic conductor: | \(1.14086\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 163^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(7.989019000\) |
| \(L(\frac12)\) | \(\approx\) | \(7.989019000\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 163 | \( ( 1 - T )^{7} \) |
| good | 2 | \( 1 - 3 T + 9 T^{2} - 17 T^{3} + 17 p T^{4} - 51 T^{5} + 21 p^{2} T^{6} - 55 p T^{7} + 21 p^{3} T^{8} - 51 p^{2} T^{9} + 17 p^{4} T^{10} - 17 p^{4} T^{11} + 9 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 - T + 10 T^{2} - 5 T^{3} + 50 T^{4} - 2 p^{2} T^{5} + 205 T^{6} - 74 T^{7} + 205 p T^{8} - 2 p^{4} T^{9} + 50 p^{3} T^{10} - 5 p^{4} T^{11} + 10 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 5 | \( 1 - 11 T + 76 T^{2} - 374 T^{3} + 1477 T^{4} - 4806 T^{5} + 13394 T^{6} - 32086 T^{7} + 13394 p T^{8} - 4806 p^{2} T^{9} + 1477 p^{3} T^{10} - 374 p^{4} T^{11} + 76 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 + 4 p T^{2} - 18 T^{3} + 398 T^{4} - 389 T^{5} + 3763 T^{6} - 3840 T^{7} + 3763 p T^{8} - 389 p^{2} T^{9} + 398 p^{3} T^{10} - 18 p^{4} T^{11} + 4 p^{6} T^{12} + p^{7} T^{14} \) | |
| 11 | \( 1 - 2 T + 57 T^{2} - 65 T^{3} + 1407 T^{4} - 731 T^{5} + 21283 T^{6} - 5664 T^{7} + 21283 p T^{8} - 731 p^{2} T^{9} + 1407 p^{3} T^{10} - 65 p^{4} T^{11} + 57 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 10 T + 102 T^{2} - 631 T^{3} + 3771 T^{4} - 17291 T^{5} + 76660 T^{6} - 280562 T^{7} + 76660 p T^{8} - 17291 p^{2} T^{9} + 3771 p^{3} T^{10} - 631 p^{4} T^{11} + 102 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 13 T + 166 T^{2} - 1329 T^{3} + 9840 T^{4} - 56274 T^{5} + 296357 T^{6} - 1272982 T^{7} + 296357 p T^{8} - 56274 p^{2} T^{9} + 9840 p^{3} T^{10} - 1329 p^{4} T^{11} + 166 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 5 T + 91 T^{2} + 469 T^{3} + 3938 T^{4} + 20122 T^{5} + 107766 T^{6} + 493646 T^{7} + 107766 p T^{8} + 20122 p^{2} T^{9} + 3938 p^{3} T^{10} + 469 p^{4} T^{11} + 91 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 2 T + 118 T^{2} - 250 T^{3} + 6608 T^{4} - 13499 T^{5} + 227743 T^{6} - 405686 T^{7} + 227743 p T^{8} - 13499 p^{2} T^{9} + 6608 p^{3} T^{10} - 250 p^{4} T^{11} + 118 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 7 T + 127 T^{2} - 646 T^{3} + 8098 T^{4} - 34888 T^{5} + 334936 T^{6} - 1194456 T^{7} + 334936 p T^{8} - 34888 p^{2} T^{9} + 8098 p^{3} T^{10} - 646 p^{4} T^{11} + 127 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 11 T + 141 T^{2} + 989 T^{3} + 7342 T^{4} + 36732 T^{5} + 225056 T^{6} + 1015190 T^{7} + 225056 p T^{8} + 36732 p^{2} T^{9} + 7342 p^{3} T^{10} + 989 p^{4} T^{11} + 141 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 3 T + 94 T^{2} - 406 T^{3} + 4983 T^{4} - 20842 T^{5} + 252248 T^{6} - 733312 T^{7} + 252248 p T^{8} - 20842 p^{2} T^{9} + 4983 p^{3} T^{10} - 406 p^{4} T^{11} + 94 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 17 T + 283 T^{2} - 2852 T^{3} + 28544 T^{4} - 214023 T^{5} + 1647302 T^{6} - 10274539 T^{7} + 1647302 p T^{8} - 214023 p^{2} T^{9} + 28544 p^{3} T^{10} - 2852 p^{4} T^{11} + 283 p^{5} T^{12} - 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 10 T + 141 T^{2} + 1062 T^{3} + 9368 T^{4} + 61649 T^{5} + 460654 T^{6} + 2932885 T^{7} + 460654 p T^{8} + 61649 p^{2} T^{9} + 9368 p^{3} T^{10} + 1062 p^{4} T^{11} + 141 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 11 T + 43 T^{2} + 314 T^{3} - 2804 T^{4} + 5383 T^{5} + 131912 T^{6} - 1116863 T^{7} + 131912 p T^{8} + 5383 p^{2} T^{9} - 2804 p^{3} T^{10} + 314 p^{4} T^{11} + 43 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 18 T + 375 T^{2} - 4005 T^{3} + 48141 T^{4} - 371978 T^{5} + 3458860 T^{6} - 22383137 T^{7} + 3458860 p T^{8} - 371978 p^{2} T^{9} + 48141 p^{3} T^{10} - 4005 p^{4} T^{11} + 375 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 11 T + 252 T^{2} - 1934 T^{3} + 28087 T^{4} - 191672 T^{5} + 2291334 T^{6} - 13940160 T^{7} + 2291334 p T^{8} - 191672 p^{2} T^{9} + 28087 p^{3} T^{10} - 1934 p^{4} T^{11} + 252 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 4 T + 187 T^{2} - 510 T^{3} + 16426 T^{4} - 42911 T^{5} + 1151882 T^{6} - 3243451 T^{7} + 1151882 p T^{8} - 42911 p^{2} T^{9} + 16426 p^{3} T^{10} - 510 p^{4} T^{11} + 187 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 18 T + 6 p T^{2} + 4949 T^{3} + 70617 T^{4} + 686165 T^{5} + 109956 p T^{6} + 57501620 T^{7} + 109956 p^{2} T^{8} + 686165 p^{2} T^{9} + 70617 p^{3} T^{10} + 4949 p^{4} T^{11} + 6 p^{6} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 3 T + 178 T^{2} + 793 T^{3} + 13251 T^{4} + 107538 T^{5} + 826217 T^{6} + 9409813 T^{7} + 826217 p T^{8} + 107538 p^{2} T^{9} + 13251 p^{3} T^{10} + 793 p^{4} T^{11} + 178 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 2 T + 363 T^{2} - 1347 T^{3} + 59673 T^{4} - 291847 T^{5} + 6116143 T^{6} - 29811958 T^{7} + 6116143 p T^{8} - 291847 p^{2} T^{9} + 59673 p^{3} T^{10} - 1347 p^{4} T^{11} + 363 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 200 T^{2} - 121 T^{3} + 21111 T^{4} - 85025 T^{5} + 1701860 T^{6} - 10725940 T^{7} + 1701860 p T^{8} - 85025 p^{2} T^{9} + 21111 p^{3} T^{10} - 121 p^{4} T^{11} + 200 p^{5} T^{12} + p^{7} T^{14} \) | |
| 83 | \( 1 - 18 T + 511 T^{2} - 6019 T^{3} + 100973 T^{4} - 894832 T^{5} + 11680108 T^{6} - 86133917 T^{7} + 11680108 p T^{8} - 894832 p^{2} T^{9} + 100973 p^{3} T^{10} - 6019 p^{4} T^{11} + 511 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 18 T + 618 T^{2} - 8390 T^{3} + 159538 T^{4} - 1707859 T^{5} + 23082447 T^{6} - 196461066 T^{7} + 23082447 p T^{8} - 1707859 p^{2} T^{9} + 159538 p^{3} T^{10} - 8390 p^{4} T^{11} + 618 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 21 T + 439 T^{2} - 4879 T^{3} + 68083 T^{4} - 712034 T^{5} + 9819956 T^{6} - 93024973 T^{7} + 9819956 p T^{8} - 712034 p^{2} T^{9} + 68083 p^{3} T^{10} - 4879 p^{4} T^{11} + 439 p^{5} T^{12} - 21 p^{6} T^{13} + p^{7} T^{14} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.12956227503021670579776316578, −6.10240573953686972810072656976, −6.08737799409859381725385339693, −6.01711916191861644758234756686, −5.81156106012233173079506868661, −5.44082983575268086345510074702, −5.37473203753946182789875963094, −5.32855104639928313335556309681, −5.09903510074009460337799975921, −5.09621504393955739036992433568, −4.92673110150428299793321804000, −4.27067351328136370108714292685, −4.20066258932315849567074386485, −4.03617004132020185873889644367, −3.65044867129466979333847764461, −3.60782419925779693835230266672, −3.47991588475019851580335949212, −3.22806043850852759037456532922, −2.93393844430639228261608141677, −2.51840699200184662531304528816, −2.31983254968895164350985553299, −2.29480665808985747365433352025, −1.85469866455491119482796785200, −1.43247209006502686423966037383, −1.20259738799241033749877060632, 1.20259738799241033749877060632, 1.43247209006502686423966037383, 1.85469866455491119482796785200, 2.29480665808985747365433352025, 2.31983254968895164350985553299, 2.51840699200184662531304528816, 2.93393844430639228261608141677, 3.22806043850852759037456532922, 3.47991588475019851580335949212, 3.60782419925779693835230266672, 3.65044867129466979333847764461, 4.03617004132020185873889644367, 4.20066258932315849567074386485, 4.27067351328136370108714292685, 4.92673110150428299793321804000, 5.09621504393955739036992433568, 5.09903510074009460337799975921, 5.32855104639928313335556309681, 5.37473203753946182789875963094, 5.44082983575268086345510074702, 5.81156106012233173079506868661, 6.01711916191861644758234756686, 6.08737799409859381725385339693, 6.10240573953686972810072656976, 6.12956227503021670579776316578
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.