Dirichlet series
| L(s) = 1 | + 2·2-s + 2·3-s + 2·5-s + 4·6-s − 9-s + 4·10-s + 7·11-s − 6·13-s + 4·15-s + 5·16-s − 19·17-s − 2·18-s − 7·19-s + 14·22-s − 23-s − 17·25-s − 12·26-s − 2·27-s + 24·29-s + 8·30-s + 4·32-s + 14·33-s − 38·34-s + 8·37-s − 14·38-s − 12·39-s + 4·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 0.894·5-s + 1.63·6-s − 1/3·9-s + 1.26·10-s + 2.11·11-s − 1.66·13-s + 1.03·15-s + 5/4·16-s − 4.60·17-s − 0.471·18-s − 1.60·19-s + 2.98·22-s − 0.208·23-s − 3.39·25-s − 2.35·26-s − 0.384·27-s + 4.45·29-s + 1.46·30-s + 0.707·32-s + 2.43·33-s − 6.51·34-s + 1.31·37-s − 2.27·38-s − 1.92·39-s + 0.624·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{14} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{14} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(7^{14} \cdot 19^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.25483\times 10^{6}\) |
| Root analytic conductor: | \(2.72654\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 7^{14} \cdot 19^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(10.46986300\) |
| \(L(\frac12)\) | \(\approx\) | \(10.46986300\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 \) |
| 19 | \( ( 1 + T )^{7} \) | |
| good | 2 | \( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + 11 T^{4} - p^{4} T^{5} + 3 p^{3} T^{6} - 13 p T^{7} + 3 p^{4} T^{8} - p^{6} T^{9} + 11 p^{3} T^{10} - p^{7} T^{11} + p^{7} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 - 2 T + 5 T^{2} - 10 T^{3} + 7 p T^{4} - 10 T^{5} + 25 T^{6} - 28 T^{7} + 25 p T^{8} - 10 p^{2} T^{9} + 7 p^{4} T^{10} - 10 p^{4} T^{11} + 5 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 5 | \( 1 - 2 T + 21 T^{2} - 28 T^{3} + 208 T^{4} - 8 p^{2} T^{5} + 1382 T^{6} - 1088 T^{7} + 1382 p T^{8} - 8 p^{4} T^{9} + 208 p^{3} T^{10} - 28 p^{4} T^{11} + 21 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 7 T + 64 T^{2} - 331 T^{3} + 1877 T^{4} - 7490 T^{5} + 31971 T^{6} - 103447 T^{7} + 31971 p T^{8} - 7490 p^{2} T^{9} + 1877 p^{3} T^{10} - 331 p^{4} T^{11} + 64 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 6 T + 67 T^{2} + 258 T^{3} + 145 p T^{4} + 5910 T^{5} + 35479 T^{6} + 94028 T^{7} + 35479 p T^{8} + 5910 p^{2} T^{9} + 145 p^{4} T^{10} + 258 p^{4} T^{11} + 67 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 19 T + 258 T^{2} + 2435 T^{3} + 18796 T^{4} + 116989 T^{5} + 620513 T^{6} + 2756938 T^{7} + 620513 p T^{8} + 116989 p^{2} T^{9} + 18796 p^{3} T^{10} + 2435 p^{4} T^{11} + 258 p^{5} T^{12} + 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + T + 76 T^{2} + 145 T^{3} + 2993 T^{4} + 5150 T^{5} + 90483 T^{6} + 110989 T^{7} + 90483 p T^{8} + 5150 p^{2} T^{9} + 2993 p^{3} T^{10} + 145 p^{4} T^{11} + 76 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 24 T + 409 T^{2} - 4844 T^{3} + 1627 p T^{4} - 372712 T^{5} + 2538615 T^{6} - 14627880 T^{7} + 2538615 p T^{8} - 372712 p^{2} T^{9} + 1627 p^{4} T^{10} - 4844 p^{4} T^{11} + 409 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 121 T^{2} - 6 T^{3} + 7597 T^{4} - 540 T^{5} + 319141 T^{6} - 30580 T^{7} + 319141 p T^{8} - 540 p^{2} T^{9} + 7597 p^{3} T^{10} - 6 p^{4} T^{11} + 121 p^{5} T^{12} + p^{7} T^{14} \) | |
| 37 | \( 1 - 8 T + 105 T^{2} - 596 T^{3} + 5511 T^{4} - 21384 T^{5} + 185359 T^{6} - 605240 T^{7} + 185359 p T^{8} - 21384 p^{2} T^{9} + 5511 p^{3} T^{10} - 596 p^{4} T^{11} + 105 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 4 T + 247 T^{2} - 856 T^{3} + 27533 T^{4} - 80636 T^{5} + 1791819 T^{6} - 4284880 T^{7} + 1791819 p T^{8} - 80636 p^{2} T^{9} + 27533 p^{3} T^{10} - 856 p^{4} T^{11} + 247 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 4 T + 153 T^{2} - 476 T^{3} + 13796 T^{4} - 38616 T^{5} + 822496 T^{6} - 1893500 T^{7} + 822496 p T^{8} - 38616 p^{2} T^{9} + 13796 p^{3} T^{10} - 476 p^{4} T^{11} + 153 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 5 T + 180 T^{2} + 997 T^{3} + 16969 T^{4} + 92822 T^{5} + 1076483 T^{6} + 5414945 T^{7} + 1076483 p T^{8} + 92822 p^{2} T^{9} + 16969 p^{3} T^{10} + 997 p^{4} T^{11} + 180 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 20 T + 427 T^{2} - 5354 T^{3} + 67181 T^{4} - 620128 T^{5} + 5729967 T^{6} - 41610284 T^{7} + 5729967 p T^{8} - 620128 p^{2} T^{9} + 67181 p^{3} T^{10} - 5354 p^{4} T^{11} + 427 p^{5} T^{12} - 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 16 T + 411 T^{2} - 4430 T^{3} + 65959 T^{4} - 543928 T^{5} + 6027485 T^{6} - 40038004 T^{7} + 6027485 p T^{8} - 543928 p^{2} T^{9} + 65959 p^{3} T^{10} - 4430 p^{4} T^{11} + 411 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 5 T + 240 T^{2} + 835 T^{3} + 32437 T^{4} + 99182 T^{5} + 2835629 T^{6} + 6874399 T^{7} + 2835629 p T^{8} + 99182 p^{2} T^{9} + 32437 p^{3} T^{10} + 835 p^{4} T^{11} + 240 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 4 T + 213 T^{2} + 424 T^{3} + 24957 T^{4} + 33948 T^{5} + 2254897 T^{6} + 3140272 T^{7} + 2254897 p T^{8} + 33948 p^{2} T^{9} + 24957 p^{3} T^{10} + 424 p^{4} T^{11} + 213 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 12 T + 241 T^{2} - 3098 T^{3} + 34397 T^{4} - 371080 T^{5} + 3742413 T^{6} - 30042540 T^{7} + 3742413 p T^{8} - 371080 p^{2} T^{9} + 34397 p^{3} T^{10} - 3098 p^{4} T^{11} + 241 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 3 T + 208 T^{2} + 369 T^{3} + 20749 T^{4} + 56754 T^{5} + 1773025 T^{6} + 6523085 T^{7} + 1773025 p T^{8} + 56754 p^{2} T^{9} + 20749 p^{3} T^{10} + 369 p^{4} T^{11} + 208 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 20 T + 489 T^{2} + 6706 T^{3} + 99149 T^{4} + 1060368 T^{5} + 11868757 T^{6} + 103403740 T^{7} + 11868757 p T^{8} + 1060368 p^{2} T^{9} + 99149 p^{3} T^{10} + 6706 p^{4} T^{11} + 489 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 11 T + 372 T^{2} - 4183 T^{3} + 73441 T^{4} - 715466 T^{5} + 9272063 T^{6} - 73855667 T^{7} + 9272063 p T^{8} - 715466 p^{2} T^{9} + 73441 p^{3} T^{10} - 4183 p^{4} T^{11} + 372 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 10 T + 601 T^{2} - 4852 T^{3} + 155531 T^{4} - 1017854 T^{5} + 22672155 T^{6} - 118422472 T^{7} + 22672155 p T^{8} - 1017854 p^{2} T^{9} + 155531 p^{3} T^{10} - 4852 p^{4} T^{11} + 601 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 4 T + 421 T^{2} - 1686 T^{3} + 75627 T^{4} - 318228 T^{5} + 8590831 T^{6} - 37310500 T^{7} + 8590831 p T^{8} - 318228 p^{2} T^{9} + 75627 p^{3} T^{10} - 1686 p^{4} T^{11} + 421 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−4.73008471518844336383240396338, −4.42742043288674521271924080101, −4.41585267168923823952323515345, −4.38777429506878055510947802380, −4.24220502466199952309004201093, −4.04460081770474790060966716943, −3.97925727722400436216140643810, −3.85468240556411564751216651775, −3.75874129122996296659290809427, −3.71277490400983914789673850677, −3.24203944256221060455529588386, −2.85210426903392303817413297017, −2.84088758892071107493637263760, −2.74365341982551060176249416356, −2.68356164098590714230111694343, −2.35480779727851790389510500162, −2.33830644632319030203966255852, −2.19975378651990435209091830791, −1.94082089896720299647658915095, −1.79783304811519867611520672411, −1.59209387417379272426075594376, −1.31810651374183008416775367719, −0.938818314941439700302789225650, −0.46960085043576933470813815231, −0.39350901278907161403994640454, 0.39350901278907161403994640454, 0.46960085043576933470813815231, 0.938818314941439700302789225650, 1.31810651374183008416775367719, 1.59209387417379272426075594376, 1.79783304811519867611520672411, 1.94082089896720299647658915095, 2.19975378651990435209091830791, 2.33830644632319030203966255852, 2.35480779727851790389510500162, 2.68356164098590714230111694343, 2.74365341982551060176249416356, 2.84088758892071107493637263760, 2.85210426903392303817413297017, 3.24203944256221060455529588386, 3.71277490400983914789673850677, 3.75874129122996296659290809427, 3.85468240556411564751216651775, 3.97925727722400436216140643810, 4.04460081770474790060966716943, 4.24220502466199952309004201093, 4.38777429506878055510947802380, 4.41585267168923823952323515345, 4.42742043288674521271924080101, 4.73008471518844336383240396338
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.