Dirichlet series
| L(s) = 1 | + 3·3-s − 4·4-s − 2·5-s + 15·7-s − 4·9-s − 3·11-s − 12·12-s + 12·13-s − 6·15-s + 4·16-s − 6·17-s + 10·19-s + 8·20-s + 45·21-s − 3·23-s − 13·25-s − 23·27-s − 60·28-s − 9·29-s + 13·31-s − 3·32-s − 9·33-s − 30·35-s + 16·36-s − 2·37-s + 36·39-s − 41-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 2·4-s − 0.894·5-s + 5.66·7-s − 4/3·9-s − 0.904·11-s − 3.46·12-s + 3.32·13-s − 1.54·15-s + 16-s − 1.45·17-s + 2.29·19-s + 1.78·20-s + 9.81·21-s − 0.625·23-s − 2.59·25-s − 4.42·27-s − 11.3·28-s − 1.67·29-s + 2.33·31-s − 0.530·32-s − 1.56·33-s − 5.07·35-s + 8/3·36-s − 0.328·37-s + 5.76·39-s − 0.156·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(137^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(137^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(137^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.87492\) |
| Root analytic conductor: | \(1.04592\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 137^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.971876350\) |
| \(L(\frac12)\) | \(\approx\) | \(1.971876350\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 137 | \( ( 1 - T )^{7} \) |
| good | 2 | \( 1 + p^{2} T^{2} + 3 p^{2} T^{4} + 3 T^{5} + 29 T^{6} + 5 T^{7} + 29 p T^{8} + 3 p^{2} T^{9} + 3 p^{5} T^{10} + p^{7} T^{12} + p^{7} T^{14} \) |
| 3 | \( 1 - p T + 13 T^{2} - 28 T^{3} + 80 T^{4} - 151 T^{5} + 340 T^{6} - 550 T^{7} + 340 p T^{8} - 151 p^{2} T^{9} + 80 p^{3} T^{10} - 28 p^{4} T^{11} + 13 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) | |
| 5 | \( 1 + 2 T + 17 T^{2} + 39 T^{3} + 178 T^{4} + 356 T^{5} + 1232 T^{6} + 2198 T^{7} + 1232 p T^{8} + 356 p^{2} T^{9} + 178 p^{3} T^{10} + 39 p^{4} T^{11} + 17 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 - 15 T + 129 T^{2} - 114 p T^{3} + 3872 T^{4} - 15429 T^{5} + 51756 T^{6} - 3020 p^{2} T^{7} + 51756 p T^{8} - 15429 p^{2} T^{9} + 3872 p^{3} T^{10} - 114 p^{5} T^{11} + 129 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 3 T + 51 T^{2} + 58 T^{3} + 892 T^{4} - 807 T^{5} + 7922 T^{6} - 23788 T^{7} + 7922 p T^{8} - 807 p^{2} T^{9} + 892 p^{3} T^{10} + 58 p^{4} T^{11} + 51 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 12 T + 123 T^{2} - 851 T^{3} + 406 p T^{4} - 26202 T^{5} + 118162 T^{6} - 445854 T^{7} + 118162 p T^{8} - 26202 p^{2} T^{9} + 406 p^{4} T^{10} - 851 p^{4} T^{11} + 123 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 6 T + 95 T^{2} + 543 T^{3} + 4214 T^{4} + 21472 T^{5} + 111662 T^{6} + 475146 T^{7} + 111662 p T^{8} + 21472 p^{2} T^{9} + 4214 p^{3} T^{10} + 543 p^{4} T^{11} + 95 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 10 T + 113 T^{2} - 823 T^{3} + 5398 T^{4} - 30598 T^{5} + 151558 T^{6} - 705714 T^{7} + 151558 p T^{8} - 30598 p^{2} T^{9} + 5398 p^{3} T^{10} - 823 p^{4} T^{11} + 113 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 3 T + 73 T^{2} + 208 T^{3} + 3372 T^{4} + 8773 T^{5} + 105956 T^{6} + 244890 T^{7} + 105956 p T^{8} + 8773 p^{2} T^{9} + 3372 p^{3} T^{10} + 208 p^{4} T^{11} + 73 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 9 T + 178 T^{2} + 1127 T^{3} + 12971 T^{4} + 64025 T^{5} + 558690 T^{6} + 2264414 T^{7} + 558690 p T^{8} + 64025 p^{2} T^{9} + 12971 p^{3} T^{10} + 1127 p^{4} T^{11} + 178 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 13 T + 188 T^{2} - 1337 T^{3} + 10113 T^{4} - 42245 T^{5} + 237912 T^{6} - 823944 T^{7} + 237912 p T^{8} - 42245 p^{2} T^{9} + 10113 p^{3} T^{10} - 1337 p^{4} T^{11} + 188 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 2 T + 157 T^{2} + 427 T^{3} + 12606 T^{4} + 34956 T^{5} + 670796 T^{6} + 1622562 T^{7} + 670796 p T^{8} + 34956 p^{2} T^{9} + 12606 p^{3} T^{10} + 427 p^{4} T^{11} + 157 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + T + 93 T^{2} - 38 T^{3} + 5590 T^{4} - 1199 T^{5} + 301732 T^{6} + 174536 T^{7} + 301732 p T^{8} - 1199 p^{2} T^{9} + 5590 p^{3} T^{10} - 38 p^{4} T^{11} + 93 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 7 T + 206 T^{2} - 1343 T^{3} + 21155 T^{4} - 119191 T^{5} + 1355836 T^{6} - 6409256 T^{7} + 1355836 p T^{8} - 119191 p^{2} T^{9} + 21155 p^{3} T^{10} - 1343 p^{4} T^{11} + 206 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 15 T + 332 T^{2} - 67 p T^{3} + 39709 T^{4} - 273693 T^{5} + 2635024 T^{6} - 14919916 T^{7} + 2635024 p T^{8} - 273693 p^{2} T^{9} + 39709 p^{3} T^{10} - 67 p^{5} T^{11} + 332 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 8 T + 288 T^{2} + 1814 T^{3} + 36997 T^{4} + 192882 T^{5} + 2904714 T^{6} + 12651936 T^{7} + 2904714 p T^{8} + 192882 p^{2} T^{9} + 36997 p^{3} T^{10} + 1814 p^{4} T^{11} + 288 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 6 T + 198 T^{2} + 1468 T^{3} + 24127 T^{4} + 171910 T^{5} + 1953030 T^{6} + 12762480 T^{7} + 1953030 p T^{8} + 171910 p^{2} T^{9} + 24127 p^{3} T^{10} + 1468 p^{4} T^{11} + 198 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - T + 12 T^{2} - 487 T^{3} + 5975 T^{4} + 10451 T^{5} + 214104 T^{6} - 2840218 T^{7} + 214104 p T^{8} + 10451 p^{2} T^{9} + 5975 p^{3} T^{10} - 487 p^{4} T^{11} + 12 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 24 T + 446 T^{2} - 6120 T^{3} + 71475 T^{4} - 729832 T^{5} + 6714526 T^{6} - 57104098 T^{7} + 6714526 p T^{8} - 729832 p^{2} T^{9} + 71475 p^{3} T^{10} - 6120 p^{4} T^{11} + 446 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 16 T + 273 T^{2} - 2406 T^{3} + 25517 T^{4} - 165400 T^{5} + 1673677 T^{6} - 10460964 T^{7} + 1673677 p T^{8} - 165400 p^{2} T^{9} + 25517 p^{3} T^{10} - 2406 p^{4} T^{11} + 273 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + T + 201 T^{2} + 1636 T^{3} + 20116 T^{4} + 273121 T^{5} + 1979882 T^{6} + 23515524 T^{7} + 1979882 p T^{8} + 273121 p^{2} T^{9} + 20116 p^{3} T^{10} + 1636 p^{4} T^{11} + 201 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 15 T + 410 T^{2} - 4519 T^{3} + 73241 T^{4} - 676989 T^{5} + 8338486 T^{6} - 65535080 T^{7} + 8338486 p T^{8} - 676989 p^{2} T^{9} + 73241 p^{3} T^{10} - 4519 p^{4} T^{11} + 410 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 21 T + 623 T^{2} - 8744 T^{3} + 150662 T^{4} - 1600903 T^{5} + 20176824 T^{6} - 169376458 T^{7} + 20176824 p T^{8} - 1600903 p^{2} T^{9} + 150662 p^{3} T^{10} - 8744 p^{4} T^{11} + 623 p^{5} T^{12} - 21 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 8 T + 106 T^{2} + 2508 T^{3} + 36713 T^{4} + 223630 T^{5} + 3464796 T^{6} + 42882076 T^{7} + 3464796 p T^{8} + 223630 p^{2} T^{9} + 36713 p^{3} T^{10} + 2508 p^{4} T^{11} + 106 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - T + 497 T^{2} - 1186 T^{3} + 114886 T^{4} - 346413 T^{5} + 16457240 T^{6} - 46762336 T^{7} + 16457240 p T^{8} - 346413 p^{2} T^{9} + 114886 p^{3} T^{10} - 1186 p^{4} T^{11} + 497 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−6.48993223476228348686244266562, −6.46140179465865521560908523453, −6.43340982828611351800783789212, −6.04144486724601369422554180740, −5.80076306903174326726839397170, −5.53598861565986631329791523114, −5.44131895034915348065645457362, −5.26375649216481914134952583789, −5.17302732944544435233038020827, −5.05852130094471935058239118442, −4.77233345701346562512742884865, −4.67723120194910132117244346598, −4.19507147314932098718166069574, −4.08808046212244216949256868960, −4.03233803230269764776571312188, −3.82424259621272110286215832650, −3.78852682147521072397297094688, −3.26491793387044866427641752405, −3.07193410602751831948435910711, −2.66417012094255280443262978977, −2.34424255897471683872894487346, −2.11275298668249495308565925735, −1.68480745966448017204875462644, −1.65237322295873900270367694163, −0.959687958358020747190974332463, 0.959687958358020747190974332463, 1.65237322295873900270367694163, 1.68480745966448017204875462644, 2.11275298668249495308565925735, 2.34424255897471683872894487346, 2.66417012094255280443262978977, 3.07193410602751831948435910711, 3.26491793387044866427641752405, 3.78852682147521072397297094688, 3.82424259621272110286215832650, 4.03233803230269764776571312188, 4.08808046212244216949256868960, 4.19507147314932098718166069574, 4.67723120194910132117244346598, 4.77233345701346562512742884865, 5.05852130094471935058239118442, 5.17302732944544435233038020827, 5.26375649216481914134952583789, 5.44131895034915348065645457362, 5.53598861565986631329791523114, 5.80076306903174326726839397170, 6.04144486724601369422554180740, 6.43340982828611351800783789212, 6.46140179465865521560908523453, 6.48993223476228348686244266562
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.