Dirichlet series
| L(s) = 1 | + 2-s + 4-s − 2·5-s + 10·7-s + 3·8-s − 2·10-s + 7·11-s − 4·13-s + 10·14-s + 2·16-s − 2·17-s + 7·19-s − 2·20-s + 7·22-s − 10·23-s − 11·25-s − 4·26-s + 10·28-s + 18·29-s + 24·31-s + 8·32-s − 2·34-s − 20·35-s + 7·38-s − 6·40-s + 12·41-s + 2·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 3.77·7-s + 1.06·8-s − 0.632·10-s + 2.11·11-s − 1.10·13-s + 2.67·14-s + 1/2·16-s − 0.485·17-s + 1.60·19-s − 0.447·20-s + 1.49·22-s − 2.08·23-s − 2.19·25-s − 0.784·26-s + 1.88·28-s + 3.34·29-s + 4.31·31-s + 1.41·32-s − 0.342·34-s − 3.38·35-s + 1.13·38-s − 0.948·40-s + 1.87·41-s + 0.304·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 11^{7} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 11^{7} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{14} \cdot 11^{7} \cdot 19^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.72449\times 10^{8}\) |
| Root analytic conductor: | \(3.87554\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 3^{14} \cdot 11^{7} \cdot 19^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(41.62895932\) |
| \(L(\frac12)\) | \(\approx\) | \(41.62895932\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 11 | \( ( 1 - T )^{7} \) | |
| 19 | \( ( 1 - T )^{7} \) | |
| good | 2 | \( 1 - T - p T^{3} + 3 T^{4} - 7 T^{5} + p^{3} T^{6} + p T^{7} + p^{4} T^{8} - 7 p^{2} T^{9} + 3 p^{3} T^{10} - p^{5} T^{11} - p^{6} T^{13} + p^{7} T^{14} \) |
| 5 | \( 1 + 2 T + 3 p T^{2} + 26 T^{3} + 113 T^{4} + 226 T^{5} + 752 T^{6} + 1466 T^{7} + 752 p T^{8} + 226 p^{2} T^{9} + 113 p^{3} T^{10} + 26 p^{4} T^{11} + 3 p^{6} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 - 10 T + 66 T^{2} - 334 T^{3} + 1439 T^{4} - 5258 T^{5} + 16844 T^{6} - 47228 T^{7} + 16844 p T^{8} - 5258 p^{2} T^{9} + 1439 p^{3} T^{10} - 334 p^{4} T^{11} + 66 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 4 T + 40 T^{2} + 118 T^{3} + 873 T^{4} + 2134 T^{5} + 13076 T^{6} + 27460 T^{7} + 13076 p T^{8} + 2134 p^{2} T^{9} + 873 p^{3} T^{10} + 118 p^{4} T^{11} + 40 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 2 T + 49 T^{2} + 160 T^{3} + 1671 T^{4} + 4814 T^{5} + 2199 p T^{6} + 107936 T^{7} + 2199 p^{2} T^{8} + 4814 p^{2} T^{9} + 1671 p^{3} T^{10} + 160 p^{4} T^{11} + 49 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 10 T + 110 T^{2} + 732 T^{3} + 4928 T^{4} + 24870 T^{5} + 5981 p T^{6} + 610984 T^{7} + 5981 p^{2} T^{8} + 24870 p^{2} T^{9} + 4928 p^{3} T^{10} + 732 p^{4} T^{11} + 110 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 18 T + 320 T^{2} - 3472 T^{3} + 35009 T^{4} - 266396 T^{5} + 1870372 T^{6} - 10488792 T^{7} + 1870372 p T^{8} - 266396 p^{2} T^{9} + 35009 p^{3} T^{10} - 3472 p^{4} T^{11} + 320 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 24 T + 431 T^{2} - 5368 T^{3} + 55269 T^{4} - 459990 T^{5} + 3278314 T^{6} - 19632048 T^{7} + 3278314 p T^{8} - 459990 p^{2} T^{9} + 55269 p^{3} T^{10} - 5368 p^{4} T^{11} + 431 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 138 T^{2} - 194 T^{3} + 9876 T^{4} - 19416 T^{5} + 504517 T^{6} - 914604 T^{7} + 504517 p T^{8} - 19416 p^{2} T^{9} + 9876 p^{3} T^{10} - 194 p^{4} T^{11} + 138 p^{5} T^{12} + p^{7} T^{14} \) | |
| 41 | \( 1 - 12 T + 282 T^{2} - 2426 T^{3} + 32453 T^{4} - 216142 T^{5} + 2107796 T^{6} - 11219712 T^{7} + 2107796 p T^{8} - 216142 p^{2} T^{9} + 32453 p^{3} T^{10} - 2426 p^{4} T^{11} + 282 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 2 T + 212 T^{2} - 366 T^{3} + 497 p T^{4} - 30878 T^{5} + 1346480 T^{6} - 1615092 T^{7} + 1346480 p T^{8} - 30878 p^{2} T^{9} + 497 p^{4} T^{10} - 366 p^{4} T^{11} + 212 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 8 T + 177 T^{2} + 912 T^{3} + 11693 T^{4} + 35256 T^{5} + 432605 T^{6} + 866144 T^{7} + 432605 p T^{8} + 35256 p^{2} T^{9} + 11693 p^{3} T^{10} + 912 p^{4} T^{11} + 177 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 2 T + 211 T^{2} + 604 T^{3} + 22621 T^{4} + 63566 T^{5} + 1684215 T^{6} + 3939464 T^{7} + 1684215 p T^{8} + 63566 p^{2} T^{9} + 22621 p^{3} T^{10} + 604 p^{4} T^{11} + 211 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 10 T + 68 T^{2} - 564 T^{3} + 7490 T^{4} - 69606 T^{5} + 453907 T^{6} - 1842328 T^{7} + 453907 p T^{8} - 69606 p^{2} T^{9} + 7490 p^{3} T^{10} - 564 p^{4} T^{11} + 68 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 14 T + 393 T^{2} - 4080 T^{3} + 66043 T^{4} - 544594 T^{5} + 105303 p T^{6} - 42469120 T^{7} + 105303 p^{2} T^{8} - 544594 p^{2} T^{9} + 66043 p^{3} T^{10} - 4080 p^{4} T^{11} + 393 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 8 T + 299 T^{2} - 1908 T^{3} + 43661 T^{4} - 221222 T^{5} + 4054526 T^{6} - 17312388 T^{7} + 4054526 p T^{8} - 221222 p^{2} T^{9} + 43661 p^{3} T^{10} - 1908 p^{4} T^{11} + 299 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 10 T + 363 T^{2} + 3316 T^{3} + 60569 T^{4} + 7042 p T^{5} + 6248102 T^{6} + 44683996 T^{7} + 6248102 p T^{8} + 7042 p^{3} T^{9} + 60569 p^{3} T^{10} + 3316 p^{4} T^{11} + 363 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 6 T + 291 T^{2} + 1036 T^{3} + 35145 T^{4} + 59322 T^{5} + 2766403 T^{6} + 2354856 T^{7} + 2766403 p T^{8} + 59322 p^{2} T^{9} + 35145 p^{3} T^{10} + 1036 p^{4} T^{11} + 291 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 52 T + 1523 T^{2} - 31800 T^{3} + 522203 T^{4} - 7037132 T^{5} + 79661705 T^{6} - 766418576 T^{7} + 79661705 p T^{8} - 7037132 p^{2} T^{9} + 522203 p^{3} T^{10} - 31800 p^{4} T^{11} + 1523 p^{5} T^{12} - 52 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 10 T + 362 T^{2} - 1618 T^{3} + 45511 T^{4} + 11482 T^{5} + 3276004 T^{6} + 12186140 T^{7} + 3276004 p T^{8} + 11482 p^{2} T^{9} + 45511 p^{3} T^{10} - 1618 p^{4} T^{11} + 362 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 222 T^{2} + 698 T^{3} + 38288 T^{4} + 87304 T^{5} + 4414265 T^{6} + 12681948 T^{7} + 4414265 p T^{8} + 87304 p^{2} T^{9} + 38288 p^{3} T^{10} + 698 p^{4} T^{11} + 222 p^{5} T^{12} + p^{7} T^{14} \) | |
| 97 | \( 1 + 24 T + 490 T^{2} + 7290 T^{3} + 114080 T^{4} + 1401624 T^{5} + 16484869 T^{6} + 161147084 T^{7} + 16484869 p T^{8} + 1401624 p^{2} T^{9} + 114080 p^{3} T^{10} + 7290 p^{4} T^{11} + 490 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−4.37668536789625739076645803103, −4.31473693523203794741502539808, −4.29953245786613256529485147732, −4.01553981921566074711944945812, −3.70335793926854116225750852487, −3.69749194964756242402871313388, −3.56537810870732679451359683539, −3.41096969379596577259637059323, −3.32987825402886191946760069002, −2.85066113680062216193465736387, −2.81663158388664963419337976073, −2.72144490756309306282780528205, −2.62949448611511245027306210203, −2.42392053261770673650481379131, −2.14614216442436132537415853784, −2.03200904359066458909134711951, −1.87891585826336042691272984039, −1.83299889348916415789579579287, −1.73207091703478115952497059605, −1.31593700445846289135337422907, −1.11810863354556702352829007086, −0.997783724846721786686513047855, −0.847447173011759445712677465534, −0.837710298294896467161867034901, −0.37616945552612121761066263649, 0.37616945552612121761066263649, 0.837710298294896467161867034901, 0.847447173011759445712677465534, 0.997783724846721786686513047855, 1.11810863354556702352829007086, 1.31593700445846289135337422907, 1.73207091703478115952497059605, 1.83299889348916415789579579287, 1.87891585826336042691272984039, 2.03200904359066458909134711951, 2.14614216442436132537415853784, 2.42392053261770673650481379131, 2.62949448611511245027306210203, 2.72144490756309306282780528205, 2.81663158388664963419337976073, 2.85066113680062216193465736387, 3.32987825402886191946760069002, 3.41096969379596577259637059323, 3.56537810870732679451359683539, 3.69749194964756242402871313388, 3.70335793926854116225750852487, 4.01553981921566074711944945812, 4.29953245786613256529485147732, 4.31473693523203794741502539808, 4.37668536789625739076645803103
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.