Dirichlet series
| L(s) = 1 | + 4·2-s + 6·4-s + 4·8-s − 4·13-s + 16-s − 16·26-s − 4·49-s − 24·52-s + 4·73-s + 4·97-s − 16·98-s − 16·104-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 16·146-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 4·2-s + 6·4-s + 4·8-s − 4·13-s + 16-s − 16·26-s − 4·49-s − 24·52-s + 4·73-s + 4·97-s − 16·98-s − 16·104-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 16·146-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128} \cdot 5^{64} \cdot 137^{64}\right)^{s/2} \, \Gamma_{\C}(s)^{64} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128} \cdot 5^{64} \cdot 137^{64}\right)^{s/2} \, \Gamma_{\C}(s)^{64} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(128\) |
| Conductor: | \(2^{128} \cdot 5^{64} \cdot 137^{64}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.98990\times 10^{8}\) |
| Root analytic conductor: | \(1.16937\) |
| Motivic weight: | \(0\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((128,\ 2^{128} \cdot 5^{64} \cdot 137^{64} ,\ ( \ : [0]^{64} ),\ 1 )\) |
Particular Values
| \(L(\frac{1}{2})\) | \(\approx\) | \(0.006783750533\) |
| \(L(\frac12)\) | \(\approx\) | \(0.006783750533\) |
| \(L(1)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )^{4} \) |
| 5 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} \) | |
| 137 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} )^{2} \) | |
| good | 3 | \( 1 - T^{8} + T^{16} - T^{24} + T^{32} - T^{40} + T^{48} - T^{56} + T^{64} - T^{72} + T^{80} - T^{88} + T^{96} - T^{104} + T^{112} - T^{120} + T^{128} \) |
| 7 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )^{4} \) | |
| 11 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} )^{2} \) | |
| 13 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} ) \) | |
| 17 | \( ( 1 + T^{4} )^{16}( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} ) \) | |
| 19 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} )^{2} \) | |
| 23 | \( 1 - T^{8} + T^{16} - T^{24} + T^{32} - T^{40} + T^{48} - T^{56} + T^{64} - T^{72} + T^{80} - T^{88} + T^{96} - T^{104} + T^{112} - T^{120} + T^{128} \) | |
| 29 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} ) \) | |
| 31 | \( 1 - T^{8} + T^{16} - T^{24} + T^{32} - T^{40} + T^{48} - T^{56} + T^{64} - T^{72} + T^{80} - T^{88} + T^{96} - T^{104} + T^{112} - T^{120} + T^{128} \) | |
| 37 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} )^{2} \) | |
| 41 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} ) \) | |
| 43 | \( 1 - T^{8} + T^{16} - T^{24} + T^{32} - T^{40} + T^{48} - T^{56} + T^{64} - T^{72} + T^{80} - T^{88} + T^{96} - T^{104} + T^{112} - T^{120} + T^{128} \) | |
| 47 | \( 1 - T^{8} + T^{16} - T^{24} + T^{32} - T^{40} + T^{48} - T^{56} + T^{64} - T^{72} + T^{80} - T^{88} + T^{96} - T^{104} + T^{112} - T^{120} + T^{128} \) | |
| 53 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} ) \) | |
| 59 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} )^{4} \) | |
| 61 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} )^{2} \) | |
| 67 | \( 1 - T^{8} + T^{16} - T^{24} + T^{32} - T^{40} + T^{48} - T^{56} + T^{64} - T^{72} + T^{80} - T^{88} + T^{96} - T^{104} + T^{112} - T^{120} + T^{128} \) | |
| 71 | \( 1 - T^{8} + T^{16} - T^{24} + T^{32} - T^{40} + T^{48} - T^{56} + T^{64} - T^{72} + T^{80} - T^{88} + T^{96} - T^{104} + T^{112} - T^{120} + T^{128} \) | |
| 73 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} )^{2} \) | |
| 79 | \( 1 - T^{8} + T^{16} - T^{24} + T^{32} - T^{40} + T^{48} - T^{56} + T^{64} - T^{72} + T^{80} - T^{88} + T^{96} - T^{104} + T^{112} - T^{120} + T^{128} \) | |
| 83 | \( 1 - T^{8} + T^{16} - T^{24} + T^{32} - T^{40} + T^{48} - T^{56} + T^{64} - T^{72} + T^{80} - T^{88} + T^{96} - T^{104} + T^{112} - T^{120} + T^{128} \) | |
| 89 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} ) \) | |
| 97 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} ) \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{128} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.01838033242510311772077862946, −0.984683061678026729738992449890, −0.969909730416675131386674352162, −0.967237763674474901442029671550, −0.960258784694577686447583486646, −0.948104909519925723915944654858, −0.874212452775505896881701150992, −0.851381939055099244737008619204, −0.822624288133893817292820990426, −0.818838585906879415648364821556, −0.805692381968485502743319479447, −0.804817366917510367202796133111, −0.77767180654633082045929563386, −0.61952025170652376716803631832, −0.61774675751212954298782140727, −0.58740971383628961419599586052, −0.46958937697875096546171925663, −0.46486079764329245209735284496, −0.45512950257694576769555391888, −0.44771882815684825692217122000, −0.44554529091858023764650774268, −0.38490453251846944840116068796, −0.35512440441971610725202233724, −0.28323741097730388500538683151, −0.00356249611192069994215029566, 0.00356249611192069994215029566, 0.28323741097730388500538683151, 0.35512440441971610725202233724, 0.38490453251846944840116068796, 0.44554529091858023764650774268, 0.44771882815684825692217122000, 0.45512950257694576769555391888, 0.46486079764329245209735284496, 0.46958937697875096546171925663, 0.58740971383628961419599586052, 0.61774675751212954298782140727, 0.61952025170652376716803631832, 0.77767180654633082045929563386, 0.804817366917510367202796133111, 0.805692381968485502743319479447, 0.818838585906879415648364821556, 0.822624288133893817292820990426, 0.851381939055099244737008619204, 0.874212452775505896881701150992, 0.948104909519925723915944654858, 0.960258784694577686447583486646, 0.967237763674474901442029671550, 0.969909730416675131386674352162, 0.984683061678026729738992449890, 1.01838033242510311772077862946
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.