Dirichlet series
| L(s) = 1 | + 7·2-s + 2·3-s + 28·4-s + 2·5-s + 14·6-s + 7-s + 84·8-s − 9-s + 14·10-s + 5·11-s + 56·12-s + 7·14-s + 4·15-s + 210·16-s + 16·17-s − 7·18-s + 7·19-s + 56·20-s + 2·21-s + 35·22-s + 3·23-s + 168·24-s − 12·25-s − 5·27-s + 28·28-s − 7·29-s + 28·30-s + ⋯ |
| L(s) = 1 | + 4.94·2-s + 1.15·3-s + 14·4-s + 0.894·5-s + 5.71·6-s + 0.377·7-s + 29.6·8-s − 1/3·9-s + 4.42·10-s + 1.50·11-s + 16.1·12-s + 1.87·14-s + 1.03·15-s + 52.5·16-s + 3.88·17-s − 1.64·18-s + 1.60·19-s + 12.5·20-s + 0.436·21-s + 7.46·22-s + 0.625·23-s + 34.2·24-s − 2.39·25-s − 0.962·27-s + 5.29·28-s − 1.29·29-s + 5.11·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 13^{14} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 13^{14} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{7} \cdot 13^{14} \cdot 19^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.32463\times 10^{11}\) |
| Root analytic conductor: | \(7.16100\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{7} \cdot 13^{14} \cdot 19^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(4683.497406\) |
| \(L(\frac12)\) | \(\approx\) | \(4683.497406\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{7} \) |
| 13 | \( 1 \) | |
| 19 | \( ( 1 - T )^{7} \) | |
| good | 3 | \( 1 - 2 T + 5 T^{2} - 7 T^{3} + p^{2} T^{4} - T^{5} - 2 T^{6} + 10 T^{7} - 2 p T^{8} - p^{2} T^{9} + p^{5} T^{10} - 7 p^{4} T^{11} + 5 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) |
| 5 | \( 1 - 2 T + 16 T^{2} - 31 T^{3} + 127 T^{4} - 48 p T^{5} + 764 T^{6} - 1334 T^{7} + 764 p T^{8} - 48 p^{3} T^{9} + 127 p^{3} T^{10} - 31 p^{4} T^{11} + 16 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 - T + 18 T^{2} - 11 T^{3} + 164 T^{4} - 167 T^{5} + 213 p T^{6} - 1930 T^{7} + 213 p^{2} T^{8} - 167 p^{2} T^{9} + 164 p^{3} T^{10} - 11 p^{4} T^{11} + 18 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 5 T + 43 T^{2} - 15 p T^{3} + 989 T^{4} - 3235 T^{5} + 14907 T^{6} - 41222 T^{7} + 14907 p T^{8} - 3235 p^{2} T^{9} + 989 p^{3} T^{10} - 15 p^{5} T^{11} + 43 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 16 T + 161 T^{2} - 1237 T^{3} + 8205 T^{4} - 46589 T^{5} + 230680 T^{6} - 1003634 T^{7} + 230680 p T^{8} - 46589 p^{2} T^{9} + 8205 p^{3} T^{10} - 1237 p^{4} T^{11} + 161 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 3 T + 74 T^{2} - 250 T^{3} + 124 p T^{4} - 12500 T^{5} + 79575 T^{6} - 368558 T^{7} + 79575 p T^{8} - 12500 p^{2} T^{9} + 124 p^{4} T^{10} - 250 p^{4} T^{11} + 74 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 7 T + 130 T^{2} + 646 T^{3} + 7556 T^{4} + 28802 T^{5} + 284695 T^{6} + 924038 T^{7} + 284695 p T^{8} + 28802 p^{2} T^{9} + 7556 p^{3} T^{10} + 646 p^{4} T^{11} + 130 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 11 T + 129 T^{2} - 831 T^{3} + 6693 T^{4} - 34921 T^{5} + 229733 T^{6} - 1050266 T^{7} + 229733 p T^{8} - 34921 p^{2} T^{9} + 6693 p^{3} T^{10} - 831 p^{4} T^{11} + 129 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 3 T + 125 T^{2} + 678 T^{3} + 8895 T^{4} + 52085 T^{5} + 470803 T^{6} + 2338196 T^{7} + 470803 p T^{8} + 52085 p^{2} T^{9} + 8895 p^{3} T^{10} + 678 p^{4} T^{11} + 125 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 15 T + 281 T^{2} - 3087 T^{3} + 33455 T^{4} - 284081 T^{5} + 2234519 T^{6} - 14979618 T^{7} + 2234519 p T^{8} - 284081 p^{2} T^{9} + 33455 p^{3} T^{10} - 3087 p^{4} T^{11} + 281 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 23 T + 395 T^{2} - 4698 T^{3} + 47751 T^{4} - 398081 T^{5} + 3044749 T^{6} - 20515756 T^{7} + 3044749 p T^{8} - 398081 p^{2} T^{9} + 47751 p^{3} T^{10} - 4698 p^{4} T^{11} + 395 p^{5} T^{12} - 23 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + T + 135 T^{2} - 302 T^{3} + 9127 T^{4} - 46801 T^{5} + 472929 T^{6} - 2965380 T^{7} + 472929 p T^{8} - 46801 p^{2} T^{9} + 9127 p^{3} T^{10} - 302 p^{4} T^{11} + 135 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 21 T + 362 T^{2} - 4604 T^{3} + 51046 T^{4} - 483304 T^{5} + 4128631 T^{6} - 31517530 T^{7} + 4128631 p T^{8} - 483304 p^{2} T^{9} + 51046 p^{3} T^{10} - 4604 p^{4} T^{11} + 362 p^{5} T^{12} - 21 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 8 T + 130 T^{2} - 1410 T^{3} + 18752 T^{4} - 123088 T^{5} + 1392795 T^{6} - 10270892 T^{7} + 1392795 p T^{8} - 123088 p^{2} T^{9} + 18752 p^{3} T^{10} - 1410 p^{4} T^{11} + 130 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 6 T + 251 T^{2} + 1852 T^{3} + 30109 T^{4} + 261034 T^{5} + 2389407 T^{6} + 20729096 T^{7} + 2389407 p T^{8} + 261034 p^{2} T^{9} + 30109 p^{3} T^{10} + 1852 p^{4} T^{11} + 251 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 28 T + 654 T^{2} - 10010 T^{3} + 139320 T^{4} - 1535116 T^{5} + 15747535 T^{6} - 133527196 T^{7} + 15747535 p T^{8} - 1535116 p^{2} T^{9} + 139320 p^{3} T^{10} - 10010 p^{4} T^{11} + 654 p^{5} T^{12} - 28 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 4 T + 292 T^{2} + 391 T^{3} + 27987 T^{4} + 279772 T^{5} + 1177472 T^{6} + 32843490 T^{7} + 1177472 p T^{8} + 279772 p^{2} T^{9} + 27987 p^{3} T^{10} + 391 p^{4} T^{11} + 292 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 12 T + 192 T^{2} - 2156 T^{3} + 22982 T^{4} - 194572 T^{5} + 1930449 T^{6} - 15455576 T^{7} + 1930449 p T^{8} - 194572 p^{2} T^{9} + 22982 p^{3} T^{10} - 2156 p^{4} T^{11} + 192 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 4 T + 289 T^{2} + 816 T^{3} + 49277 T^{4} + 129852 T^{5} + 5490301 T^{6} + 11393056 T^{7} + 5490301 p T^{8} + 129852 p^{2} T^{9} + 49277 p^{3} T^{10} + 816 p^{4} T^{11} + 289 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 13 T + 303 T^{2} - 1237 T^{3} + 27765 T^{4} - 44291 T^{5} + 3497727 T^{6} - 11433782 T^{7} + 3497727 p T^{8} - 44291 p^{2} T^{9} + 27765 p^{3} T^{10} - 1237 p^{4} T^{11} + 303 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 15 T + 147 T^{2} + 2045 T^{3} + 28871 T^{4} + 298893 T^{5} + 2740621 T^{6} + 28440382 T^{7} + 2740621 p T^{8} + 298893 p^{2} T^{9} + 28871 p^{3} T^{10} + 2045 p^{4} T^{11} + 147 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 37 T + 1115 T^{2} + 22575 T^{3} + 398679 T^{4} + 5597907 T^{5} + 70078109 T^{6} + 728630674 T^{7} + 70078109 p T^{8} + 5597907 p^{2} T^{9} + 398679 p^{3} T^{10} + 22575 p^{4} T^{11} + 1115 p^{5} T^{12} + 37 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.65890127618121867886108082729, −3.48457885128111647317672453411, −3.44650618691759367171548860124, −3.35672511143208404597969635977, −3.20577449498229238405295108441, −3.11601360745786456419090229567, −2.87343376954607028625081004971, −2.86879123812297882393324160382, −2.73293380923226548192313510320, −2.71614236828118323630135048154, −2.69338153977192364317071162618, −2.24941068701778727502933135978, −2.16589498899022442133958509025, −2.11616903193882868817993136737, −2.05737234482029047602580386788, −2.00294163812052367702028365700, −1.73125918287641820139170665768, −1.57050759252099668953343562586, −1.46563475183157943734836024312, −1.19272421327097452460447221798, −1.03680704775940595712036422388, −0.865085936445304537548457529125, −0.852680760968564892575035601725, −0.826193564310661629607336569353, −0.41277719194510712231915903265, 0.41277719194510712231915903265, 0.826193564310661629607336569353, 0.852680760968564892575035601725, 0.865085936445304537548457529125, 1.03680704775940595712036422388, 1.19272421327097452460447221798, 1.46563475183157943734836024312, 1.57050759252099668953343562586, 1.73125918287641820139170665768, 2.00294163812052367702028365700, 2.05737234482029047602580386788, 2.11616903193882868817993136737, 2.16589498899022442133958509025, 2.24941068701778727502933135978, 2.69338153977192364317071162618, 2.71614236828118323630135048154, 2.73293380923226548192313510320, 2.86879123812297882393324160382, 2.87343376954607028625081004971, 3.11601360745786456419090229567, 3.20577449498229238405295108441, 3.35672511143208404597969635977, 3.44650618691759367171548860124, 3.48457885128111647317672453411, 3.65890127618121867886108082729
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.