Dirichlet series
| L(s) = 1 | − 2·2-s + 4-s + 6·7-s + 2·8-s + 9-s − 2·11-s − 12·14-s − 4·16-s − 2·18-s + 4·22-s − 23-s + 25-s + 6·28-s − 29-s + 4·32-s + 36-s − 5·37-s + 43-s − 2·44-s + 2·46-s + 21·49-s − 2·50-s + 10·53-s + 12·56-s + 2·58-s + 6·63-s − 64-s + ⋯ |
| L(s) = 1 | − 2·2-s + 4-s + 6·7-s + 2·8-s + 9-s − 2·11-s − 12·14-s − 4·16-s − 2·18-s + 4·22-s − 23-s + 25-s + 6·28-s − 29-s + 4·32-s + 36-s − 5·37-s + 43-s − 2·44-s + 2·46-s + 21·49-s − 2·50-s + 10·53-s + 12·56-s + 2·58-s + 6·63-s − 64-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{48} \cdot 11^{48} \cdot 43^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{48} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{48} \cdot 11^{48} \cdot 43^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{48} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(96\) |
| Conductor: | \(7^{48} \cdot 11^{48} \cdot 43^{48}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.94837\times 10^{10}\) |
| Root analytic conductor: | \(1.28545\) |
| Motivic weight: | \(0\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((96,\ 7^{48} \cdot 11^{48} \cdot 43^{48} ,\ ( \ : [0]^{48} ),\ 1 )\) |
Particular Values
| \(L(\frac{1}{2})\) | \(\approx\) | \(0.1056755913\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1056755913\) |
| \(L(1)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{6} \) |
| 11 | \( ( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} )^{2} \) | |
| 43 | \( 1 - T + T^{2} + T^{5} - T^{6} + 2 T^{7} - T^{8} + T^{9} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} - T^{20} - T^{22} - T^{24} - T^{26} - T^{28} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} + T^{39} - T^{40} + 2 T^{41} - T^{42} + T^{43} + T^{46} - T^{47} + T^{48} \) | |
| good | 2 | \( ( 1 + T + T^{2} - T^{5} - T^{6} - p T^{7} - T^{8} - T^{9} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} - T^{20} - T^{22} - T^{24} - T^{26} - T^{28} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} - T^{39} - T^{40} - p T^{41} - T^{42} - T^{43} + T^{46} + T^{47} + T^{48} )^{2} \) |
| 3 | \( 1 - T^{2} + T^{4} + T^{10} - T^{12} + 2 T^{14} - T^{16} + T^{18} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} - T^{34} - T^{40} - T^{44} - T^{48} - T^{52} - T^{56} - T^{62} + T^{64} - T^{66} + T^{68} - T^{70} + T^{72} + T^{78} - T^{80} + 2 T^{82} - T^{84} + T^{86} + T^{92} - T^{94} + T^{96} \) | |
| 5 | \( 1 - T^{2} + T^{4} + T^{10} - T^{12} + 2 T^{14} - T^{16} + T^{18} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} - T^{34} - T^{40} - T^{44} - T^{48} - T^{52} - T^{56} - T^{62} + T^{64} - T^{66} + T^{68} - T^{70} + T^{72} + T^{78} - T^{80} + 2 T^{82} - T^{84} + T^{86} + T^{92} - T^{94} + T^{96} \) | |
| 13 | \( 1 - T^{2} + T^{4} + T^{10} - T^{12} + 2 T^{14} - T^{16} + T^{18} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} - T^{34} - T^{40} - T^{44} - T^{48} - T^{52} - T^{56} - T^{62} + T^{64} - T^{66} + T^{68} - T^{70} + T^{72} + T^{78} - T^{80} + 2 T^{82} - T^{84} + T^{86} + T^{92} - T^{94} + T^{96} \) | |
| 17 | \( 1 - T^{2} + T^{4} + T^{10} - T^{12} + 2 T^{14} - T^{16} + T^{18} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} - T^{34} - T^{40} - T^{44} - T^{48} - T^{52} - T^{56} - T^{62} + T^{64} - T^{66} + T^{68} - T^{70} + T^{72} + T^{78} - T^{80} + 2 T^{82} - T^{84} + T^{86} + T^{92} - T^{94} + T^{96} \) | |
| 19 | \( ( 1 - T + T^{2} + T^{5} - T^{6} + 2 T^{7} - T^{8} + T^{9} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} - T^{20} - T^{22} - T^{24} - T^{26} - T^{28} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} + T^{39} - T^{40} + 2 T^{41} - T^{42} + T^{43} + T^{46} - T^{47} + T^{48} )( 1 + T + T^{2} - T^{5} - T^{6} - 2 T^{7} - T^{8} - T^{9} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} - T^{20} - T^{22} - T^{24} - T^{26} - T^{28} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} - T^{39} - T^{40} - 2 T^{41} - T^{42} - T^{43} + T^{46} + T^{47} + T^{48} ) \) | |
| 23 | \( ( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} )^{2}( 1 - T + T^{2} + T^{5} - T^{6} + 2 T^{7} - T^{8} + T^{9} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} - T^{20} - T^{22} - T^{24} - T^{26} - T^{28} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} + T^{39} - T^{40} + 2 T^{41} - T^{42} + T^{43} + T^{46} - T^{47} + T^{48} ) \) | |
| 29 | \( ( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} )^{2}( 1 - T + T^{2} + T^{5} - T^{6} + 2 T^{7} - T^{8} + T^{9} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} - T^{20} - T^{22} - T^{24} - T^{26} - T^{28} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} + T^{39} - T^{40} + 2 T^{41} - T^{42} + T^{43} + T^{46} - T^{47} + T^{48} ) \) | |
| 31 | \( ( 1 - T + T^{2} + T^{5} - T^{6} + 2 T^{7} - T^{8} + T^{9} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} - T^{20} - T^{22} - T^{24} - T^{26} - T^{28} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} + T^{39} - T^{40} + 2 T^{41} - T^{42} + T^{43} + T^{46} - T^{47} + T^{48} )( 1 + T + T^{2} - T^{5} - T^{6} - 2 T^{7} - T^{8} - T^{9} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} - T^{20} - T^{22} - T^{24} - T^{26} - T^{28} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} - T^{39} - T^{40} - 2 T^{41} - T^{42} - T^{43} + T^{46} + T^{47} + T^{48} ) \) | |
| 37 | \( ( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{4}( 1 + T + T^{2} - T^{5} - T^{6} - 2 T^{7} - T^{8} - T^{9} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} - T^{20} - T^{22} - T^{24} - T^{26} - T^{28} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} - T^{39} - T^{40} - 2 T^{41} - T^{42} - T^{43} + T^{46} + T^{47} + T^{48} ) \) | |
| 41 | \( ( 1 + T^{2} - T^{10} - T^{12} - T^{14} - T^{16} + T^{20} + T^{22} + T^{24} + T^{26} + T^{28} - T^{32} - T^{34} - T^{36} - T^{38} + T^{46} + T^{48} )^{2} \) | |
| 47 | \( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )^{2}( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} )^{2} \) | |
| 53 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{6}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{4} \) | |
| 59 | \( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )^{2}( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} )^{2} \) | |
| 61 | \( ( 1 - T + T^{2} + T^{5} - T^{6} + 2 T^{7} - T^{8} + T^{9} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} - T^{20} - T^{22} - T^{24} - T^{26} - T^{28} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} + T^{39} - T^{40} + 2 T^{41} - T^{42} + T^{43} + T^{46} - T^{47} + T^{48} )( 1 + T + T^{2} - T^{5} - T^{6} - 2 T^{7} - T^{8} - T^{9} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} - T^{20} - T^{22} - T^{24} - T^{26} - T^{28} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} - T^{39} - T^{40} - 2 T^{41} - T^{42} - T^{43} + T^{46} + T^{47} + T^{48} ) \) | |
| 67 | \( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )^{2}( 1 + T + T^{2} - T^{5} - T^{6} - 2 T^{7} - T^{8} - T^{9} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} - T^{20} - T^{22} - T^{24} - T^{26} - T^{28} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} - T^{39} - T^{40} - 2 T^{41} - T^{42} - T^{43} + T^{46} + T^{47} + T^{48} ) \) | |
| 71 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{8}( 1 - T + T^{2} + T^{5} - T^{6} + 2 T^{7} - T^{8} + T^{9} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} - T^{20} - T^{22} - T^{24} - T^{26} - T^{28} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} + T^{39} - T^{40} + 2 T^{41} - T^{42} + T^{43} + T^{46} - T^{47} + T^{48} ) \) | |
| 73 | \( ( 1 - T + T^{2} + T^{5} - T^{6} + 2 T^{7} - T^{8} + T^{9} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} - T^{20} - T^{22} - T^{24} - T^{26} - T^{28} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} + T^{39} - T^{40} + 2 T^{41} - T^{42} + T^{43} + T^{46} - T^{47} + T^{48} )( 1 + T + T^{2} - T^{5} - T^{6} - 2 T^{7} - T^{8} - T^{9} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} - T^{20} - T^{22} - T^{24} - T^{26} - T^{28} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} - T^{39} - T^{40} - 2 T^{41} - T^{42} - T^{43} + T^{46} + T^{47} + T^{48} ) \) | |
| 79 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{4}( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} )^{2} \) | |
| 83 | \( 1 - T^{2} + T^{4} + T^{10} - T^{12} + 2 T^{14} - T^{16} + T^{18} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} - T^{34} - T^{40} - T^{44} - T^{48} - T^{52} - T^{56} - T^{62} + T^{64} - T^{66} + T^{68} - T^{70} + T^{72} + T^{78} - T^{80} + 2 T^{82} - T^{84} + T^{86} + T^{92} - T^{94} + T^{96} \) | |
| 89 | \( ( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{4} \) | |
| 97 | \( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )^{2}( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{96} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.08204371802755680847856303075, −1.07206090678834801327697315172, −1.06399193300827268716943363617, −1.05553531672994746094489272706, −1.04583225367151963292817996475, −1.03925081926698751182129038198, −0.992601431247783988039548838842, −0.969136962757825995245941935252, −0.893707151201485122832732877616, −0.889239620016045217468450834875, −0.839703680472295738208274461584, −0.808595954716565554915711426547, −0.797813005435433457103783057949, −0.78700084947404000536410000752, −0.78363096289737004515204092076, −0.76389445382928560280063080540, −0.75976940101413194929045880748, −0.64945520157678066743724473144, −0.63599564825727959640845903409, −0.54046053600426233440273192316, −0.45492935297896685515071441205, −0.36457938986718500799643042908, −0.28437177316875692390713447733, −0.24105590630640827800994882164, −0.02883430124494093012985665418, 0.02883430124494093012985665418, 0.24105590630640827800994882164, 0.28437177316875692390713447733, 0.36457938986718500799643042908, 0.45492935297896685515071441205, 0.54046053600426233440273192316, 0.63599564825727959640845903409, 0.64945520157678066743724473144, 0.75976940101413194929045880748, 0.76389445382928560280063080540, 0.78363096289737004515204092076, 0.78700084947404000536410000752, 0.797813005435433457103783057949, 0.808595954716565554915711426547, 0.839703680472295738208274461584, 0.889239620016045217468450834875, 0.893707151201485122832732877616, 0.969136962757825995245941935252, 0.992601431247783988039548838842, 1.03925081926698751182129038198, 1.04583225367151963292817996475, 1.05553531672994746094489272706, 1.06399193300827268716943363617, 1.07206090678834801327697315172, 1.08204371802755680847856303075
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.