Dirichlet series
| L(s) = 1 | + 2-s − 4-s − 3·7-s − 3·9-s − 11-s + 3·13-s − 3·14-s − 2·16-s − 10·17-s − 3·18-s + 15·19-s − 22-s + 7·23-s + 3·26-s + 3·28-s + 3·29-s + 14·31-s − 10·32-s − 10·34-s + 3·36-s + 10·37-s + 15·38-s + 19·41-s − 5·43-s + 44-s + 7·46-s + 14·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.13·7-s − 9-s − 0.301·11-s + 0.832·13-s − 0.801·14-s − 1/2·16-s − 2.42·17-s − 0.707·18-s + 3.44·19-s − 0.213·22-s + 1.45·23-s + 0.588·26-s + 0.566·28-s + 0.557·29-s + 2.51·31-s − 1.76·32-s − 1.71·34-s + 1/2·36-s + 1.64·37-s + 2.43·38-s + 2.96·41-s − 0.762·43-s + 0.150·44-s + 1.03·46-s + 2.04·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{14} \cdot 23^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{14} \cdot 23^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(5^{14} \cdot 23^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(43014.4\) |
| Root analytic conductor: | \(2.14275\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 5^{14} \cdot 23^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.475510798\) |
| \(L(\frac12)\) | \(\approx\) | \(2.475510798\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 23 | \( ( 1 - T )^{7} \) | |
| good | 2 | \( 1 - T + p T^{2} - 3 T^{3} + 7 T^{4} - p T^{5} + 9 T^{6} - 9 T^{7} + 9 p T^{8} - p^{3} T^{9} + 7 p^{3} T^{10} - 3 p^{4} T^{11} + p^{6} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 + p T^{2} + 4 T^{4} + 8 T^{5} + 25 T^{6} + 28 T^{7} + 25 p T^{8} + 8 p^{2} T^{9} + 4 p^{3} T^{10} + p^{6} T^{12} + p^{7} T^{14} \) | |
| 7 | \( 1 + 3 T + 9 T^{2} + 2 p T^{3} + 45 T^{4} - 5 p T^{5} - 171 T^{6} - 1276 T^{7} - 171 p T^{8} - 5 p^{3} T^{9} + 45 p^{3} T^{10} + 2 p^{5} T^{11} + 9 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + T + 19 T^{2} - 18 T^{3} + 247 T^{4} - 153 T^{5} + 3637 T^{6} - 1148 T^{7} + 3637 p T^{8} - 153 p^{2} T^{9} + 247 p^{3} T^{10} - 18 p^{4} T^{11} + 19 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 3 T + 45 T^{2} - 158 T^{3} + 1198 T^{4} - 4246 T^{5} + 21399 T^{6} - 68537 T^{7} + 21399 p T^{8} - 4246 p^{2} T^{9} + 1198 p^{3} T^{10} - 158 p^{4} T^{11} + 45 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 10 T + 83 T^{2} + 508 T^{3} + 193 p T^{4} + 17046 T^{5} + 82619 T^{6} + 338120 T^{7} + 82619 p T^{8} + 17046 p^{2} T^{9} + 193 p^{4} T^{10} + 508 p^{4} T^{11} + 83 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 15 T + 163 T^{2} - 1158 T^{3} + 6863 T^{4} - 31817 T^{5} + 140765 T^{6} - 580340 T^{7} + 140765 p T^{8} - 31817 p^{2} T^{9} + 6863 p^{3} T^{10} - 1158 p^{4} T^{11} + 163 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 3 T + 81 T^{2} - 226 T^{3} + 2202 T^{4} - 6878 T^{5} + 11647 T^{6} - 168501 T^{7} + 11647 p T^{8} - 6878 p^{2} T^{9} + 2202 p^{3} T^{10} - 226 p^{4} T^{11} + 81 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 14 T + 231 T^{2} - 2080 T^{3} + 20508 T^{4} - 140424 T^{5} + 1028881 T^{6} - 5567026 T^{7} + 1028881 p T^{8} - 140424 p^{2} T^{9} + 20508 p^{3} T^{10} - 2080 p^{4} T^{11} + 231 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 10 T + 143 T^{2} - 1100 T^{3} + 8905 T^{4} - 53670 T^{5} + 340191 T^{6} - 1982056 T^{7} + 340191 p T^{8} - 53670 p^{2} T^{9} + 8905 p^{3} T^{10} - 1100 p^{4} T^{11} + 143 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 19 T + 309 T^{2} - 3190 T^{3} + 30654 T^{4} - 228554 T^{5} + 1692151 T^{6} - 10621429 T^{7} + 1692151 p T^{8} - 228554 p^{2} T^{9} + 30654 p^{3} T^{10} - 3190 p^{4} T^{11} + 309 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 5 T + 157 T^{2} + 826 T^{3} + 13117 T^{4} + 70451 T^{5} + 759257 T^{6} + 3813708 T^{7} + 759257 p T^{8} + 70451 p^{2} T^{9} + 13117 p^{3} T^{10} + 826 p^{4} T^{11} + 157 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 14 T + 311 T^{2} - 2944 T^{3} + 38680 T^{4} - 282800 T^{5} + 2790681 T^{6} - 16514822 T^{7} + 2790681 p T^{8} - 282800 p^{2} T^{9} + 38680 p^{3} T^{10} - 2944 p^{4} T^{11} + 311 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 4 T + 247 T^{2} + 1232 T^{3} + 28177 T^{4} + 159580 T^{5} + 2044527 T^{6} + 11188576 T^{7} + 2044527 p T^{8} + 159580 p^{2} T^{9} + 28177 p^{3} T^{10} + 1232 p^{4} T^{11} + 247 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 16 T + 322 T^{2} + 2460 T^{3} + 24083 T^{4} + 27416 T^{5} + 111413 T^{6} - 7169964 T^{7} + 111413 p T^{8} + 27416 p^{2} T^{9} + 24083 p^{3} T^{10} + 2460 p^{4} T^{11} + 322 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 40 T + 971 T^{2} - 17008 T^{3} + 237341 T^{4} - 2736312 T^{5} + 26811983 T^{6} - 225335392 T^{7} + 26811983 p T^{8} - 2736312 p^{2} T^{9} + 237341 p^{3} T^{10} - 17008 p^{4} T^{11} + 971 p^{5} T^{12} - 40 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 4 T + 245 T^{2} - 680 T^{3} + 28669 T^{4} - 85884 T^{5} + 2488785 T^{6} - 7855600 T^{7} + 2488785 p T^{8} - 85884 p^{2} T^{9} + 28669 p^{3} T^{10} - 680 p^{4} T^{11} + 245 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 14 T + 479 T^{2} + 4960 T^{3} + 1352 p T^{4} + 781136 T^{5} + 10923513 T^{6} + 70967270 T^{7} + 10923513 p T^{8} + 781136 p^{2} T^{9} + 1352 p^{4} T^{10} + 4960 p^{4} T^{11} + 479 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 3 T + 325 T^{2} - 658 T^{3} + 54034 T^{4} - 90182 T^{5} + 5763275 T^{6} - 7883729 T^{7} + 5763275 p T^{8} - 90182 p^{2} T^{9} + 54034 p^{3} T^{10} - 658 p^{4} T^{11} + 325 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + T + 321 T^{2} + 1538 T^{3} + 46557 T^{4} + 4737 p T^{5} + 4556893 T^{6} + 40938716 T^{7} + 4556893 p T^{8} + 4737 p^{3} T^{9} + 46557 p^{3} T^{10} + 1538 p^{4} T^{11} + 321 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 17 T + 541 T^{2} + 7850 T^{3} + 130949 T^{4} + 1547895 T^{5} + 17976145 T^{6} + 168233836 T^{7} + 17976145 p T^{8} + 1547895 p^{2} T^{9} + 130949 p^{3} T^{10} + 7850 p^{4} T^{11} + 541 p^{5} T^{12} + 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 16 T + 379 T^{2} - 4856 T^{3} + 79377 T^{4} - 825904 T^{5} + 10350019 T^{6} - 90162448 T^{7} + 10350019 p T^{8} - 825904 p^{2} T^{9} + 79377 p^{3} T^{10} - 4856 p^{4} T^{11} + 379 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 24 T + 683 T^{2} - 11736 T^{3} + 196873 T^{4} - 2578984 T^{5} + 31528651 T^{6} - 323282256 T^{7} + 31528651 p T^{8} - 2578984 p^{2} T^{9} + 196873 p^{3} T^{10} - 11736 p^{4} T^{11} + 683 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−5.12505070508469272458451546294, −5.07218841678203078397728099413, −4.88713994690620178771252990418, −4.72346799451179089600394555806, −4.48165522719323949150322040306, −4.36028730265763810091908506123, −4.26067854315737860964606266990, −4.08915542784928689963665117526, −3.86711200900401930215065076642, −3.81219646137762222784405398822, −3.72666843377489168954290458487, −3.37876735365718811996321996948, −3.01365546360221628032844896980, −2.97044616453819613150272902616, −2.88667435643997365809850131275, −2.86429866651778834166509848089, −2.48682347996296556621568818795, −2.48372054734298308501582278122, −2.15605379065299438089137493238, −2.00160818778932676829724142468, −1.48837419994344034417830819797, −1.02145549070198253562436699316, −0.996550874545041937901103527487, −0.944767757366656453830495277657, −0.28501340393134311584626698659, 0.28501340393134311584626698659, 0.944767757366656453830495277657, 0.996550874545041937901103527487, 1.02145549070198253562436699316, 1.48837419994344034417830819797, 2.00160818778932676829724142468, 2.15605379065299438089137493238, 2.48372054734298308501582278122, 2.48682347996296556621568818795, 2.86429866651778834166509848089, 2.88667435643997365809850131275, 2.97044616453819613150272902616, 3.01365546360221628032844896980, 3.37876735365718811996321996948, 3.72666843377489168954290458487, 3.81219646137762222784405398822, 3.86711200900401930215065076642, 4.08915542784928689963665117526, 4.26067854315737860964606266990, 4.36028730265763810091908506123, 4.48165522719323949150322040306, 4.72346799451179089600394555806, 4.88713994690620178771252990418, 5.07218841678203078397728099413, 5.12505070508469272458451546294
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.