Dirichlet series
| L(s) = 1 | − 3-s − 5-s − 6·9-s + 3·11-s − 6·13-s + 15-s − 10·17-s − 7·19-s − 2·23-s − 13·25-s + 8·27-s + 3·29-s + 6·31-s − 3·33-s − 7·37-s + 6·39-s − 9·41-s + 43-s + 6·45-s − 15·47-s + 10·51-s + 5·53-s − 3·55-s + 7·57-s + 9·59-s − 13·61-s + 6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2·9-s + 0.904·11-s − 1.66·13-s + 0.258·15-s − 2.42·17-s − 1.60·19-s − 0.417·23-s − 2.59·25-s + 1.53·27-s + 0.557·29-s + 1.07·31-s − 0.522·33-s − 1.15·37-s + 0.960·39-s − 1.40·41-s + 0.152·43-s + 0.894·45-s − 2.18·47-s + 1.40·51-s + 0.686·53-s − 0.404·55-s + 0.927·57-s + 1.17·59-s − 1.66·61-s + 0.744·65-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 7^{14} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 7^{14} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{21} \cdot 7^{14} \cdot 19^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(2.63158\times 10^{12}\) |
| Root analytic conductor: | \(7.71184\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 2^{21} \cdot 7^{14} \cdot 19^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
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 | \( 1 \) | |
| 19 | \( ( 1 + T )^{7} \) | |
| good | 3 | \( 1 + T + 7 T^{2} + 5 T^{3} + 29 T^{4} + 32 T^{5} + 110 T^{6} + 128 T^{7} + 110 p T^{8} + 32 p^{2} T^{9} + 29 p^{3} T^{10} + 5 p^{4} T^{11} + 7 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) |
| 5 | \( 1 + T + 14 T^{2} + 2 T^{3} + 91 T^{4} - 102 T^{5} + 71 p T^{6} - 178 p T^{7} + 71 p^{2} T^{8} - 102 p^{2} T^{9} + 91 p^{3} T^{10} + 2 p^{4} T^{11} + 14 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 3 T + 45 T^{2} - 97 T^{3} + 909 T^{4} - 1328 T^{5} + 12108 T^{6} - 13628 T^{7} + 12108 p T^{8} - 1328 p^{2} T^{9} + 909 p^{3} T^{10} - 97 p^{4} T^{11} + 45 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 6 T + 67 T^{2} + 366 T^{3} + 2260 T^{4} + 10006 T^{5} + 46632 T^{6} + 162684 T^{7} + 46632 p T^{8} + 10006 p^{2} T^{9} + 2260 p^{3} T^{10} + 366 p^{4} T^{11} + 67 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 10 T + 112 T^{2} + 762 T^{3} + 5133 T^{4} + 26688 T^{5} + 136106 T^{6} + 565856 T^{7} + 136106 p T^{8} + 26688 p^{2} T^{9} + 5133 p^{3} T^{10} + 762 p^{4} T^{11} + 112 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 2 T + 83 T^{2} + 32 T^{3} + 3692 T^{4} - 240 T^{5} + 118042 T^{6} - 16708 T^{7} + 118042 p T^{8} - 240 p^{2} T^{9} + 3692 p^{3} T^{10} + 32 p^{4} T^{11} + 83 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 3 T + 107 T^{2} - 257 T^{3} + 5943 T^{4} - 13412 T^{5} + 234288 T^{6} - 489326 T^{7} + 234288 p T^{8} - 13412 p^{2} T^{9} + 5943 p^{3} T^{10} - 257 p^{4} T^{11} + 107 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 6 T + 142 T^{2} - 840 T^{3} + 10215 T^{4} - 54236 T^{5} + 472346 T^{6} - 2104844 T^{7} + 472346 p T^{8} - 54236 p^{2} T^{9} + 10215 p^{3} T^{10} - 840 p^{4} T^{11} + 142 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 7 T + 126 T^{2} + 746 T^{3} + 8141 T^{4} + 36660 T^{5} + 337331 T^{6} + 1430712 T^{7} + 337331 p T^{8} + 36660 p^{2} T^{9} + 8141 p^{3} T^{10} + 746 p^{4} T^{11} + 126 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 9 T + 201 T^{2} + 45 p T^{3} + 20093 T^{4} + 168956 T^{5} + 30316 p T^{6} + 8912308 T^{7} + 30316 p^{2} T^{8} + 168956 p^{2} T^{9} + 20093 p^{3} T^{10} + 45 p^{5} T^{11} + 201 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - T + 182 T^{2} - 218 T^{3} + 16844 T^{4} - 19671 T^{5} + 1036531 T^{6} - 1043276 T^{7} + 1036531 p T^{8} - 19671 p^{2} T^{9} + 16844 p^{3} T^{10} - 218 p^{4} T^{11} + 182 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 15 T + 224 T^{2} + 1922 T^{3} + 18209 T^{4} + 121906 T^{5} + 945783 T^{6} + 5714242 T^{7} + 945783 p T^{8} + 121906 p^{2} T^{9} + 18209 p^{3} T^{10} + 1922 p^{4} T^{11} + 224 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 5 T + 157 T^{2} - 829 T^{3} + 11547 T^{4} - 53344 T^{5} + 668902 T^{6} - 2485262 T^{7} + 668902 p T^{8} - 53344 p^{2} T^{9} + 11547 p^{3} T^{10} - 829 p^{4} T^{11} + 157 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 9 T + 186 T^{2} - 1198 T^{3} + 20517 T^{4} - 121334 T^{5} + 1651753 T^{6} - 8336746 T^{7} + 1651753 p T^{8} - 121334 p^{2} T^{9} + 20517 p^{3} T^{10} - 1198 p^{4} T^{11} + 186 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 13 T + 298 T^{2} + 3606 T^{3} + 45179 T^{4} + 461934 T^{5} + 4192407 T^{6} + 35565742 T^{7} + 4192407 p T^{8} + 461934 p^{2} T^{9} + 45179 p^{3} T^{10} + 3606 p^{4} T^{11} + 298 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 2 T + 296 T^{2} + 192 T^{3} + 43335 T^{4} + 4626 T^{5} + 4185784 T^{6} + 43160 T^{7} + 4185784 p T^{8} + 4626 p^{2} T^{9} + 43335 p^{3} T^{10} + 192 p^{4} T^{11} + 296 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + T + 262 T^{2} - 384 T^{3} + 28597 T^{4} - 158140 T^{5} + 1977913 T^{6} - 17852870 T^{7} + 1977913 p T^{8} - 158140 p^{2} T^{9} + 28597 p^{3} T^{10} - 384 p^{4} T^{11} + 262 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 42 T + 1098 T^{2} + 21114 T^{3} + 324387 T^{4} + 4129728 T^{5} + 44505698 T^{6} + 410621920 T^{7} + 44505698 p T^{8} + 4129728 p^{2} T^{9} + 324387 p^{3} T^{10} + 21114 p^{4} T^{11} + 1098 p^{5} T^{12} + 42 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 3 T + 508 T^{2} + 1326 T^{3} + 113890 T^{4} + 251213 T^{5} + 14588391 T^{6} + 26098308 T^{7} + 14588391 p T^{8} + 251213 p^{2} T^{9} + 113890 p^{3} T^{10} + 1326 p^{4} T^{11} + 508 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 12 T + 336 T^{2} - 2788 T^{3} + 45625 T^{4} - 253516 T^{5} + 3685578 T^{6} - 17373720 T^{7} + 3685578 p T^{8} - 253516 p^{2} T^{9} + 45625 p^{3} T^{10} - 2788 p^{4} T^{11} + 336 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 41 T + 1156 T^{2} + 23234 T^{3} + 383262 T^{4} + 5212647 T^{5} + 61258037 T^{6} + 617487548 T^{7} + 61258037 p T^{8} + 5212647 p^{2} T^{9} + 383262 p^{3} T^{10} + 23234 p^{4} T^{11} + 1156 p^{5} T^{12} + 41 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 5 T + 386 T^{2} + 1858 T^{3} + 79769 T^{4} + 324422 T^{5} + 10781099 T^{6} + 38136614 T^{7} + 10781099 p T^{8} + 324422 p^{2} T^{9} + 79769 p^{3} T^{10} + 1858 p^{4} T^{11} + 386 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.87345476498182212315065881715, −3.87066652500247498777413638520, −3.81955395799156122817363021318, −3.77566766607275378084739557709, −3.60532894495688111545453566714, −3.17673541548510467442271456974, −3.12971419921599556329429941823, −3.10358445148559779950181261112, −3.06952284517280482482597338341, −2.95605182937106734468023450883, −2.71327303288960919899472453448, −2.63363311433670156875393507634, −2.44821760121683802811007662478, −2.37149961156804437910638292276, −2.30138642366607745263560870633, −2.24282268720171340847015363200, −2.11139529140974214251916375536, −1.90206873783794701925164849660, −1.63123258421354632963394552681, −1.57222126765185362242193136912, −1.45052122443862501107392573360, −1.27990606945342916641619244975, −1.25156295569949067054141038458, −0.956556406745872160110082077235, −0.859997741606716928547514637424, 0, 0, 0, 0, 0, 0, 0, 0.859997741606716928547514637424, 0.956556406745872160110082077235, 1.25156295569949067054141038458, 1.27990606945342916641619244975, 1.45052122443862501107392573360, 1.57222126765185362242193136912, 1.63123258421354632963394552681, 1.90206873783794701925164849660, 2.11139529140974214251916375536, 2.24282268720171340847015363200, 2.30138642366607745263560870633, 2.37149961156804437910638292276, 2.44821760121683802811007662478, 2.63363311433670156875393507634, 2.71327303288960919899472453448, 2.95605182937106734468023450883, 3.06952284517280482482597338341, 3.10358445148559779950181261112, 3.12971419921599556329429941823, 3.17673541548510467442271456974, 3.60532894495688111545453566714, 3.77566766607275378084739557709, 3.81955395799156122817363021318, 3.87066652500247498777413638520, 3.87345476498182212315065881715
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.