Dirichlet series
| L(s) = 1 | − 2·2-s − 7·3-s + 14·6-s + 7-s + 3·8-s + 28·9-s + 4·11-s + 13-s − 2·14-s − 4·16-s − 8·17-s − 56·18-s + 15·19-s − 7·21-s − 8·22-s − 14·23-s − 21·24-s − 2·26-s − 84·27-s − 7·29-s + 5·31-s − 28·33-s + 16·34-s + 6·37-s − 30·38-s − 7·39-s + 22·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 4.04·3-s + 5.71·6-s + 0.377·7-s + 1.06·8-s + 28/3·9-s + 1.20·11-s + 0.277·13-s − 0.534·14-s − 16-s − 1.94·17-s − 13.1·18-s + 3.44·19-s − 1.52·21-s − 1.70·22-s − 2.91·23-s − 4.28·24-s − 0.392·26-s − 16.1·27-s − 1.29·29-s + 0.898·31-s − 4.87·33-s + 2.74·34-s + 0.986·37-s − 4.86·38-s − 1.12·39-s + 3.43·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 5^{14} \cdot 29^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 5^{14} \cdot 29^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{7} \cdot 5^{14} \cdot 29^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.76600\times 10^{8}\) |
| Root analytic conductor: | \(4.16742\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 3^{7} \cdot 5^{14} \cdot 29^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.6403376762\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6403376762\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| 29 | \( ( 1 + T )^{7} \) | |
| good | 2 | \( 1 + p T + p^{2} T^{2} + 5 T^{3} + p^{3} T^{4} + 3 p^{2} T^{5} + 21 T^{6} + 13 p T^{7} + 21 p T^{8} + 3 p^{4} T^{9} + p^{6} T^{10} + 5 p^{4} T^{11} + p^{7} T^{12} + p^{7} T^{13} + p^{7} T^{14} \) |
| 7 | \( 1 - T + 19 T^{2} - 4 T^{3} + 4 p^{2} T^{4} - 8 T^{5} + 1809 T^{6} - 405 T^{7} + 1809 p T^{8} - 8 p^{2} T^{9} + 4 p^{5} T^{10} - 4 p^{4} T^{11} + 19 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 4 T + 17 T^{2} - 74 T^{3} + 346 T^{4} - 1348 T^{5} + 4717 T^{6} - 11650 T^{7} + 4717 p T^{8} - 1348 p^{2} T^{9} + 346 p^{3} T^{10} - 74 p^{4} T^{11} + 17 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - T + 33 T^{2} + 46 T^{3} + 534 T^{4} + 2130 T^{5} + 5519 T^{6} + 42241 T^{7} + 5519 p T^{8} + 2130 p^{2} T^{9} + 534 p^{3} T^{10} + 46 p^{4} T^{11} + 33 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 8 T + 43 T^{2} + 236 T^{3} + 1582 T^{4} + 8242 T^{5} + 35495 T^{6} + 136294 T^{7} + 35495 p T^{8} + 8242 p^{2} T^{9} + 1582 p^{3} T^{10} + 236 p^{4} T^{11} + 43 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 15 T + 161 T^{2} - 1258 T^{3} + 8505 T^{4} - 49369 T^{5} + 258353 T^{6} - 1189196 T^{7} + 258353 p T^{8} - 49369 p^{2} T^{9} + 8505 p^{3} T^{10} - 1258 p^{4} T^{11} + 161 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 14 T + 153 T^{2} + 996 T^{3} + 5629 T^{4} + 21330 T^{5} + 86229 T^{6} + 295096 T^{7} + 86229 p T^{8} + 21330 p^{2} T^{9} + 5629 p^{3} T^{10} + 996 p^{4} T^{11} + 153 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 5 T + 97 T^{2} - 10 p T^{3} + 3325 T^{4} - 277 p T^{5} + 74429 T^{6} - 246580 T^{7} + 74429 p T^{8} - 277 p^{3} T^{9} + 3325 p^{3} T^{10} - 10 p^{5} T^{11} + 97 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 6 T + 183 T^{2} - 996 T^{3} + 16657 T^{4} - 78442 T^{5} + 934479 T^{6} - 3671800 T^{7} + 934479 p T^{8} - 78442 p^{2} T^{9} + 16657 p^{3} T^{10} - 996 p^{4} T^{11} + 183 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 22 T + 326 T^{2} - 3328 T^{3} + 29567 T^{4} - 228704 T^{5} + 1694159 T^{6} - 11224148 T^{7} + 1694159 p T^{8} - 228704 p^{2} T^{9} + 29567 p^{3} T^{10} - 3328 p^{4} T^{11} + 326 p^{5} T^{12} - 22 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 19 T + 347 T^{2} - 4166 T^{3} + 46407 T^{4} - 409893 T^{5} + 3355901 T^{6} - 22823892 T^{7} + 3355901 p T^{8} - 409893 p^{2} T^{9} + 46407 p^{3} T^{10} - 4166 p^{4} T^{11} + 347 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 22 T + 421 T^{2} + 5464 T^{3} + 63328 T^{4} + 592886 T^{5} + 5040195 T^{6} + 36162386 T^{7} + 5040195 p T^{8} + 592886 p^{2} T^{9} + 63328 p^{3} T^{10} + 5464 p^{4} T^{11} + 421 p^{5} T^{12} + 22 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 10 T + 171 T^{2} + 1924 T^{3} + 23061 T^{4} + 180870 T^{5} + 1739015 T^{6} + 12570552 T^{7} + 1739015 p T^{8} + 180870 p^{2} T^{9} + 23061 p^{3} T^{10} + 1924 p^{4} T^{11} + 171 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 6 T + 241 T^{2} - 1540 T^{3} + 31113 T^{4} - 182410 T^{5} + 2656289 T^{6} - 13355128 T^{7} + 2656289 p T^{8} - 182410 p^{2} T^{9} + 31113 p^{3} T^{10} - 1540 p^{4} T^{11} + 241 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 23 T + 411 T^{2} - 4614 T^{3} + 46205 T^{4} - 339329 T^{5} + 2592159 T^{6} - 17450100 T^{7} + 2592159 p T^{8} - 339329 p^{2} T^{9} + 46205 p^{3} T^{10} - 4614 p^{4} T^{11} + 411 p^{5} T^{12} - 23 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 13 T + 295 T^{2} - 3096 T^{3} + 39964 T^{4} - 349620 T^{5} + 3421013 T^{6} - 26910289 T^{7} + 3421013 p T^{8} - 349620 p^{2} T^{9} + 39964 p^{3} T^{10} - 3096 p^{4} T^{11} + 295 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 26 T + 585 T^{2} - 9492 T^{3} + 131485 T^{4} - 1533958 T^{5} + 15838501 T^{6} - 140769112 T^{7} + 15838501 p T^{8} - 1533958 p^{2} T^{9} + 131485 p^{3} T^{10} - 9492 p^{4} T^{11} + 585 p^{5} T^{12} - 26 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 24 T + 459 T^{2} - 5928 T^{3} + 73905 T^{4} - 770216 T^{5} + 7983475 T^{6} - 69602480 T^{7} + 7983475 p T^{8} - 770216 p^{2} T^{9} + 73905 p^{3} T^{10} - 5928 p^{4} T^{11} + 459 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 14 T + 449 T^{2} - 4540 T^{3} + 1011 p T^{4} - 628562 T^{5} + 8354141 T^{6} - 56447048 T^{7} + 8354141 p T^{8} - 628562 p^{2} T^{9} + 1011 p^{4} T^{10} - 4540 p^{4} T^{11} + 449 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 10 T + 501 T^{2} + 4476 T^{3} + 114237 T^{4} + 864230 T^{5} + 15185521 T^{6} + 93232840 T^{7} + 15185521 p T^{8} + 864230 p^{2} T^{9} + 114237 p^{3} T^{10} + 4476 p^{4} T^{11} + 501 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 14 T + 421 T^{2} - 4676 T^{3} + 83234 T^{4} - 766332 T^{5} + 10594021 T^{6} - 81837236 T^{7} + 10594021 p T^{8} - 766332 p^{2} T^{9} + 83234 p^{3} T^{10} - 4676 p^{4} T^{11} + 421 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 31 T + 609 T^{2} - 9602 T^{3} + 123511 T^{4} - 1501713 T^{5} + 17079559 T^{6} - 173792924 T^{7} + 17079559 p T^{8} - 1501713 p^{2} T^{9} + 123511 p^{3} T^{10} - 9602 p^{4} T^{11} + 609 p^{5} T^{12} - 31 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−4.19289818021193808504136536984, −4.11307252849259335785251141687, −4.06740684127135385280676452325, −3.98402738369101836133915587932, −3.75242842147108719678119899357, −3.54866664353302200250335193421, −3.45827037235196307659753610624, −3.41867480863150989055727545481, −3.37472649750434607845588471185, −2.90763899115558260939117157652, −2.61023099584197387327062384380, −2.44212949335232541260235104019, −2.38344205188089769363062483193, −2.34394616893858795940599577437, −2.03738678077670244844773244886, −1.78161934335717031107631019572, −1.71974356643536391713853410367, −1.53708590639601758346834133363, −1.39056736795133797244020417178, −1.07489796017433509545714203180, −0.834733188292331876571569817971, −0.822724460421109999941447272957, −0.58370932640766198592224533619, −0.41388092543812425358550677346, −0.34138934485228128137122512189, 0.34138934485228128137122512189, 0.41388092543812425358550677346, 0.58370932640766198592224533619, 0.822724460421109999941447272957, 0.834733188292331876571569817971, 1.07489796017433509545714203180, 1.39056736795133797244020417178, 1.53708590639601758346834133363, 1.71974356643536391713853410367, 1.78161934335717031107631019572, 2.03738678077670244844773244886, 2.34394616893858795940599577437, 2.38344205188089769363062483193, 2.44212949335232541260235104019, 2.61023099584197387327062384380, 2.90763899115558260939117157652, 3.37472649750434607845588471185, 3.41867480863150989055727545481, 3.45827037235196307659753610624, 3.54866664353302200250335193421, 3.75242842147108719678119899357, 3.98402738369101836133915587932, 4.06740684127135385280676452325, 4.11307252849259335785251141687, 4.19289818021193808504136536984
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.