Dirichlet series
| L(s) = 1 | + 2·2-s + 2·4-s + 5·7-s + 4·8-s − 12·11-s + 2·13-s + 10·14-s + 5·16-s + 11·17-s + 16·19-s − 24·22-s + 7·23-s + 4·26-s + 10·28-s + 29-s + 5·31-s + 22·34-s + 17·37-s + 32·38-s − 19·41-s + 14·43-s − 24·44-s + 14·46-s − 6·47-s + 3·49-s + 4·52-s − 15·53-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.88·7-s + 1.41·8-s − 3.61·11-s + 0.554·13-s + 2.67·14-s + 5/4·16-s + 2.66·17-s + 3.67·19-s − 5.11·22-s + 1.45·23-s + 0.784·26-s + 1.88·28-s + 0.185·29-s + 0.898·31-s + 3.77·34-s + 2.79·37-s + 5.19·38-s − 2.96·41-s + 2.13·43-s − 3.61·44-s + 2.06·46-s − 0.875·47-s + 3/7·49-s + 0.554·52-s − 2.06·53-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 5^{14} \cdot 23^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 5^{14} \cdot 23^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{14} \cdot 5^{14} \cdot 23^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.05736\times 10^{11}\) |
| Root analytic conductor: | \(6.42826\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 3^{14} \cdot 5^{14} \cdot 23^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(43.58523449\) |
| \(L(\frac12)\) | \(\approx\) | \(43.58523449\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) | |
| 23 | \( ( 1 - T )^{7} \) | |
| good | 2 | \( 1 - p T + p T^{2} - p^{2} T^{3} + 7 T^{4} - p^{2} T^{5} + p T^{6} - 3 p T^{7} + p^{2} T^{8} - p^{4} T^{9} + 7 p^{3} T^{10} - p^{6} T^{11} + p^{6} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) |
| 7 | \( 1 - 5 T + 22 T^{2} - 69 T^{3} + 242 T^{4} - 795 T^{5} + 2389 T^{6} - 6782 T^{7} + 2389 p T^{8} - 795 p^{2} T^{9} + 242 p^{3} T^{10} - 69 p^{4} T^{11} + 22 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 12 T + 85 T^{2} + 454 T^{3} + 2117 T^{4} + 8636 T^{5} + 31593 T^{6} + 106948 T^{7} + 31593 p T^{8} + 8636 p^{2} T^{9} + 2117 p^{3} T^{10} + 454 p^{4} T^{11} + 85 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 2 T + 5 p T^{2} - 138 T^{3} + 2003 T^{4} - 4230 T^{5} + 38379 T^{6} - 71996 T^{7} + 38379 p T^{8} - 4230 p^{2} T^{9} + 2003 p^{3} T^{10} - 138 p^{4} T^{11} + 5 p^{6} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 11 T + 118 T^{2} - 847 T^{3} + 5650 T^{4} - 1797 p T^{5} + 154799 T^{6} - 657626 T^{7} + 154799 p T^{8} - 1797 p^{3} T^{9} + 5650 p^{3} T^{10} - 847 p^{4} T^{11} + 118 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 16 T + 215 T^{2} - 1878 T^{3} + 14471 T^{4} - 87496 T^{5} + 480449 T^{6} - 2186420 T^{7} + 480449 p T^{8} - 87496 p^{2} T^{9} + 14471 p^{3} T^{10} - 1878 p^{4} T^{11} + 215 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - T + 128 T^{2} - 245 T^{3} + 8366 T^{4} - 17091 T^{5} + 357125 T^{6} - 637966 T^{7} + 357125 p T^{8} - 17091 p^{2} T^{9} + 8366 p^{3} T^{10} - 245 p^{4} T^{11} + 128 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 5 T + 136 T^{2} - 741 T^{3} + 9446 T^{4} - 47311 T^{5} + 432575 T^{6} - 1806862 T^{7} + 432575 p T^{8} - 47311 p^{2} T^{9} + 9446 p^{3} T^{10} - 741 p^{4} T^{11} + 136 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 17 T + 326 T^{2} - 3777 T^{3} + 40884 T^{4} - 349491 T^{5} + 2661449 T^{6} - 17243126 T^{7} + 2661449 p T^{8} - 349491 p^{2} T^{9} + 40884 p^{3} T^{10} - 3777 p^{4} T^{11} + 326 p^{5} T^{12} - 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 19 T + 394 T^{2} + 4701 T^{3} + 55916 T^{4} + 481409 T^{5} + 4052625 T^{6} + 26296414 T^{7} + 4052625 p T^{8} + 481409 p^{2} T^{9} + 55916 p^{3} T^{10} + 4701 p^{4} T^{11} + 394 p^{5} T^{12} + 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 14 T + 225 T^{2} - 2344 T^{3} + 23113 T^{4} - 197474 T^{5} + 1484753 T^{6} - 10496528 T^{7} + 1484753 p T^{8} - 197474 p^{2} T^{9} + 23113 p^{3} T^{10} - 2344 p^{4} T^{11} + 225 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 6 T + 113 T^{2} + 766 T^{3} + 7517 T^{4} + 50706 T^{5} + 433805 T^{6} + 2517332 T^{7} + 433805 p T^{8} + 50706 p^{2} T^{9} + 7517 p^{3} T^{10} + 766 p^{4} T^{11} + 113 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 15 T + 208 T^{2} + 1457 T^{3} + 14012 T^{4} + 92953 T^{5} + 981327 T^{6} + 6221462 T^{7} + 981327 p T^{8} + 92953 p^{2} T^{9} + 14012 p^{3} T^{10} + 1457 p^{4} T^{11} + 208 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 11 T + 230 T^{2} + 1599 T^{3} + 18416 T^{4} + 66169 T^{5} + 690007 T^{6} + 1175722 T^{7} + 690007 p T^{8} + 66169 p^{2} T^{9} + 18416 p^{3} T^{10} + 1599 p^{4} T^{11} + 230 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 8 T + 145 T^{2} - 942 T^{3} + 15371 T^{4} - 104880 T^{5} + 1138851 T^{6} - 6201236 T^{7} + 1138851 p T^{8} - 104880 p^{2} T^{9} + 15371 p^{3} T^{10} - 942 p^{4} T^{11} + 145 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 3 T + 118 T^{2} + 155 T^{3} + 8162 T^{4} + 19373 T^{5} + 678793 T^{6} + 2355010 T^{7} + 678793 p T^{8} + 19373 p^{2} T^{9} + 8162 p^{3} T^{10} + 155 p^{4} T^{11} + 118 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - T + 232 T^{2} + 193 T^{3} + 30842 T^{4} + 36365 T^{5} + 3021963 T^{6} + 3211222 T^{7} + 3021963 p T^{8} + 36365 p^{2} T^{9} + 30842 p^{3} T^{10} + 193 p^{4} T^{11} + 232 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 18 T + 493 T^{2} - 6482 T^{3} + 103019 T^{4} - 1050182 T^{5} + 12177991 T^{6} - 98243980 T^{7} + 12177991 p T^{8} - 1050182 p^{2} T^{9} + 103019 p^{3} T^{10} - 6482 p^{4} T^{11} + 493 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 2 T + 265 T^{2} - 12 T^{3} + 29517 T^{4} - 114626 T^{5} + 2111557 T^{6} - 15982568 T^{7} + 2111557 p T^{8} - 114626 p^{2} T^{9} + 29517 p^{3} T^{10} - 12 p^{4} T^{11} + 265 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + T + 346 T^{2} + 1005 T^{3} + 59632 T^{4} + 224715 T^{5} + 6904107 T^{6} + 24596814 T^{7} + 6904107 p T^{8} + 224715 p^{2} T^{9} + 59632 p^{3} T^{10} + 1005 p^{4} T^{11} + 346 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 16 T + 455 T^{2} + 5800 T^{3} + 94517 T^{4} + 1025952 T^{5} + 12462835 T^{6} + 113567824 T^{7} + 12462835 p T^{8} + 1025952 p^{2} T^{9} + 94517 p^{3} T^{10} + 5800 p^{4} T^{11} + 455 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 26 T + 727 T^{2} - 12432 T^{3} + 210405 T^{4} - 2712758 T^{5} + 33692675 T^{6} - 338833216 T^{7} + 33692675 p T^{8} - 2712758 p^{2} T^{9} + 210405 p^{3} T^{10} - 12432 p^{4} T^{11} + 727 p^{5} T^{12} - 26 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.67622770805796589384485524624, −3.56469555456623493417008919666, −3.45716728962821934041412097839, −3.28995321715952124922929882832, −3.23262147138743849076422661964, −3.16699766154011162100316583276, −2.99330207757845276578569083605, −2.96362940323382753446566629888, −2.85971139080819872327909040418, −2.58322594850782075950632283470, −2.44652476204408738712624434938, −2.38344957877817988169252468426, −2.26870317908304011734157154213, −2.25880784801975331959579619232, −1.74875604923383414949773294600, −1.69201810164252663486875508493, −1.59181251169268922075121550421, −1.58807710115916272239700248441, −1.38393619385922293234686809489, −1.12755690715627160119586567899, −0.970801217109611142879721505640, −0.73997415199205457410465997546, −0.72491032411176141644869221386, −0.62605450261852490887545447240, −0.20894264582800039865576681169, 0.20894264582800039865576681169, 0.62605450261852490887545447240, 0.72491032411176141644869221386, 0.73997415199205457410465997546, 0.970801217109611142879721505640, 1.12755690715627160119586567899, 1.38393619385922293234686809489, 1.58807710115916272239700248441, 1.59181251169268922075121550421, 1.69201810164252663486875508493, 1.74875604923383414949773294600, 2.25880784801975331959579619232, 2.26870317908304011734157154213, 2.38344957877817988169252468426, 2.44652476204408738712624434938, 2.58322594850782075950632283470, 2.85971139080819872327909040418, 2.96362940323382753446566629888, 2.99330207757845276578569083605, 3.16699766154011162100316583276, 3.23262147138743849076422661964, 3.28995321715952124922929882832, 3.45716728962821934041412097839, 3.56469555456623493417008919666, 3.67622770805796589384485524624
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.