Dirichlet series
| L(s) = 1 | − 2·2-s − 4·4-s − 13·5-s − 8·7-s + 8·8-s − 11·9-s + 26·10-s − 10·11-s + 5·13-s + 16·14-s + 13·16-s − 17·17-s + 22·18-s − 19·19-s + 52·20-s + 20·22-s + 15·23-s + 76·25-s − 10·26-s + 3·27-s + 32·28-s − 24·29-s + 4·31-s − 15·32-s + 34·34-s + 104·35-s + 44·36-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2·4-s − 5.81·5-s − 3.02·7-s + 2.82·8-s − 3.66·9-s + 8.22·10-s − 3.01·11-s + 1.38·13-s + 4.27·14-s + 13/4·16-s − 4.12·17-s + 5.18·18-s − 4.35·19-s + 11.6·20-s + 4.26·22-s + 3.12·23-s + 76/5·25-s − 1.96·26-s + 0.577·27-s + 6.04·28-s − 4.45·29-s + 0.718·31-s − 2.65·32-s + 5.83·34-s + 17.5·35-s + 22/3·36-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(331^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(331^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(331^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(901.022\) |
| Root analytic conductor: | \(1.62574\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 331^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
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 | 331 | \( ( 1 + T )^{7} \) |
| good | 2 | \( 1 + p T + p^{3} T^{2} + p^{4} T^{3} + 35 T^{4} + 59 T^{5} + 101 T^{6} + 141 T^{7} + 101 p T^{8} + 59 p^{2} T^{9} + 35 p^{3} T^{10} + p^{8} T^{11} + p^{8} T^{12} + p^{7} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 + 11 T^{2} - p T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + 149 T^{6} - 161 T^{7} + 149 p T^{8} - 4 p^{4} T^{9} + 17 p^{4} T^{10} - p^{5} T^{11} + 11 p^{5} T^{12} + p^{7} T^{14} \) | |
| 5 | \( 1 + 13 T + 93 T^{2} + 469 T^{3} + 1861 T^{4} + 1224 p T^{5} + 17137 T^{6} + 41257 T^{7} + 17137 p T^{8} + 1224 p^{3} T^{9} + 1861 p^{3} T^{10} + 469 p^{4} T^{11} + 93 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 + 8 T + 43 T^{2} + 152 T^{3} + 10 p^{2} T^{4} + 1562 T^{5} + 5144 T^{6} + 14837 T^{7} + 5144 p T^{8} + 1562 p^{2} T^{9} + 10 p^{5} T^{10} + 152 p^{4} T^{11} + 43 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 10 T + 81 T^{2} + 454 T^{3} + 2330 T^{4} + 9951 T^{5} + 40053 T^{6} + 137531 T^{7} + 40053 p T^{8} + 9951 p^{2} T^{9} + 2330 p^{3} T^{10} + 454 p^{4} T^{11} + 81 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 5 T + 61 T^{2} - 201 T^{3} + 1533 T^{4} - 3662 T^{5} + 24527 T^{6} - 49323 T^{7} + 24527 p T^{8} - 3662 p^{2} T^{9} + 1533 p^{3} T^{10} - 201 p^{4} T^{11} + 61 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + p T + 194 T^{2} + 1454 T^{3} + 514 p T^{4} + 41212 T^{5} + 178145 T^{6} + 714777 T^{7} + 178145 p T^{8} + 41212 p^{2} T^{9} + 514 p^{4} T^{10} + 1454 p^{4} T^{11} + 194 p^{5} T^{12} + p^{7} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + p T + 11 p T^{2} + 1458 T^{3} + 6991 T^{4} + 20115 T^{5} + 21335 T^{6} - 57427 T^{7} + 21335 p T^{8} + 20115 p^{2} T^{9} + 6991 p^{3} T^{10} + 1458 p^{4} T^{11} + 11 p^{6} T^{12} + p^{7} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 15 T + 204 T^{2} - 1807 T^{3} + 14762 T^{4} - 94746 T^{5} + 567210 T^{6} - 2809977 T^{7} + 567210 p T^{8} - 94746 p^{2} T^{9} + 14762 p^{3} T^{10} - 1807 p^{4} T^{11} + 204 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 24 T + 366 T^{2} + 4151 T^{3} + 38243 T^{4} + 295772 T^{5} + 1968148 T^{6} + 11346327 T^{7} + 1968148 p T^{8} + 295772 p^{2} T^{9} + 38243 p^{3} T^{10} + 4151 p^{4} T^{11} + 366 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 4 T + 127 T^{2} - 561 T^{3} + 8464 T^{4} - 35039 T^{5} + 380728 T^{6} - 1333903 T^{7} + 380728 p T^{8} - 35039 p^{2} T^{9} + 8464 p^{3} T^{10} - 561 p^{4} T^{11} + 127 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 2 T + 100 T^{2} - 410 T^{3} + 7076 T^{4} - 29679 T^{5} + 336420 T^{6} - 1458419 T^{7} + 336420 p T^{8} - 29679 p^{2} T^{9} + 7076 p^{3} T^{10} - 410 p^{4} T^{11} + 100 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 18 T + 337 T^{2} + 3894 T^{3} + 43458 T^{4} + 370537 T^{5} + 3010001 T^{6} + 19768195 T^{7} + 3010001 p T^{8} + 370537 p^{2} T^{9} + 43458 p^{3} T^{10} + 3894 p^{4} T^{11} + 337 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 8 T + 196 T^{2} + 1098 T^{3} + 18040 T^{4} + 84403 T^{5} + 1113386 T^{6} + 4413629 T^{7} + 1113386 p T^{8} + 84403 p^{2} T^{9} + 18040 p^{3} T^{10} + 1098 p^{4} T^{11} + 196 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 10 T + 208 T^{2} - 1642 T^{3} + 18814 T^{4} - 118269 T^{5} + 1078328 T^{6} - 5930611 T^{7} + 1078328 p T^{8} - 118269 p^{2} T^{9} + 18814 p^{3} T^{10} - 1642 p^{4} T^{11} + 208 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 21 T + 455 T^{2} + 6146 T^{3} + 79431 T^{4} + 780215 T^{5} + 7272823 T^{6} + 54429927 T^{7} + 7272823 p T^{8} + 780215 p^{2} T^{9} + 79431 p^{3} T^{10} + 6146 p^{4} T^{11} + 455 p^{5} T^{12} + 21 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 17 T + 6 p T^{2} + 4010 T^{3} + 51984 T^{4} + 469870 T^{5} + 4678683 T^{6} + 34557975 T^{7} + 4678683 p T^{8} + 469870 p^{2} T^{9} + 51984 p^{3} T^{10} + 4010 p^{4} T^{11} + 6 p^{6} T^{12} + 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 12 T + 368 T^{2} + 3764 T^{3} + 60441 T^{4} + 524570 T^{5} + 5817191 T^{6} + 41440589 T^{7} + 5817191 p T^{8} + 524570 p^{2} T^{9} + 60441 p^{3} T^{10} + 3764 p^{4} T^{11} + 368 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 16 T + 365 T^{2} + 4079 T^{3} + 55226 T^{4} + 492317 T^{5} + 5128224 T^{6} + 38957685 T^{7} + 5128224 p T^{8} + 492317 p^{2} T^{9} + 55226 p^{3} T^{10} + 4079 p^{4} T^{11} + 365 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 4 T + 415 T^{2} - 1408 T^{3} + 77695 T^{4} - 218868 T^{5} + 8593520 T^{6} - 19746705 T^{7} + 8593520 p T^{8} - 218868 p^{2} T^{9} + 77695 p^{3} T^{10} - 1408 p^{4} T^{11} + 415 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 11 T + 316 T^{2} - 2549 T^{3} + 39292 T^{4} - 233203 T^{5} + 2915717 T^{6} - 15425577 T^{7} + 2915717 p T^{8} - 233203 p^{2} T^{9} + 39292 p^{3} T^{10} - 2549 p^{4} T^{11} + 316 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 2 T + 115 T^{2} - 592 T^{3} + 8335 T^{4} - 56980 T^{5} + 1314090 T^{6} - 2374757 T^{7} + 1314090 p T^{8} - 56980 p^{2} T^{9} + 8335 p^{3} T^{10} - 592 p^{4} T^{11} + 115 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 4 T + 487 T^{2} + 1925 T^{3} + 107166 T^{4} + 391575 T^{5} + 13910480 T^{6} + 43052877 T^{7} + 13910480 p T^{8} + 391575 p^{2} T^{9} + 107166 p^{3} T^{10} + 1925 p^{4} T^{11} + 487 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 28 T + 672 T^{2} + 11271 T^{3} + 172835 T^{4} + 2135588 T^{5} + 24657610 T^{6} + 240694563 T^{7} + 24657610 p T^{8} + 2135588 p^{2} T^{9} + 172835 p^{3} T^{10} + 11271 p^{4} T^{11} + 672 p^{5} T^{12} + 28 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 32 T + 839 T^{2} + 155 p T^{3} + 240474 T^{4} + 3118861 T^{5} + 37380542 T^{6} + 381409039 T^{7} + 37380542 p T^{8} + 3118861 p^{2} T^{9} + 240474 p^{3} T^{10} + 155 p^{5} T^{11} + 839 p^{5} T^{12} + 32 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−6.36733042281788659000048027120, −6.08907796768265436522593812621, −6.00142656730110974625264576177, −5.83581951036131831360905659844, −5.59522617518486457083399528847, −5.19892539182472013561609564687, −5.17277097597729667408897274154, −5.15067372631621535490949901493, −4.82638889324460360942563347040, −4.69736847090945016814935920602, −4.43605909102832151720271521877, −4.24571261269716188674625672061, −4.18757031588260210727021047479, −4.02686948513383774217182407276, −3.93861652172860571574890250319, −3.70994131291777073965429940903, −3.51571663451794488476863352206, −3.25949366408291738058265364600, −3.19591230146160039132200220920, −3.13080672791918484137053871621, −2.77715779561736493733193869525, −2.62550230790099230352345352558, −2.62406039114773398932219084232, −2.15776505728865061108693266681, −1.47585138495525055001141603202, 0, 0, 0, 0, 0, 0, 0, 1.47585138495525055001141603202, 2.15776505728865061108693266681, 2.62406039114773398932219084232, 2.62550230790099230352345352558, 2.77715779561736493733193869525, 3.13080672791918484137053871621, 3.19591230146160039132200220920, 3.25949366408291738058265364600, 3.51571663451794488476863352206, 3.70994131291777073965429940903, 3.93861652172860571574890250319, 4.02686948513383774217182407276, 4.18757031588260210727021047479, 4.24571261269716188674625672061, 4.43605909102832151720271521877, 4.69736847090945016814935920602, 4.82638889324460360942563347040, 5.15067372631621535490949901493, 5.17277097597729667408897274154, 5.19892539182472013561609564687, 5.59522617518486457083399528847, 5.83581951036131831360905659844, 6.00142656730110974625264576177, 6.08907796768265436522593812621, 6.36733042281788659000048027120
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.