Dirichlet series
| L(s) = 1 | + 3·3-s − 2·5-s − 4·9-s − 12·11-s − 2·13-s − 6·15-s − 7·17-s + 3·19-s − 18·23-s − 8·25-s − 18·27-s + 5·29-s + 10·31-s − 36·33-s + 11·37-s − 6·39-s − 15·43-s + 8·45-s − 15·47-s − 21·51-s + 8·53-s + 24·55-s + 9·57-s − 9·59-s + 4·61-s + 4·65-s − 30·67-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.894·5-s − 4/3·9-s − 3.61·11-s − 0.554·13-s − 1.54·15-s − 1.69·17-s + 0.688·19-s − 3.75·23-s − 8/5·25-s − 3.46·27-s + 0.928·29-s + 1.79·31-s − 6.26·33-s + 1.80·37-s − 0.960·39-s − 2.28·43-s + 1.19·45-s − 2.18·47-s − 2.94·51-s + 1.09·53-s + 3.23·55-s + 1.19·57-s − 1.17·59-s + 0.512·61-s + 0.496·65-s − 3.66·67-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 7^{14} \cdot 17^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 7^{14} \cdot 17^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{21} \cdot 7^{14} \cdot 17^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.20804\times 10^{12}\) |
| Root analytic conductor: | \(7.29467\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 2^{21} \cdot 7^{14} \cdot 17^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) | |
| 17 | \( ( 1 + T )^{7} \) | |
| good | 3 | \( 1 - p T + 13 T^{2} - 11 p T^{3} + 94 T^{4} - 194 T^{5} + 422 T^{6} - 713 T^{7} + 422 p T^{8} - 194 p^{2} T^{9} + 94 p^{3} T^{10} - 11 p^{5} T^{11} + 13 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) |
| 5 | \( 1 + 2 T + 12 T^{2} + p T^{3} + 2 p^{2} T^{4} + 2 T^{5} + 381 T^{6} + 302 T^{7} + 381 p T^{8} + 2 p^{2} T^{9} + 2 p^{5} T^{10} + p^{5} T^{11} + 12 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 12 T + 118 T^{2} + 801 T^{3} + 425 p T^{4} + 22104 T^{5} + 92229 T^{6} + 324399 T^{7} + 92229 p T^{8} + 22104 p^{2} T^{9} + 425 p^{4} T^{10} + 801 p^{4} T^{11} + 118 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 2 T + 22 T^{2} + 115 T^{3} + 383 T^{4} + 1594 T^{5} + 7707 T^{6} + 16485 T^{7} + 7707 p T^{8} + 1594 p^{2} T^{9} + 383 p^{3} T^{10} + 115 p^{4} T^{11} + 22 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 3 T + 52 T^{2} - 159 T^{3} + 2010 T^{4} - 5793 T^{5} + 50579 T^{6} - 127050 T^{7} + 50579 p T^{8} - 5793 p^{2} T^{9} + 2010 p^{3} T^{10} - 159 p^{4} T^{11} + 52 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 18 T + 225 T^{2} + 2012 T^{3} + 15593 T^{4} + 101398 T^{5} + 591689 T^{6} + 2984472 T^{7} + 591689 p T^{8} + 101398 p^{2} T^{9} + 15593 p^{3} T^{10} + 2012 p^{4} T^{11} + 225 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 5 T + 102 T^{2} - 353 T^{3} + 3780 T^{4} - 8367 T^{5} + 69425 T^{6} - 121062 T^{7} + 69425 p T^{8} - 8367 p^{2} T^{9} + 3780 p^{3} T^{10} - 353 p^{4} T^{11} + 102 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 10 T + 126 T^{2} - 843 T^{3} + 6328 T^{4} - 31826 T^{5} + 195855 T^{6} - 907790 T^{7} + 195855 p T^{8} - 31826 p^{2} T^{9} + 6328 p^{3} T^{10} - 843 p^{4} T^{11} + 126 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 11 T + 153 T^{2} - 1207 T^{3} + 9987 T^{4} - 74209 T^{5} + 488419 T^{6} - 3326986 T^{7} + 488419 p T^{8} - 74209 p^{2} T^{9} + 9987 p^{3} T^{10} - 1207 p^{4} T^{11} + 153 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 126 T^{2} - 261 T^{3} + 10356 T^{4} - 19860 T^{5} + 589781 T^{6} - 1117278 T^{7} + 589781 p T^{8} - 19860 p^{2} T^{9} + 10356 p^{3} T^{10} - 261 p^{4} T^{11} + 126 p^{5} T^{12} + p^{7} T^{14} \) | |
| 43 | \( 1 + 15 T + 228 T^{2} + 2231 T^{3} + 21390 T^{4} + 160269 T^{5} + 1211599 T^{6} + 7845354 T^{7} + 1211599 p T^{8} + 160269 p^{2} T^{9} + 21390 p^{3} T^{10} + 2231 p^{4} T^{11} + 228 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 15 T + 220 T^{2} + 2199 T^{3} + 23706 T^{4} + 195453 T^{5} + 1616079 T^{6} + 10823946 T^{7} + 1616079 p T^{8} + 195453 p^{2} T^{9} + 23706 p^{3} T^{10} + 2199 p^{4} T^{11} + 220 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 8 T + 190 T^{2} - 1371 T^{3} + 16499 T^{4} - 113350 T^{5} + 960923 T^{6} - 6612841 T^{7} + 960923 p T^{8} - 113350 p^{2} T^{9} + 16499 p^{3} T^{10} - 1371 p^{4} T^{11} + 190 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 9 T + 343 T^{2} + 2647 T^{3} + 53851 T^{4} + 348931 T^{5} + 4992069 T^{6} + 26430058 T^{7} + 4992069 p T^{8} + 348931 p^{2} T^{9} + 53851 p^{3} T^{10} + 2647 p^{4} T^{11} + 343 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 4 T + 212 T^{2} - 1299 T^{3} + 23798 T^{4} - 156616 T^{5} + 1944473 T^{6} - 11416290 T^{7} + 1944473 p T^{8} - 156616 p^{2} T^{9} + 23798 p^{3} T^{10} - 1299 p^{4} T^{11} + 212 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 30 T + 590 T^{2} + 7997 T^{3} + 90760 T^{4} + 857278 T^{5} + 7593807 T^{6} + 62110478 T^{7} + 7593807 p T^{8} + 857278 p^{2} T^{9} + 90760 p^{3} T^{10} + 7997 p^{4} T^{11} + 590 p^{5} T^{12} + 30 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 27 T + 686 T^{2} + 11085 T^{3} + 165905 T^{4} + 1913700 T^{5} + 20556189 T^{6} + 179316035 T^{7} + 20556189 p T^{8} + 1913700 p^{2} T^{9} + 165905 p^{3} T^{10} + 11085 p^{4} T^{11} + 686 p^{5} T^{12} + 27 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 5 T + 199 T^{2} - 167 T^{3} + 17481 T^{4} + 40177 T^{5} + 1371695 T^{6} + 3731590 T^{7} + 1371695 p T^{8} + 40177 p^{2} T^{9} + 17481 p^{3} T^{10} - 167 p^{4} T^{11} + 199 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 3 T + 177 T^{2} - 1055 T^{3} + 22562 T^{4} - 122658 T^{5} + 1953358 T^{6} - 13117795 T^{7} + 1953358 p T^{8} - 122658 p^{2} T^{9} + 22562 p^{3} T^{10} - 1055 p^{4} T^{11} + 177 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 12 T + 480 T^{2} - 4457 T^{3} + 101546 T^{4} - 759728 T^{5} + 12787335 T^{6} - 78372422 T^{7} + 12787335 p T^{8} - 759728 p^{2} T^{9} + 101546 p^{3} T^{10} - 4457 p^{4} T^{11} + 480 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 32 T + 969 T^{2} + 18396 T^{3} + 320536 T^{4} + 4264988 T^{5} + 52140586 T^{6} + 512856896 T^{7} + 52140586 p T^{8} + 4264988 p^{2} T^{9} + 320536 p^{3} T^{10} + 18396 p^{4} T^{11} + 969 p^{5} T^{12} + 32 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 39 T + 1183 T^{2} - 25359 T^{3} + 446417 T^{4} - 6530253 T^{5} + 80626447 T^{6} - 860866202 T^{7} + 80626447 p T^{8} - 6530253 p^{2} T^{9} + 446417 p^{3} T^{10} - 25359 p^{4} T^{11} + 1183 p^{5} T^{12} - 39 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.92224433562041406017331357335, −3.89173219557213306869218651812, −3.84133073023448731353651896776, −3.59464785821756544772234647451, −3.38081820174129260487125976221, −3.34840191373700383456344382874, −3.28727282403221470659834706783, −3.06583545444811522943990444568, −2.98601344202859453158311659618, −2.96370305770854525798402959383, −2.74505540690296328935330625576, −2.67167880952196412568472262994, −2.65077582722546873169726588590, −2.39165555752981310981417738096, −2.29402150483888745347557253671, −2.29331204409351757444769279111, −2.24709165664352740776105292483, −2.19449993579561168063591517067, −1.82811353926958865849415936112, −1.81217062564505939973711898687, −1.36139577277888562786440379830, −1.31970446111376064636666757036, −1.19104261757454437749061592403, −1.08819633604190994707398467851, −0.923684486171913800812987993741, 0, 0, 0, 0, 0, 0, 0, 0.923684486171913800812987993741, 1.08819633604190994707398467851, 1.19104261757454437749061592403, 1.31970446111376064636666757036, 1.36139577277888562786440379830, 1.81217062564505939973711898687, 1.82811353926958865849415936112, 2.19449993579561168063591517067, 2.24709165664352740776105292483, 2.29331204409351757444769279111, 2.29402150483888745347557253671, 2.39165555752981310981417738096, 2.65077582722546873169726588590, 2.67167880952196412568472262994, 2.74505540690296328935330625576, 2.96370305770854525798402959383, 2.98601344202859453158311659618, 3.06583545444811522943990444568, 3.28727282403221470659834706783, 3.34840191373700383456344382874, 3.38081820174129260487125976221, 3.59464785821756544772234647451, 3.84133073023448731353651896776, 3.89173219557213306869218651812, 3.92224433562041406017331357335
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.