Dirichlet series
| L(s) = 1 | + 2·2-s + 3·4-s + 2·8-s + 9-s − 11·11-s + 16-s + 2·18-s − 22·22-s + 2·23-s + 25-s − 4·29-s + 3·36-s + 2·37-s − 4·43-s − 33·44-s + 4·46-s + 2·50-s − 11·53-s − 8·58-s + 2·67-s − 4·71-s + 2·72-s + 4·74-s + 79-s + 81-s − 8·86-s − 22·88-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s + 2·8-s + 9-s − 11·11-s + 16-s + 2·18-s − 22·22-s + 2·23-s + 25-s − 4·29-s + 3·36-s + 2·37-s − 4·43-s − 33·44-s + 4·46-s + 2·50-s − 11·53-s − 8·58-s + 2·67-s − 4·71-s + 2·72-s + 4·74-s + 79-s + 81-s − 8·86-s − 22·88-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{48} \cdot 79^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{48} \cdot 79^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(48\) |
| Conductor: | \(7^{48} \cdot 79^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.30343\times 10^{6}\) |
| Root analytic conductor: | \(1.38992\) |
| Motivic weight: | \(0\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((48,\ 7^{48} \cdot 79^{24} ,\ ( \ : [0]^{24} ),\ 1 )\) |
Particular Values
| \(L(\frac{1}{2})\) | \(\approx\) | \(0.006642784366\) |
| \(L(\frac12)\) | \(\approx\) | \(0.006642784366\) |
| \(L(1)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 \) |
| 79 | \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{14} + T^{15} - T^{17} + T^{18} - T^{20} + T^{21} - T^{23} + T^{24} \) | |
| good | 2 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{14} + T^{15} - T^{17} + T^{18} - T^{20} + T^{21} - T^{23} + T^{24} )^{2} \) |
| 3 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{14} + T^{15} - T^{17} + T^{18} - T^{20} + T^{21} - T^{23} + T^{24} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} - T^{14} - T^{15} + T^{17} + T^{18} - T^{20} - T^{21} + T^{23} + T^{24} ) \) | |
| 5 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{14} + T^{15} - T^{17} + T^{18} - T^{20} + T^{21} - T^{23} + T^{24} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} - T^{14} - T^{15} + T^{17} + T^{18} - T^{20} - T^{21} + T^{23} + T^{24} ) \) | |
| 11 | \( ( 1 + T + T^{2} )^{12}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{14} + T^{15} - T^{17} + T^{18} - T^{20} + T^{21} - T^{23} + T^{24} ) \) | |
| 13 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2} \) | |
| 17 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{14} + T^{15} - T^{17} + T^{18} - T^{20} + T^{21} - T^{23} + T^{24} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} - T^{14} - T^{15} + T^{17} + T^{18} - T^{20} - T^{21} + T^{23} + T^{24} ) \) | |
| 19 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{14} + T^{15} - T^{17} + T^{18} - T^{20} + T^{21} - T^{23} + T^{24} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} - T^{14} - T^{15} + T^{17} + T^{18} - T^{20} - T^{21} + T^{23} + T^{24} ) \) | |
| 23 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{14} + T^{15} - T^{17} + T^{18} - T^{20} + T^{21} - T^{23} + T^{24} )^{2} \) | |
| 29 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{4} \) | |
| 31 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{14} + T^{15} - T^{17} + T^{18} - T^{20} + T^{21} - T^{23} + T^{24} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} - T^{14} - T^{15} + T^{17} + T^{18} - T^{20} - T^{21} + T^{23} + T^{24} ) \) | |
| 37 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{14} + T^{15} - T^{17} + T^{18} - T^{20} + T^{21} - T^{23} + T^{24} )^{2} \) | |
| 41 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2} \) | |
| 43 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{4} \) | |
| 47 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{14} + T^{15} - T^{17} + T^{18} - T^{20} + T^{21} - T^{23} + T^{24} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} - T^{14} - T^{15} + T^{17} + T^{18} - T^{20} - T^{21} + T^{23} + T^{24} ) \) | |
| 53 | \( ( 1 + T + T^{2} )^{12}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{14} + T^{15} - T^{17} + T^{18} - T^{20} + T^{21} - T^{23} + T^{24} ) \) | |
| 59 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{14} + T^{15} - T^{17} + T^{18} - T^{20} + T^{21} - T^{23} + T^{24} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} - T^{14} - T^{15} + T^{17} + T^{18} - T^{20} - T^{21} + T^{23} + T^{24} ) \) | |
| 61 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{14} + T^{15} - T^{17} + T^{18} - T^{20} + T^{21} - T^{23} + T^{24} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} - T^{14} - T^{15} + T^{17} + T^{18} - T^{20} - T^{21} + T^{23} + T^{24} ) \) | |
| 67 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{14} + T^{15} - T^{17} + T^{18} - T^{20} + T^{21} - T^{23} + T^{24} )^{2} \) | |
| 71 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{4} \) | |
| 73 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{14} + T^{15} - T^{17} + T^{18} - T^{20} + T^{21} - T^{23} + T^{24} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} - T^{14} - T^{15} + T^{17} + T^{18} - T^{20} - T^{21} + T^{23} + T^{24} ) \) | |
| 83 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2} \) | |
| 89 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{14} + T^{15} - T^{17} + T^{18} - T^{20} + T^{21} - T^{23} + T^{24} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} - T^{14} - T^{15} + T^{17} + T^{18} - T^{20} - T^{21} + T^{23} + T^{24} ) \) | |
| 97 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{48} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.76447057053797995469928310387, −1.74766981490873180826448064011, −1.73480047398945739364261811829, −1.71294251450463248090125756537, −1.67739266961256975639268382519, −1.55346998676396176869116343141, −1.53234396984097683258901354779, −1.41168979521748207234957154383, −1.39988702654315535890665161363, −1.36868344785586983529525569064, −1.27711261027246942143264630869, −1.22061092791992141730578050350, −1.18890998726257884162513671777, −1.18756735063081972846037068301, −1.07969457062629045044411247911, −1.00436436613886064939918080265, −0.77407503199687616767547278833, −0.69340601176719338492940193887, −0.65677029537615908598749756349, −0.56482749744519270425344110137, −0.52039554997253915103711568766, −0.47994425554680791114407763729, −0.35463081202802459905876258731, −0.33238277873308189365136350690, −0.01345343807104193562484764774, 0.01345343807104193562484764774, 0.33238277873308189365136350690, 0.35463081202802459905876258731, 0.47994425554680791114407763729, 0.52039554997253915103711568766, 0.56482749744519270425344110137, 0.65677029537615908598749756349, 0.69340601176719338492940193887, 0.77407503199687616767547278833, 1.00436436613886064939918080265, 1.07969457062629045044411247911, 1.18756735063081972846037068301, 1.18890998726257884162513671777, 1.22061092791992141730578050350, 1.27711261027246942143264630869, 1.36868344785586983529525569064, 1.39988702654315535890665161363, 1.41168979521748207234957154383, 1.53234396984097683258901354779, 1.55346998676396176869116343141, 1.67739266961256975639268382519, 1.71294251450463248090125756537, 1.73480047398945739364261811829, 1.74766981490873180826448064011, 1.76447057053797995469928310387
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.