Properties

Degree 88
Conductor $ 2^{88} \cdot 3^{88} $
Sign $1$
Motivic weight 4
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·5-s + 41·8-s − 30·9-s + 2·10-s − 2·13-s − 41·16-s − 56·17-s + 30·18-s + 6.00e3·25-s + 2·26-s + 526·29-s − 24·32-s + 56·34-s − 8·37-s − 82·40-s − 2.76e3·41-s + 60·45-s − 2.46e4·49-s − 6.00e3·50-s − 1.00e4·53-s − 526·58-s − 2·61-s + 2.36e3·64-s + 4·65-s − 1.23e3·72-s − 3.41e3·73-s + ⋯
L(s)  = 1  − 1/4·2-s − 0.0799·5-s + 0.640·8-s − 0.370·9-s + 1/50·10-s − 0.0118·13-s − 0.160·16-s − 0.193·17-s + 5/54·18-s + 9.60·25-s + 0.00295·26-s + 0.625·29-s − 0.0234·32-s + 0.0484·34-s − 0.00584·37-s − 0.0512·40-s − 1.64·41-s + 0.0296·45-s − 10.2·49-s − 2.40·50-s − 3.59·53-s − 0.156·58-s − 0.000537·61-s + 0.578·64-s + 0.000946·65-s − 0.237·72-s − 0.641·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{88} \cdot 3^{88}\right)^{s/2} \, \Gamma_{\C}(s)^{44} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{88} \cdot 3^{88}\right)^{s/2} \, \Gamma_{\C}(s+2)^{44} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(88\)
\( N \)  =  \(2^{88} \cdot 3^{88}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(4\)
character  :  induced by $\chi_{36} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((88,\ 2^{88} \cdot 3^{88} ,\ ( \ : [2]^{44} ),\ 1 )\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.822528\)
\(L(\frac12)\)  \(\approx\)  \(0.822528\)
\(L(3)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 88. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 87.
$p$$F_p(T)$
bad2 \( 1 + T + T^{2} - 5 p^{3} T^{3} - 5 p^{3} T^{4} - p^{4} T^{5} - 85 p^{3} T^{6} + 247 p^{6} T^{7} + 547 p^{4} T^{8} + 3939 p^{5} T^{9} - 2013 p^{7} T^{10} - 12087 p^{7} T^{11} - 4341 p^{8} T^{12} - 1287 p^{14} T^{13} + 87423 p^{11} T^{14} - 103275 p^{14} T^{15} + 25305 p^{15} T^{16} + 29703 p^{17} T^{17} + 1135705 p^{16} T^{18} + 623695 p^{18} T^{19} - 208175 p^{20} T^{20} - 6155 p^{23} T^{21} - 942233 p^{25} T^{22} - 6155 p^{27} T^{23} - 208175 p^{28} T^{24} + 623695 p^{30} T^{25} + 1135705 p^{32} T^{26} + 29703 p^{37} T^{27} + 25305 p^{39} T^{28} - 103275 p^{42} T^{29} + 87423 p^{43} T^{30} - 1287 p^{50} T^{31} - 4341 p^{48} T^{32} - 12087 p^{51} T^{33} - 2013 p^{55} T^{34} + 3939 p^{57} T^{35} + 547 p^{60} T^{36} + 247 p^{66} T^{37} - 85 p^{67} T^{38} - p^{72} T^{39} - 5 p^{75} T^{40} - 5 p^{79} T^{41} + p^{80} T^{42} + p^{84} T^{43} + p^{88} T^{44} \)
3 \( 1 + 10 p T^{2} + 247 p T^{4} - 49516 p^{3} T^{6} - 350534 p^{4} T^{8} - 5953112 p^{6} T^{10} + 123880432 p^{8} T^{12} - 21687668 p^{11} T^{14} + 2584337665 p^{13} T^{16} - 2885051554 p^{17} T^{18} + 5654051053 p^{20} T^{20} - 7872384512 p^{24} T^{22} + 5654051053 p^{28} T^{24} - 2885051554 p^{33} T^{26} + 2584337665 p^{37} T^{28} - 21687668 p^{43} T^{30} + 123880432 p^{48} T^{32} - 5953112 p^{54} T^{34} - 350534 p^{60} T^{36} - 49516 p^{67} T^{38} + 247 p^{73} T^{40} + 10 p^{81} T^{42} + p^{88} T^{44} \)
good5 \( 1 + 2T - 5.99e3T^{2} - 2.30e4T^{3} + 1.76e7T^{4} + 1.27e8T^{5} - 3.42e10T^{6} - 4.32e11T^{7} + 4.85e13T^{8} + 1.02e15T^{9} - 5.27e16T^{10} - 1.79e18T^{11} + 4.26e19T^{12} + 2.49e21T^{13} - 1.98e22T^{14} - 2.82e24T^{15} - 9.31e24T^{16} + 2.65e27T^{17} + 3.62e28T^{18} - 2.08e30T^{19} - 5.38e31T^{20} + 1.29e33T^{21} + 5.92e34T^{22} - 5.22e35T^{23} - 5.40e37T^{24} - 8.55e37T^{25} + 4.23e40T^{26} + 4.57e41T^{27} - 2.85e43T^{28} - 6.05e44T^{29} + 1.60e46T^{30} + 5.89e47T^{31} - 6.61e48T^{32} - 4.82e50T^{33} + 5.68e50T^{34} + 3.44e53T^{35} + 2.51e54T^{36} - 2.15e56T^{37} - 3.50e57T^{38} + 1.15e59T^{39} + 3.27e60T^{40}+O(T^{41}) \)
7 \( 1 + 2.46e4T^{2} + 3.02e8T^{4} + 2.51e12T^{6} + 1.63e16T^{8} + 9.06e19T^{10} + 4.45e23T^{12} + 2.00e27T^{14} + 8.33e30T^{16} + 3.25e34T^{18} + 1.20e38T^{20} + 4.24e41T^{22} + 1.42e45T^{24} + 4.61e48T^{26} + 1.43e52T^{28} + 4.32e55T^{30} + 1.25e59T^{32}+O(T^{34}) \)
11 \( 1 + 1.61e5T^{2} + 1.34e10T^{4} + 7.74e14T^{6} + 3.44e19T^{8} + 1.25e24T^{10} + 3.90e28T^{12} + 1.04e33T^{14} + 2.47e37T^{16} + 5.12e41T^{18} + 9.27e45T^{20} + 1.44e50T^{22} + 1.83e54T^{24} + 1.62e58T^{26}+O(T^{28}) \)
13 \( 1 + 2T - 3.16e5T^{2} - 2.82e6T^{3} + 5.03e10T^{4} + 4.73e11T^{5} - 5.37e15T^{6} - 9.32e15T^{7} + 4.30e20T^{8} - 6.58e21T^{9} - 2.76e25T^{10} + 1.18e27T^{11} + 1.46e30T^{12} - 1.20e32T^{13} - 6.54e34T^{14} + 9.00e36T^{15} + 2.38e39T^{16} - 5.35e41T^{17} - 6.37e43T^{18} + 2.64e46T^{19} + 6.16e47T^{20} - 1.10e51T^{21} + 6.10e52T^{22} + 3.82e55T^{23} - 5.33e57T^{24} - 1.05e60T^{25} + 2.79e62T^{26}+O(T^{27}) \)
17 \( 1 + 56T + 1.92e6T^{2} - 3.51e7T^{3} + 1.84e12T^{4} - 1.61e14T^{5} + 1.18e18T^{6} - 1.78e20T^{7} + 5.80e23T^{8} - 1.19e26T^{9} + 2.30e29T^{10} - 5.81e31T^{11} + 7.77e34T^{12} - 2.23e37T^{13} + 2.28e40T^{14} - 7.17e42T^{15} + 5.92e45T^{16} - 1.97e48T^{17} + 1.38e51T^{18} - 4.75e53T^{19} + 2.92e56T^{20} - 1.01e59T^{21} + 5.64e61T^{22} - 1.96e64T^{23}+O(T^{24}) \)
19 \( 1 - 3.05e6T^{2} + 4.61e12T^{4} - 4.61e18T^{6} + 3.44e24T^{8} - 2.03e30T^{10} + 9.97e35T^{12} - 4.17e41T^{14} + 1.52e47T^{16} - 4.92e52T^{18} + 1.43e58T^{20} - 3.78e63T^{22}+O(T^{23}) \)
23 \( 1 + 3.09e6T^{2} + 4.82e12T^{4} + 5.07e18T^{6} + 4.06e24T^{8} + 2.63e30T^{10} + 1.43e36T^{12} + 6.62e41T^{14} + 2.62e47T^{16} + 8.80e52T^{18} + 2.40e58T^{20}+O(T^{22}) \)
29 \( 1 - 526T - 7.51e6T^{2} + 5.23e9T^{3} + 2.63e13T^{4} - 2.33e16T^{5} - 5.73e19T^{6} + 6.40e22T^{7} + 8.82e25T^{8} - 1.23e29T^{9} - 1.03e32T^{10} + 1.84e35T^{11} + 9.90e37T^{12} - 2.23e41T^{13} - 8.18e43T^{14} + 2.29e47T^{15} + 6.50e49T^{16} - 2.04e53T^{17} - 5.97e55T^{18} + 1.62e59T^{19} + 6.83e61T^{20}+O(T^{21}) \)
31 \( 1 + 1.05e7T^{2} + 5.42e13T^{4} + 1.83e20T^{6} + 4.62e26T^{8} + 9.44e32T^{10} + 1.63e39T^{12} + 2.48e45T^{14} + 3.39e51T^{16} + 4.26e57T^{18}+O(T^{20}) \)
37 \( 1 + 8T + 4.26e7T^{2} - 1.81e9T^{3} + 9.22e14T^{4} - 9.71e16T^{5} + 1.34e22T^{6} - 2.36e24T^{7} + 1.48e29T^{8} - 3.73e31T^{9} + 1.32e36T^{10} - 4.33e38T^{11} + 9.87e42T^{12} - 3.99e45T^{13} + 6.36e49T^{14} - 3.03e52T^{15} + 3.61e56T^{16} - 1.96e59T^{17} + 1.82e63T^{18}+O(T^{19}) \)
41 \( 1 + 2.76e3T - 2.70e7T^{2} - 4.54e10T^{3} + 4.61e14T^{4} + 2.55e17T^{5} - 5.34e21T^{6} + 2.36e24T^{7} + 4.38e28T^{8} - 6.11e31T^{9} - 2.46e35T^{10} + 6.50e38T^{11} + 7.84e41T^{12} - 4.45e45T^{13} + 8.83e47T^{14} + 2.07e52T^{15} - 2.84e55T^{16} - 5.74e58T^{17} + 1.92e62T^{18}+O(T^{19}) \)
43 \( 1 + 3.81e7T^{2} + 7.22e14T^{4} + 9.09e21T^{6} + 8.59e28T^{8} + 6.54e35T^{10} + 4.22e42T^{12} + 2.39e49T^{14} + 1.22e56T^{16} + 5.81e62T^{18}+O(T^{19}) \)
47 \( 1 + 4.81e7T^{2} + 1.13e15T^{4} + 1.77e22T^{6} + 2.10e29T^{8} + 2.05e36T^{10} + 1.75e43T^{12} + 1.36e50T^{14} + 1.00e57T^{16}+O(T^{18}) \)
53 \( 1 + 1.00e4T + 2.11e8T^{2} + 1.80e12T^{3} + 2.14e16T^{4} + 1.58e20T^{5} + 1.37e24T^{6} + 8.92e27T^{7} + 6.35e31T^{8} + 3.65e35T^{9} + 2.22e39T^{10} + 1.15e43T^{11} + 6.19e46T^{12} + 2.91e50T^{13} + 1.40e54T^{14} + 6.02e57T^{15} + 2.65e61T^{16} + 1.04e65T^{17}+O(T^{18}) \)
59 \( 1 + 1.33e8T^{2} + 8.87e15T^{4} + 3.92e23T^{6} + 1.28e31T^{8} + 3.26e38T^{10} + 6.60e45T^{12} + 1.06e53T^{14} + 1.34e60T^{16}+O(T^{17}) \)
61 \( 1 + 2T - 1.88e8T^{2} + 1.16e11T^{3} + 1.82e16T^{4} - 2.05e19T^{5} - 1.18e24T^{6} + 1.82e27T^{7} + 5.76e31T^{8} - 1.07e35T^{9} - 2.21e39T^{10} + 4.62e42T^{11} + 6.95e46T^{12} - 1.52e50T^{13} - 1.83e54T^{14} + 3.91e57T^{15} + 4.12e61T^{16}+O(T^{17}) \)
67 \( 1 + 2.73e8T^{2} + 3.82e16T^{4} + 3.60e24T^{6} + 2.55e32T^{8} + 1.43e40T^{10} + 6.64e47T^{12} + 2.57e55T^{14} + 8.46e62T^{16}+O(T^{17}) \)
71 \( 1 - 6.20e8T^{2} + 1.91e17T^{4} - 3.89e25T^{6} + 5.88e33T^{8} - 7.04e41T^{10} + 6.93e49T^{12} - 5.76e57T^{14} + 4.13e65T^{16}+O(T^{17}) \)
73 \( 1 + 3.41e3T + 8.52e8T^{2} + 2.10e12T^{3} + 3.55e17T^{4} + 5.44e20T^{5} + 9.68e25T^{6} + 5.77e28T^{7} + 1.94e34T^{8} - 6.60e36T^{9} + 3.09e42T^{10} - 3.90e45T^{11} + 4.04e50T^{12} - 8.74e53T^{13} + 4.50e58T^{14} - 1.35e62T^{15} + 4.36e66T^{16}+O(T^{17}) \)
79 \( 1 + 5.82e8T^{2} + 1.70e17T^{4} + 3.37e25T^{6} + 5.10e33T^{8} + 6.30e41T^{10} + 6.64e49T^{12} + 6.15e57T^{14}+O(T^{16}) \)
83 \( 1 + 7.57e8T^{2} + 2.92e17T^{4} + 7.64e25T^{6} + 1.52e34T^{8} + 2.47e42T^{10} + 3.41e50T^{12} + 4.09e58T^{14}+O(T^{16}) \)
89 \( 1 - 1.30e4T + 2.04e9T^{2} - 2.57e13T^{3} + 2.06e18T^{4} - 2.51e22T^{5} + 1.38e27T^{6} - 1.62e31T^{7} + 6.85e35T^{8} - 7.78e39T^{9} + 2.69e44T^{10} - 2.95e48T^{11} + 8.74e52T^{12} - 9.27e56T^{13} + 2.40e61T^{14} - 2.46e65T^{15}+O(T^{16}) \)
97 \( 1 - 5.63e3T - 9.11e8T^{2} + 9.88e12T^{3} + 3.92e17T^{4} - 6.67e21T^{5} - 9.39e25T^{6} + 2.65e30T^{7} + 8.72e33T^{8} - 7.14e38T^{9} + 2.51e42T^{10} + 1.35e47T^{11} - 1.33e51T^{12} - 1.67e55T^{13} + 3.35e59T^{14} + 7.15e62T^{15}+O(T^{16}) \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{88} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.16089593979534874751017280059, −2.14373438311457880157550492481, −2.01099162173915924132802087843, −1.93021883501458398461808737503, −1.78236657500407336433791857894, −1.71285057113933243888958110916, −1.58555972493759118783091986684, −1.54599612164484243544478582671, −1.53990652692313975715810553501, −1.53319744853450624304702618224, −1.49093064112495424875125885165, −1.44019910810240842522800285201, −1.21567446135793135053540431719, −1.18674798133047241269007210662, −1.03695869917525662800809481985, −1.02998462060073872800935734061, −0.952064631723440077198451708152, −0.837272819582801971131818610122, −0.70367603191526192561688938739, −0.61499935531518778601015653380, −0.46226052549709142310936958066, −0.31381882753901200297430851551, −0.29976144366264578282958425453, −0.083892274039083929075577699879, −0.06864290152603763806765706979, 0.06864290152603763806765706979, 0.083892274039083929075577699879, 0.29976144366264578282958425453, 0.31381882753901200297430851551, 0.46226052549709142310936958066, 0.61499935531518778601015653380, 0.70367603191526192561688938739, 0.837272819582801971131818610122, 0.952064631723440077198451708152, 1.02998462060073872800935734061, 1.03695869917525662800809481985, 1.18674798133047241269007210662, 1.21567446135793135053540431719, 1.44019910810240842522800285201, 1.49093064112495424875125885165, 1.53319744853450624304702618224, 1.53990652692313975715810553501, 1.54599612164484243544478582671, 1.58555972493759118783091986684, 1.71285057113933243888958110916, 1.78236657500407336433791857894, 1.93021883501458398461808737503, 2.01099162173915924132802087843, 2.14373438311457880157550492481, 2.16089593979534874751017280059

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.