Dirichlet series
| L(s) = 1 | + 5·2-s + 5·3-s + 9·4-s − 5-s + 25·6-s + 7-s + 4·8-s + 5·9-s − 5·10-s + 10·11-s + 45·12-s − 5·13-s + 5·14-s − 5·15-s − 11·16-s + 5·17-s + 25·18-s − 3·19-s − 9·20-s + 5·21-s + 50·22-s + 15·23-s + 20·24-s − 18·25-s − 25·26-s − 16·27-s + 9·28-s + ⋯ |
| L(s) = 1 | + 3.53·2-s + 2.88·3-s + 9/2·4-s − 0.447·5-s + 10.2·6-s + 0.377·7-s + 1.41·8-s + 5/3·9-s − 1.58·10-s + 3.01·11-s + 12.9·12-s − 1.38·13-s + 1.33·14-s − 1.29·15-s − 2.75·16-s + 1.21·17-s + 5.89·18-s − 0.688·19-s − 2.01·20-s + 1.09·21-s + 10.6·22-s + 3.12·23-s + 4.08·24-s − 3.59·25-s − 4.90·26-s − 3.07·27-s + 1.70·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(157^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(157^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(157^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.86673\) |
| Root analytic conductor: | \(1.11966\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 157^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(15.70219039\) |
| \(L(\frac12)\) | \(\approx\) | \(15.70219039\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 157 | \( ( 1 - T )^{7} \) |
| good | 2 | \( 1 - 5 T + p^{4} T^{2} - 39 T^{3} + 41 p T^{4} - 153 T^{5} + 255 T^{6} - 381 T^{7} + 255 p T^{8} - 153 p^{2} T^{9} + 41 p^{4} T^{10} - 39 p^{4} T^{11} + p^{9} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 - 5 T + 20 T^{2} - 59 T^{3} + 154 T^{4} - 116 p T^{5} + 719 T^{6} - 1300 T^{7} + 719 p T^{8} - 116 p^{3} T^{9} + 154 p^{3} T^{10} - 59 p^{4} T^{11} + 20 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 5 | \( 1 + T + 19 T^{2} + 33 T^{3} + 198 T^{4} + 348 T^{5} + 1478 T^{6} + 2096 T^{7} + 1478 p T^{8} + 348 p^{2} T^{9} + 198 p^{3} T^{10} + 33 p^{4} T^{11} + 19 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 - T + 33 T^{2} - 23 T^{3} + 75 p T^{4} - 278 T^{5} + 5360 T^{6} - 2325 T^{7} + 5360 p T^{8} - 278 p^{2} T^{9} + 75 p^{4} T^{10} - 23 p^{4} T^{11} + 33 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 10 T + 105 T^{2} - 669 T^{3} + 367 p T^{4} - 153 p^{2} T^{5} + 79021 T^{6} - 272016 T^{7} + 79021 p T^{8} - 153 p^{4} T^{9} + 367 p^{4} T^{10} - 669 p^{4} T^{11} + 105 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 5 T + 75 T^{2} + 327 T^{3} + 2637 T^{4} + 9586 T^{5} + 54440 T^{6} + 160793 T^{7} + 54440 p T^{8} + 9586 p^{2} T^{9} + 2637 p^{3} T^{10} + 327 p^{4} T^{11} + 75 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 5 T + 75 T^{2} - 358 T^{3} + 2922 T^{4} - 12229 T^{5} + 72654 T^{6} - 257579 T^{7} + 72654 p T^{8} - 12229 p^{2} T^{9} + 2922 p^{3} T^{10} - 358 p^{4} T^{11} + 75 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 3 T + 2 p T^{2} - 26 T^{3} + 273 T^{4} - 2538 T^{5} + 7536 T^{6} - 31222 T^{7} + 7536 p T^{8} - 2538 p^{2} T^{9} + 273 p^{3} T^{10} - 26 p^{4} T^{11} + 2 p^{6} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 15 T + 215 T^{2} - 1880 T^{3} + 15790 T^{4} - 99819 T^{5} + 611356 T^{6} - 2977717 T^{7} + 611356 p T^{8} - 99819 p^{2} T^{9} + 15790 p^{3} T^{10} - 1880 p^{4} T^{11} + 215 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 8 T + 127 T^{2} - 509 T^{3} + 5411 T^{4} - 8821 T^{5} + 140125 T^{6} - 64696 T^{7} + 140125 p T^{8} - 8821 p^{2} T^{9} + 5411 p^{3} T^{10} - 509 p^{4} T^{11} + 127 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 13 T + 250 T^{2} + 2268 T^{3} + 24683 T^{4} + 168824 T^{5} + 1303958 T^{6} + 6882122 T^{7} + 1303958 p T^{8} + 168824 p^{2} T^{9} + 24683 p^{3} T^{10} + 2268 p^{4} T^{11} + 250 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 15 T + 213 T^{2} + 2310 T^{3} + 22312 T^{4} + 176965 T^{5} + 1308308 T^{6} + 8329759 T^{7} + 1308308 p T^{8} + 176965 p^{2} T^{9} + 22312 p^{3} T^{10} + 2310 p^{4} T^{11} + 213 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 3 T + 112 T^{2} - 661 T^{3} + 7104 T^{4} - 56254 T^{5} + 344411 T^{6} - 2851476 T^{7} + 344411 p T^{8} - 56254 p^{2} T^{9} + 7104 p^{3} T^{10} - 661 p^{4} T^{11} + 112 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 11 T + 201 T^{2} - 1642 T^{3} + 20522 T^{4} - 141273 T^{5} + 1313286 T^{6} - 7351345 T^{7} + 1313286 p T^{8} - 141273 p^{2} T^{9} + 20522 p^{3} T^{10} - 1642 p^{4} T^{11} + 201 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 8 T + 259 T^{2} - 1933 T^{3} + 31813 T^{4} - 205345 T^{5} + 2338935 T^{6} - 12437048 T^{7} + 2338935 p T^{8} - 205345 p^{2} T^{9} + 31813 p^{3} T^{10} - 1933 p^{4} T^{11} + 259 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 9 T + 348 T^{2} - 2512 T^{3} + 52679 T^{4} - 308048 T^{5} + 4532360 T^{6} - 21211846 T^{7} + 4532360 p T^{8} - 308048 p^{2} T^{9} + 52679 p^{3} T^{10} - 2512 p^{4} T^{11} + 348 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 31 T + 589 T^{2} - 8315 T^{3} + 100355 T^{4} - 1037156 T^{5} + 9464584 T^{6} - 76364959 T^{7} + 9464584 p T^{8} - 1037156 p^{2} T^{9} + 100355 p^{3} T^{10} - 8315 p^{4} T^{11} + 589 p^{5} T^{12} - 31 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 6 T + 261 T^{2} + 1153 T^{3} + 34599 T^{4} + 127145 T^{5} + 3017167 T^{6} + 9269064 T^{7} + 3017167 p T^{8} + 127145 p^{2} T^{9} + 34599 p^{3} T^{10} + 1153 p^{4} T^{11} + 261 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 4 T + 308 T^{2} - 946 T^{3} + 47468 T^{4} - 124081 T^{5} + 4666809 T^{6} - 10222038 T^{7} + 4666809 p T^{8} - 124081 p^{2} T^{9} + 47468 p^{3} T^{10} - 946 p^{4} T^{11} + 308 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 14 T + 277 T^{2} - 3509 T^{3} + 44345 T^{4} - 481161 T^{5} + 4580589 T^{6} - 41284704 T^{7} + 4580589 p T^{8} - 481161 p^{2} T^{9} + 44345 p^{3} T^{10} - 3509 p^{4} T^{11} + 277 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 3 T + 333 T^{2} + 928 T^{3} + 50348 T^{4} + 130880 T^{5} + 4861730 T^{6} + 11540394 T^{7} + 4861730 p T^{8} + 130880 p^{2} T^{9} + 50348 p^{3} T^{10} + 928 p^{4} T^{11} + 333 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 6 T + 340 T^{2} - 2460 T^{3} + 58064 T^{4} - 437381 T^{5} + 6460897 T^{6} - 44486270 T^{7} + 6460897 p T^{8} - 437381 p^{2} T^{9} + 58064 p^{3} T^{10} - 2460 p^{4} T^{11} + 340 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 41 T + 1164 T^{2} - 22843 T^{3} + 368084 T^{4} - 4814190 T^{5} + 54727361 T^{6} - 530257936 T^{7} + 54727361 p T^{8} - 4814190 p^{2} T^{9} + 368084 p^{3} T^{10} - 22843 p^{4} T^{11} + 1164 p^{5} T^{12} - 41 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 13 T + 398 T^{2} + 3990 T^{3} + 78293 T^{4} + 659483 T^{5} + 10135357 T^{6} + 72512537 T^{7} + 10135357 p T^{8} + 659483 p^{2} T^{9} + 78293 p^{3} T^{10} + 3990 p^{4} T^{11} + 398 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 12 T + 258 T^{2} + 1390 T^{3} + 35960 T^{4} + 211287 T^{5} + 4748997 T^{6} + 22810082 T^{7} + 4748997 p T^{8} + 211287 p^{2} T^{9} + 35960 p^{3} T^{10} + 1390 p^{4} T^{11} + 258 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−6.35643571276732089574283470939, −6.33043602521163815693579035267, −6.06390076680058332691486347320, −6.04170621854353454474850843197, −5.37072203061166073956700889271, −5.34084698490557611830014438278, −5.15468271830851728727218536675, −5.14058764213596964295844181247, −5.13239257920608523935991467058, −5.03619395190714830382523048384, −4.40824342342276800003147825747, −4.37307710788445396032747942439, −3.99939917894198008883440703932, −3.98760215246287456376166079673, −3.86378540586457465268555061198, −3.62784556958079648766352848722, −3.61025517240893661998855700688, −3.36164583102802848944757193784, −3.17538844354642237534680566510, −2.72506629368423132351716950968, −2.71670644043858193315242388972, −2.34625139942426303591916673269, −2.00756072425055896609633704244, −1.87531208490645868203172380126, −1.12834443970132858726477947479, 1.12834443970132858726477947479, 1.87531208490645868203172380126, 2.00756072425055896609633704244, 2.34625139942426303591916673269, 2.71670644043858193315242388972, 2.72506629368423132351716950968, 3.17538844354642237534680566510, 3.36164583102802848944757193784, 3.61025517240893661998855700688, 3.62784556958079648766352848722, 3.86378540586457465268555061198, 3.98760215246287456376166079673, 3.99939917894198008883440703932, 4.37307710788445396032747942439, 4.40824342342276800003147825747, 5.03619395190714830382523048384, 5.13239257920608523935991467058, 5.14058764213596964295844181247, 5.15468271830851728727218536675, 5.34084698490557611830014438278, 5.37072203061166073956700889271, 6.04170621854353454474850843197, 6.06390076680058332691486347320, 6.33043602521163815693579035267, 6.35643571276732089574283470939
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.