Dirichlet series
| L(s) = 1 | + 2·3-s + 2·7-s + 4·11-s + 4·13-s + 4·17-s − 4·19-s + 4·21-s + 8·23-s − 4·25-s − 8·27-s + 6·29-s + 4·31-s + 8·33-s + 32·37-s + 8·39-s − 14·41-s − 24·43-s − 4·47-s − 6·49-s + 8·51-s + 6·53-s − 8·57-s − 30·59-s + 10·61-s − 2·67-s + 16·69-s − 14·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.755·7-s + 1.20·11-s + 1.10·13-s + 0.970·17-s − 0.917·19-s + 0.872·21-s + 1.66·23-s − 4/5·25-s − 1.53·27-s + 1.11·29-s + 0.718·31-s + 1.39·33-s + 5.26·37-s + 1.28·39-s − 2.18·41-s − 3.65·43-s − 0.583·47-s − 6/7·49-s + 1.12·51-s + 0.824·53-s − 1.05·57-s − 3.90·59-s + 1.28·61-s − 0.244·67-s + 1.92·69-s − 1.63·73-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 101^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 101^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{14} \cdot 101^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3635.87\) |
| Root analytic conductor: | \(1.79609\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{14} \cdot 101^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(6.595559524\) |
| \(L(\frac12)\) | \(\approx\) | \(6.595559524\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 101 | \( ( 1 + T )^{7} \) | |
| good | 3 | \( 1 - 2 T + 4 T^{2} - 2 T^{4} + 14 T^{5} + 2 T^{6} + 34 T^{7} + 2 p T^{8} + 14 p^{2} T^{9} - 2 p^{3} T^{10} + 4 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) |
| 5 | \( 1 + 4 T^{2} + 8 T^{3} + 12 T^{4} + 52 T^{6} - 6 p^{2} T^{7} + 52 p T^{8} + 12 p^{3} T^{10} + 8 p^{4} T^{11} + 4 p^{5} T^{12} + p^{7} T^{14} \) | |
| 7 | \( 1 - 2 T + 10 T^{2} - 22 T^{3} + 102 T^{4} - 208 T^{5} + 104 p T^{6} - 1430 T^{7} + 104 p^{2} T^{8} - 208 p^{2} T^{9} + 102 p^{3} T^{10} - 22 p^{4} T^{11} + 10 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 4 T + 34 T^{2} - 142 T^{3} + 724 T^{4} - 3070 T^{5} + 10554 T^{6} - 40342 T^{7} + 10554 p T^{8} - 3070 p^{2} T^{9} + 724 p^{3} T^{10} - 142 p^{4} T^{11} + 34 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 4 T + 36 T^{2} - 68 T^{3} + 508 T^{4} + 20 T^{5} + 4796 T^{6} + 5986 T^{7} + 4796 p T^{8} + 20 p^{2} T^{9} + 508 p^{3} T^{10} - 68 p^{4} T^{11} + 36 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 4 T + 60 T^{2} - 128 T^{3} + 1036 T^{4} + 788 T^{5} + 968 T^{6} + 61994 T^{7} + 968 p T^{8} + 788 p^{2} T^{9} + 1036 p^{3} T^{10} - 128 p^{4} T^{11} + 60 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 4 T + 37 T^{2} + 104 T^{3} + 1005 T^{4} + 2588 T^{5} + 19889 T^{6} + 1552 p T^{7} + 19889 p T^{8} + 2588 p^{2} T^{9} + 1005 p^{3} T^{10} + 104 p^{4} T^{11} + 37 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 8 T + 73 T^{2} - 208 T^{3} + 1581 T^{4} - 3256 T^{5} + 45909 T^{6} - 104928 T^{7} + 45909 p T^{8} - 3256 p^{2} T^{9} + 1581 p^{3} T^{10} - 208 p^{4} T^{11} + 73 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 6 T + 79 T^{2} - 460 T^{3} + 4401 T^{4} - 22698 T^{5} + 171175 T^{6} - 767976 T^{7} + 171175 p T^{8} - 22698 p^{2} T^{9} + 4401 p^{3} T^{10} - 460 p^{4} T^{11} + 79 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 4 T + 125 T^{2} - 536 T^{3} + 7073 T^{4} - 32636 T^{5} + 260517 T^{6} - 1235408 T^{7} + 260517 p T^{8} - 32636 p^{2} T^{9} + 7073 p^{3} T^{10} - 536 p^{4} T^{11} + 125 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 32 T + 632 T^{2} - 8844 T^{3} + 97896 T^{4} - 887560 T^{5} + 6807908 T^{6} - 44621714 T^{7} + 6807908 p T^{8} - 887560 p^{2} T^{9} + 97896 p^{3} T^{10} - 8844 p^{4} T^{11} + 632 p^{5} T^{12} - 32 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 14 T + 255 T^{2} + 2292 T^{3} + 23621 T^{4} + 155506 T^{5} + 1238923 T^{6} + 6974552 T^{7} + 1238923 p T^{8} + 155506 p^{2} T^{9} + 23621 p^{3} T^{10} + 2292 p^{4} T^{11} + 255 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 24 T + 445 T^{2} + 5904 T^{3} + 66445 T^{4} + 617448 T^{5} + 5027305 T^{6} + 35089760 T^{7} + 5027305 p T^{8} + 617448 p^{2} T^{9} + 66445 p^{3} T^{10} + 5904 p^{4} T^{11} + 445 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 4 T + 145 T^{2} + 1016 T^{3} + 11901 T^{4} + 86012 T^{5} + 816909 T^{6} + 4438800 T^{7} + 816909 p T^{8} + 86012 p^{2} T^{9} + 11901 p^{3} T^{10} + 1016 p^{4} T^{11} + 145 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 6 T + 215 T^{2} - 1068 T^{3} + 22897 T^{4} - 93162 T^{5} + 1610415 T^{6} - 5591656 T^{7} + 1610415 p T^{8} - 93162 p^{2} T^{9} + 22897 p^{3} T^{10} - 1068 p^{4} T^{11} + 215 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 30 T + 758 T^{2} + 12498 T^{3} + 179602 T^{4} + 2013660 T^{5} + 20032836 T^{6} + 162921770 T^{7} + 20032836 p T^{8} + 2013660 p^{2} T^{9} + 179602 p^{3} T^{10} + 12498 p^{4} T^{11} + 758 p^{5} T^{12} + 30 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 10 T + 327 T^{2} - 2852 T^{3} + 50953 T^{4} - 381574 T^{5} + 4792631 T^{6} - 29705912 T^{7} + 4792631 p T^{8} - 381574 p^{2} T^{9} + 50953 p^{3} T^{10} - 2852 p^{4} T^{11} + 327 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 2 T + 296 T^{2} - 228 T^{3} + 38178 T^{4} - 124910 T^{5} + 3130490 T^{6} - 13549054 T^{7} + 3130490 p T^{8} - 124910 p^{2} T^{9} + 38178 p^{3} T^{10} - 228 p^{4} T^{11} + 296 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 101 T^{2} + 288 T^{3} + 8161 T^{4} + 3264 T^{5} + 666093 T^{6} - 1236160 T^{7} + 666093 p T^{8} + 3264 p^{2} T^{9} + 8161 p^{3} T^{10} + 288 p^{4} T^{11} + 101 p^{5} T^{12} + p^{7} T^{14} \) | |
| 73 | \( 1 + 14 T + 247 T^{2} + 1636 T^{3} + 22845 T^{4} + 121138 T^{5} + 1925147 T^{6} + 9330680 T^{7} + 1925147 p T^{8} + 121138 p^{2} T^{9} + 22845 p^{3} T^{10} + 1636 p^{4} T^{11} + 247 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 24 T + 433 T^{2} - 5600 T^{3} + 69405 T^{4} - 763304 T^{5} + 8070605 T^{6} - 74220480 T^{7} + 8070605 p T^{8} - 763304 p^{2} T^{9} + 69405 p^{3} T^{10} - 5600 p^{4} T^{11} + 433 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 24 T + 642 T^{2} + 10454 T^{3} + 164952 T^{4} + 2003006 T^{5} + 23020014 T^{6} + 215918586 T^{7} + 23020014 p T^{8} + 2003006 p^{2} T^{9} + 164952 p^{3} T^{10} + 10454 p^{4} T^{11} + 642 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 6 T + 431 T^{2} + 2388 T^{3} + 88693 T^{4} + 450138 T^{5} + 11520651 T^{6} + 50758744 T^{7} + 11520651 p T^{8} + 450138 p^{2} T^{9} + 88693 p^{3} T^{10} + 2388 p^{4} T^{11} + 431 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 8 T + 376 T^{2} + 2512 T^{3} + 54864 T^{4} + 323944 T^{5} + 4645544 T^{6} + 30104318 T^{7} + 4645544 p T^{8} + 323944 p^{2} T^{9} + 54864 p^{3} T^{10} + 2512 p^{4} T^{11} + 376 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.52840428039505134066052145686, −5.35576445263390180201625461348, −5.34097473051533066280544314597, −4.95392978146401902622407194490, −4.59251612925786368258399693347, −4.53051803313444179858534735697, −4.49738198117458821064183307511, −4.35262145178879409531825680418, −4.28188382652582662478779576652, −4.05929188644758752253783986916, −3.85822391614139294354044170128, −3.42592001757907954914412036991, −3.35651003875549618138230561323, −3.13314678157888466418953152107, −3.10074737903976626998515460457, −3.00004050003455034197329621413, −2.89036232258287131081240559644, −2.55158403593650148063785858685, −1.95350167476068991619778553790, −1.94526040054808889426625525241, −1.89744859313734932851218576876, −1.60257798651602535394174697008, −1.14764883708721778625331193315, −1.06510304956672361848653142317, −0.52239323161849339045406216456, 0.52239323161849339045406216456, 1.06510304956672361848653142317, 1.14764883708721778625331193315, 1.60257798651602535394174697008, 1.89744859313734932851218576876, 1.94526040054808889426625525241, 1.95350167476068991619778553790, 2.55158403593650148063785858685, 2.89036232258287131081240559644, 3.00004050003455034197329621413, 3.10074737903976626998515460457, 3.13314678157888466418953152107, 3.35651003875549618138230561323, 3.42592001757907954914412036991, 3.85822391614139294354044170128, 4.05929188644758752253783986916, 4.28188382652582662478779576652, 4.35262145178879409531825680418, 4.49738198117458821064183307511, 4.53051803313444179858534735697, 4.59251612925786368258399693347, 4.95392978146401902622407194490, 5.34097473051533066280544314597, 5.35576445263390180201625461348, 5.52840428039505134066052145686
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.