Dirichlet series
| L(s) = 1 | + 2-s − 7·3-s − 4·4-s − 7·6-s + 11·7-s − 3·8-s + 28·9-s − 8·11-s + 28·12-s + 5·13-s + 11·14-s + 9·16-s + 10·17-s + 28·18-s + 7·19-s − 77·21-s − 8·22-s + 4·23-s + 21·24-s + 5·26-s − 84·27-s − 44·28-s − 11·29-s + 3·31-s + 7·32-s + 56·33-s + 10·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 4.04·3-s − 2·4-s − 2.85·6-s + 4.15·7-s − 1.06·8-s + 28/3·9-s − 2.41·11-s + 8.08·12-s + 1.38·13-s + 2.93·14-s + 9/4·16-s + 2.42·17-s + 6.59·18-s + 1.60·19-s − 16.8·21-s − 1.70·22-s + 0.834·23-s + 4.28·24-s + 0.980·26-s − 16.1·27-s − 8.31·28-s − 2.04·29-s + 0.538·31-s + 1.23·32-s + 9.74·33-s + 1.71·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 5^{14} \cdot 47^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 5^{14} \cdot 47^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{7} \cdot 5^{14} \cdot 47^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.39976\times 10^{10}\) |
| Root analytic conductor: | \(5.30539\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 3^{7} \cdot 5^{14} \cdot 47^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(5.737611320\) |
| \(L(\frac12)\) | \(\approx\) | \(5.737611320\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T )^{7} \) |
| 5 | \( 1 \) | |
| 47 | \( ( 1 + T )^{7} \) | |
| good | 2 | \( 1 - T + 5 T^{2} - 3 p T^{3} + 7 p T^{4} - 21 T^{5} + 7 p^{2} T^{6} - 49 T^{7} + 7 p^{3} T^{8} - 21 p^{2} T^{9} + 7 p^{4} T^{10} - 3 p^{5} T^{11} + 5 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) |
| 7 | \( 1 - 11 T + 78 T^{2} - 417 T^{3} + 1843 T^{4} - 6856 T^{5} + 22200 T^{6} - 62641 T^{7} + 22200 p T^{8} - 6856 p^{2} T^{9} + 1843 p^{3} T^{10} - 417 p^{4} T^{11} + 78 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 8 T + 74 T^{2} + 376 T^{3} + 2011 T^{4} + 7616 T^{5} + 30931 T^{6} + 97861 T^{7} + 30931 p T^{8} + 7616 p^{2} T^{9} + 2011 p^{3} T^{10} + 376 p^{4} T^{11} + 74 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 5 T + 4 p T^{2} - 251 T^{3} + 1524 T^{4} - 6527 T^{5} + 28941 T^{6} - 104051 T^{7} + 28941 p T^{8} - 6527 p^{2} T^{9} + 1524 p^{3} T^{10} - 251 p^{4} T^{11} + 4 p^{6} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 10 T + 89 T^{2} - 433 T^{3} + 2247 T^{4} - 8066 T^{5} + 39931 T^{6} - 139797 T^{7} + 39931 p T^{8} - 8066 p^{2} T^{9} + 2247 p^{3} T^{10} - 433 p^{4} T^{11} + 89 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 7 T + 5 p T^{2} - 375 T^{3} + 3003 T^{4} - 5634 T^{5} + 48423 T^{6} - 39383 T^{7} + 48423 p T^{8} - 5634 p^{2} T^{9} + 3003 p^{3} T^{10} - 375 p^{4} T^{11} + 5 p^{6} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 4 T + 84 T^{2} - 177 T^{3} + 2263 T^{4} + 1195 T^{5} + 887 p T^{6} + 145021 T^{7} + 887 p^{2} T^{8} + 1195 p^{2} T^{9} + 2263 p^{3} T^{10} - 177 p^{4} T^{11} + 84 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 11 T + 177 T^{2} + 1442 T^{3} + 13709 T^{4} + 86785 T^{5} + 616247 T^{6} + 3145469 T^{7} + 616247 p T^{8} + 86785 p^{2} T^{9} + 13709 p^{3} T^{10} + 1442 p^{4} T^{11} + 177 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 3 T + 98 T^{2} - 64 T^{3} + 4035 T^{4} + 9032 T^{5} + 101121 T^{6} + 17655 p T^{7} + 101121 p T^{8} + 9032 p^{2} T^{9} + 4035 p^{3} T^{10} - 64 p^{4} T^{11} + 98 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 11 T + 219 T^{2} - 1926 T^{3} + 21958 T^{4} - 157015 T^{5} + 1288533 T^{6} - 7430041 T^{7} + 1288533 p T^{8} - 157015 p^{2} T^{9} + 21958 p^{3} T^{10} - 1926 p^{4} T^{11} + 219 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 20 T + 308 T^{2} + 2913 T^{3} + 24059 T^{4} + 144612 T^{5} + 21950 p T^{6} + 4956839 T^{7} + 21950 p^{2} T^{8} + 144612 p^{2} T^{9} + 24059 p^{3} T^{10} + 2913 p^{4} T^{11} + 308 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 18 T + 247 T^{2} - 2592 T^{3} + 23786 T^{4} - 187998 T^{5} + 1405468 T^{6} - 9447185 T^{7} + 1405468 p T^{8} - 187998 p^{2} T^{9} + 23786 p^{3} T^{10} - 2592 p^{4} T^{11} + 247 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 12 T + 231 T^{2} - 1933 T^{3} + 23051 T^{4} - 157156 T^{5} + 1502211 T^{6} - 9081685 T^{7} + 1502211 p T^{8} - 157156 p^{2} T^{9} + 23051 p^{3} T^{10} - 1933 p^{4} T^{11} + 231 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 18 T + 352 T^{2} - 4385 T^{3} + 55815 T^{4} - 542525 T^{5} + 5186115 T^{6} - 40100349 T^{7} + 5186115 p T^{8} - 542525 p^{2} T^{9} + 55815 p^{3} T^{10} - 4385 p^{4} T^{11} + 352 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 4 T + 65 T^{2} + 50 T^{3} - 76 T^{4} - 24518 T^{5} + 27180 T^{6} - 2030737 T^{7} + 27180 p T^{8} - 24518 p^{2} T^{9} - 76 p^{3} T^{10} + 50 p^{4} T^{11} + 65 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 22 T + 449 T^{2} - 5357 T^{3} + 62404 T^{4} - 517179 T^{5} + 4739326 T^{6} - 34712951 T^{7} + 4739326 p T^{8} - 517179 p^{2} T^{9} + 62404 p^{3} T^{10} - 5357 p^{4} T^{11} + 449 p^{5} T^{12} - 22 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 14 T + 262 T^{2} + 2175 T^{3} + 24599 T^{4} + 195343 T^{5} + 2045093 T^{6} + 16804621 T^{7} + 2045093 p T^{8} + 195343 p^{2} T^{9} + 24599 p^{3} T^{10} + 2175 p^{4} T^{11} + 262 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 30 T + 597 T^{2} - 8032 T^{3} + 87198 T^{4} - 748197 T^{5} + 5982949 T^{6} - 47076885 T^{7} + 5982949 p T^{8} - 748197 p^{2} T^{9} + 87198 p^{3} T^{10} - 8032 p^{4} T^{11} + 597 p^{5} T^{12} - 30 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + T + 302 T^{2} + 183 T^{3} + 48716 T^{4} + 16072 T^{5} + 5351120 T^{6} + 1298973 T^{7} + 5351120 p T^{8} + 16072 p^{2} T^{9} + 48716 p^{3} T^{10} + 183 p^{4} T^{11} + 302 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 54 T + 1710 T^{2} - 38182 T^{3} + 664206 T^{4} - 9366003 T^{5} + 110077016 T^{6} - 1089541311 T^{7} + 110077016 p T^{8} - 9366003 p^{2} T^{9} + 664206 p^{3} T^{10} - 38182 p^{4} T^{11} + 1710 p^{5} T^{12} - 54 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 14 T + 531 T^{2} + 5863 T^{3} + 127228 T^{4} + 1138413 T^{5} + 17879114 T^{6} + 129373675 T^{7} + 17879114 p T^{8} + 1138413 p^{2} T^{9} + 127228 p^{3} T^{10} + 5863 p^{4} T^{11} + 531 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 24 T + 359 T^{2} - 4229 T^{3} + 21354 T^{4} + 87651 T^{5} - 3644948 T^{6} + 53341371 T^{7} - 3644948 p T^{8} + 87651 p^{2} T^{9} + 21354 p^{3} T^{10} - 4229 p^{4} T^{11} + 359 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−4.01748458811797478022949985160, −3.99282263586458520926766584178, −3.89851835604653948866413068559, −3.76977951184430136527162231076, −3.51058559966936228698681372082, −3.30953378889317122394385057218, −3.30862211487190631109574683427, −3.21697805021018013450589912455, −3.01363933175262424474553220735, −2.66587614747222675829342006208, −2.56174655159250141592739258145, −2.41034132936887952999522160601, −2.10575739678551597017643424208, −1.93724195079232779647912257076, −1.91716083683401017232295861719, −1.85539381528345775172404851013, −1.58248822444760307823797921346, −1.40696685453284279978639312081, −1.21886947335270998255421917943, −1.03500544979135913362926746855, −0.926499068439030938976291892448, −0.70592320993948094705608667879, −0.67275962926564992869686514572, −0.60952903061554276227710860370, −0.28415652470090668336756602724, 0.28415652470090668336756602724, 0.60952903061554276227710860370, 0.67275962926564992869686514572, 0.70592320993948094705608667879, 0.926499068439030938976291892448, 1.03500544979135913362926746855, 1.21886947335270998255421917943, 1.40696685453284279978639312081, 1.58248822444760307823797921346, 1.85539381528345775172404851013, 1.91716083683401017232295861719, 1.93724195079232779647912257076, 2.10575739678551597017643424208, 2.41034132936887952999522160601, 2.56174655159250141592739258145, 2.66587614747222675829342006208, 3.01363933175262424474553220735, 3.21697805021018013450589912455, 3.30862211487190631109574683427, 3.30953378889317122394385057218, 3.51058559966936228698681372082, 3.76977951184430136527162231076, 3.89851835604653948866413068559, 3.99282263586458520926766584178, 4.01748458811797478022949985160
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.