Dirichlet series
| L(s) = 1 | + 2·3-s − 6·4-s − 7·7-s + 2·8-s − 8·9-s − 6·11-s − 12·12-s + 7·13-s + 16·16-s − 2·17-s − 10·19-s − 14·21-s − 10·23-s + 4·24-s − 18·27-s + 42·28-s − 26·29-s − 6·31-s − 13·32-s − 12·33-s + 48·36-s + 4·37-s + 14·39-s − 28·41-s − 2·43-s + 36·44-s + 20·47-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3·4-s − 2.64·7-s + 0.707·8-s − 8/3·9-s − 1.80·11-s − 3.46·12-s + 1.94·13-s + 4·16-s − 0.485·17-s − 2.29·19-s − 3.05·21-s − 2.08·23-s + 0.816·24-s − 3.46·27-s + 7.93·28-s − 4.82·29-s − 1.07·31-s − 2.29·32-s − 2.08·33-s + 8·36-s + 0.657·37-s + 2.24·39-s − 4.37·41-s − 0.304·43-s + 5.42·44-s + 2.91·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{14} \cdot 7^{7} \cdot 13^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{14} \cdot 7^{7} \cdot 13^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(5^{14} \cdot 7^{7} \cdot 13^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(6.52844\times 10^{8}\) |
| Root analytic conductor: | \(4.26215\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 5^{14} \cdot 7^{7} \cdot 13^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 7 | \( ( 1 + T )^{7} \) | |
| 13 | \( ( 1 - T )^{7} \) | |
| good | 2 | \( 1 + 3 p T^{2} - p T^{3} + 5 p^{2} T^{4} - 11 T^{5} + 25 p T^{6} - 15 p T^{7} + 25 p^{2} T^{8} - 11 p^{2} T^{9} + 5 p^{5} T^{10} - p^{5} T^{11} + 3 p^{6} T^{12} + p^{7} T^{14} \) |
| 3 | \( 1 - 2 T + 4 p T^{2} - 22 T^{3} + p^{4} T^{4} - 128 T^{5} + 352 T^{6} - 472 T^{7} + 352 p T^{8} - 128 p^{2} T^{9} + p^{7} T^{10} - 22 p^{4} T^{11} + 4 p^{6} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 6 T + 63 T^{2} + 300 T^{3} + 1843 T^{4} + 7082 T^{5} + 31925 T^{6} + 98664 T^{7} + 31925 p T^{8} + 7082 p^{2} T^{9} + 1843 p^{3} T^{10} + 300 p^{4} T^{11} + 63 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 2 T + 72 T^{2} + 78 T^{3} + 2241 T^{4} + 10 p T^{5} + 44628 T^{6} - 19656 T^{7} + 44628 p T^{8} + 10 p^{3} T^{9} + 2241 p^{3} T^{10} + 78 p^{4} T^{11} + 72 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 10 T + 101 T^{2} + 518 T^{3} + 3057 T^{4} + 10822 T^{5} + 57810 T^{6} + 190202 T^{7} + 57810 p T^{8} + 10822 p^{2} T^{9} + 3057 p^{3} T^{10} + 518 p^{4} T^{11} + 101 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 10 T + 90 T^{2} + 502 T^{3} + 3052 T^{4} + 14822 T^{5} + 88283 T^{6} + 406868 T^{7} + 88283 p T^{8} + 14822 p^{2} T^{9} + 3052 p^{3} T^{10} + 502 p^{4} T^{11} + 90 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 26 T + 455 T^{2} + 5618 T^{3} + 56049 T^{4} + 454862 T^{5} + 3132276 T^{6} + 18201050 T^{7} + 3132276 p T^{8} + 454862 p^{2} T^{9} + 56049 p^{3} T^{10} + 5618 p^{4} T^{11} + 455 p^{5} T^{12} + 26 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 6 T + 117 T^{2} + 726 T^{3} + 239 p T^{4} + 46042 T^{5} + 314214 T^{6} + 1772826 T^{7} + 314214 p T^{8} + 46042 p^{2} T^{9} + 239 p^{4} T^{10} + 726 p^{4} T^{11} + 117 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 4 T + 134 T^{2} - 768 T^{3} + 7 p^{2} T^{4} - 63592 T^{5} + 468016 T^{6} - 3037916 T^{7} + 468016 p T^{8} - 63592 p^{2} T^{9} + 7 p^{5} T^{10} - 768 p^{4} T^{11} + 134 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 28 T + 480 T^{2} + 5700 T^{3} + 54095 T^{4} + 424348 T^{5} + 2987124 T^{6} + 19446504 T^{7} + 2987124 p T^{8} + 424348 p^{2} T^{9} + 54095 p^{3} T^{10} + 5700 p^{4} T^{11} + 480 p^{5} T^{12} + 28 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 2 T + 150 T^{2} + 294 T^{3} + 11376 T^{4} + 28094 T^{5} + 628263 T^{6} + 1624468 T^{7} + 628263 p T^{8} + 28094 p^{2} T^{9} + 11376 p^{3} T^{10} + 294 p^{4} T^{11} + 150 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 20 T + 338 T^{2} - 3912 T^{3} + 39832 T^{4} - 334060 T^{5} + 2614727 T^{6} - 18273392 T^{7} + 2614727 p T^{8} - 334060 p^{2} T^{9} + 39832 p^{3} T^{10} - 3912 p^{4} T^{11} + 338 p^{5} T^{12} - 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 8 T + 232 T^{2} + 1144 T^{3} + 24346 T^{4} + 93288 T^{5} + 1802985 T^{6} + 5939120 T^{7} + 1802985 p T^{8} + 93288 p^{2} T^{9} + 24346 p^{3} T^{10} + 1144 p^{4} T^{11} + 232 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 28 T + 516 T^{2} + 6072 T^{3} + 55067 T^{4} + 350154 T^{5} + 1924156 T^{6} + 10600980 T^{7} + 1924156 p T^{8} + 350154 p^{2} T^{9} + 55067 p^{3} T^{10} + 6072 p^{4} T^{11} + 516 p^{5} T^{12} + 28 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 26 T + 451 T^{2} + 6028 T^{3} + 71957 T^{4} + 740598 T^{5} + 6799999 T^{6} + 55327848 T^{7} + 6799999 p T^{8} + 740598 p^{2} T^{9} + 71957 p^{3} T^{10} + 6028 p^{4} T^{11} + 451 p^{5} T^{12} + 26 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 2 T + 80 T^{2} - 146 T^{3} + 8639 T^{4} - 26982 T^{5} + 602840 T^{6} - 1723284 T^{7} + 602840 p T^{8} - 26982 p^{2} T^{9} + 8639 p^{3} T^{10} - 146 p^{4} T^{11} + 80 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 14 T + 215 T^{2} + 1316 T^{3} + 113 p T^{4} - 36270 T^{5} - 391231 T^{6} - 7602120 T^{7} - 391231 p T^{8} - 36270 p^{2} T^{9} + 113 p^{4} T^{10} + 1316 p^{4} T^{11} + 215 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 2 T + 288 T^{2} + 418 T^{3} + 36678 T^{4} + 2306 p T^{5} + 3137729 T^{6} + 18581052 T^{7} + 3137729 p T^{8} + 2306 p^{3} T^{9} + 36678 p^{3} T^{10} + 418 p^{4} T^{11} + 288 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 14 T + 291 T^{2} - 4506 T^{3} + 51665 T^{4} - 616944 T^{5} + 6389544 T^{6} - 55474464 T^{7} + 6389544 p T^{8} - 616944 p^{2} T^{9} + 51665 p^{3} T^{10} - 4506 p^{4} T^{11} + 291 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 28 T + 390 T^{2} - 2320 T^{3} + 2012 T^{4} + 1324 T^{5} + 2079019 T^{6} - 29365376 T^{7} + 2079019 p T^{8} + 1324 p^{2} T^{9} + 2012 p^{3} T^{10} - 2320 p^{4} T^{11} + 390 p^{5} T^{12} - 28 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 30 T + 649 T^{2} + 10378 T^{3} + 142773 T^{4} + 1694580 T^{5} + 18292930 T^{6} + 179557292 T^{7} + 18292930 p T^{8} + 1694580 p^{2} T^{9} + 142773 p^{3} T^{10} + 10378 p^{4} T^{11} + 649 p^{5} T^{12} + 30 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 18 T + 500 T^{2} - 7622 T^{3} + 125566 T^{4} - 1539206 T^{5} + 19086125 T^{6} - 186892100 T^{7} + 19086125 p T^{8} - 1539206 p^{2} T^{9} + 125566 p^{3} T^{10} - 7622 p^{4} T^{11} + 500 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−4.58237862371352855916400724246, −4.51006909424309762983777832583, −4.27957852245877597477580014738, −4.13161677951033351868160000415, −3.89091866453475024972536871481, −3.85805955930307854124418912686, −3.75112989860465817622184587031, −3.75010091118653692364447715788, −3.67955713389459524802223365869, −3.55408255664486375514320732909, −3.48821226968291707291438708172, −3.11096523278524776425243204055, −3.00809422413114179607591884750, −2.85486947443942860797707756146, −2.78863101453598738858342471141, −2.73488244601827373642796235847, −2.49535443681621444533367883576, −2.29136702245211981739292986478, −2.23749905434210055217015271629, −2.01209518576945037197250983863, −1.76828029353944311013836292500, −1.68140029993492590726987097299, −1.29550729632649451064626672753, −1.25294580133689516146825860771, −1.14231956884557865479670071912, 0, 0, 0, 0, 0, 0, 0, 1.14231956884557865479670071912, 1.25294580133689516146825860771, 1.29550729632649451064626672753, 1.68140029993492590726987097299, 1.76828029353944311013836292500, 2.01209518576945037197250983863, 2.23749905434210055217015271629, 2.29136702245211981739292986478, 2.49535443681621444533367883576, 2.73488244601827373642796235847, 2.78863101453598738858342471141, 2.85486947443942860797707756146, 3.00809422413114179607591884750, 3.11096523278524776425243204055, 3.48821226968291707291438708172, 3.55408255664486375514320732909, 3.67955713389459524802223365869, 3.75010091118653692364447715788, 3.75112989860465817622184587031, 3.85805955930307854124418912686, 3.89091866453475024972536871481, 4.13161677951033351868160000415, 4.27957852245877597477580014738, 4.51006909424309762983777832583, 4.58237862371352855916400724246
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.