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Properties

Label 2-45-1.1-c21-0-5
Degree 22
Conductor 4545
Sign 11
Analytic cond. 125.764125.764
Root an. cond. 11.214411.2144
Motivic weight 2121
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.71e3·2-s + 8.50e5·4-s − 9.76e6·5-s − 6.24e8·7-s − 2.14e9·8-s − 1.67e10·10-s − 1.47e11·11-s − 5.50e11·13-s − 1.07e12·14-s − 5.45e12·16-s + 4.04e12·17-s + 1.22e12·19-s − 8.30e12·20-s − 2.53e14·22-s + 2.50e14·23-s + 9.53e13·25-s − 9.45e14·26-s − 5.31e14·28-s + 3.49e15·29-s − 2.93e15·31-s − 4.88e15·32-s + 6.95e15·34-s + 6.09e15·35-s + 4.06e16·37-s + 2.10e15·38-s + 2.09e16·40-s − 1.45e17·41-s + ⋯
L(s)  = 1  + 1.18·2-s + 0.405·4-s − 0.447·5-s − 0.835·7-s − 0.704·8-s − 0.530·10-s − 1.71·11-s − 1.10·13-s − 0.990·14-s − 1.24·16-s + 0.487·17-s + 0.0459·19-s − 0.181·20-s − 2.03·22-s + 1.26·23-s + 0.199·25-s − 1.31·26-s − 0.338·28-s + 1.54·29-s − 0.642·31-s − 0.766·32-s + 0.577·34-s + 0.373·35-s + 1.38·37-s + 0.0544·38-s + 0.315·40-s − 1.68·41-s + ⋯

Functional equation

Λ(s)=(45s/2 ΓC(s) L(s)=( Λ(22−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}
Λ(s)=(45s/2 ΓC(s+21/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4545    =    32⋅53^{2} \cdot 5
Sign: 11
Analytic conductor: 125.764125.764
Root analytic conductor: 11.214411.2144
Motivic weight: 2121
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 45, ( :21/2), 1)(2,\ 45,\ (\ :21/2),\ 1)

Particular Values

L(11)L(11) ≈\approx 1.4914231261.491423126
L(12)L(\frac12) ≈\approx 1.4914231261.491423126
L(232)L(\frac{23}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
5 1+9.76e6T 1 + 9.76e6T
good2 1−1.71e3T+2.09e6T2 1 - 1.71e3T + 2.09e6T^{2}
7 1+6.24e8T+5.58e17T2 1 + 6.24e8T + 5.58e17T^{2}
11 1+1.47e11T+7.40e21T2 1 + 1.47e11T + 7.40e21T^{2}
13 1+5.50e11T+2.47e23T2 1 + 5.50e11T + 2.47e23T^{2}
17 1−4.04e12T+6.90e25T2 1 - 4.04e12T + 6.90e25T^{2}
19 1−1.22e12T+7.14e26T2 1 - 1.22e12T + 7.14e26T^{2}
23 1−2.50e14T+3.94e28T2 1 - 2.50e14T + 3.94e28T^{2}
29 1−3.49e15T+5.13e30T2 1 - 3.49e15T + 5.13e30T^{2}
31 1+2.93e15T+2.08e31T2 1 + 2.93e15T + 2.08e31T^{2}
37 1−4.06e16T+8.55e32T2 1 - 4.06e16T + 8.55e32T^{2}
41 1+1.45e17T+7.38e33T2 1 + 1.45e17T + 7.38e33T^{2}
43 1−8.05e16T+2.00e34T2 1 - 8.05e16T + 2.00e34T^{2}
47 1+4.95e16T+1.30e35T2 1 + 4.95e16T + 1.30e35T^{2}
53 1−1.24e18T+1.62e36T2 1 - 1.24e18T + 1.62e36T^{2}
59 1+7.20e18T+1.54e37T2 1 + 7.20e18T + 1.54e37T^{2}
61 1+4.77e18T+3.10e37T2 1 + 4.77e18T + 3.10e37T^{2}
67 1−7.70e18T+2.22e38T2 1 - 7.70e18T + 2.22e38T^{2}
71 1−1.63e19T+7.52e38T2 1 - 1.63e19T + 7.52e38T^{2}
73 1+1.29e19T+1.34e39T2 1 + 1.29e19T + 1.34e39T^{2}
79 1−9.55e19T+7.08e39T2 1 - 9.55e19T + 7.08e39T^{2}
83 1−2.20e20T+1.99e40T2 1 - 2.20e20T + 1.99e40T^{2}
89 1+3.15e19T+8.65e40T2 1 + 3.15e19T + 8.65e40T^{2}
97 1−2.46e20T+5.27e41T2 1 - 2.46e20T + 5.27e41T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−12.08752824832366251081019899665, −10.62162541619406821808652968527, −9.434043916472290209117824583624, −7.925661808048971603484989154936, −6.71548747807764158673202294660, −5.37802426682904202908343187039, −4.64948348735575554198424198864, −3.20942321412360675521358032589, −2.63281472626392836416860681181, −0.44074888670186003444212388732, 0.44074888670186003444212388732, 2.63281472626392836416860681181, 3.20942321412360675521358032589, 4.64948348735575554198424198864, 5.37802426682904202908343187039, 6.71548747807764158673202294660, 7.925661808048971603484989154936, 9.434043916472290209117824583624, 10.62162541619406821808652968527, 12.08752824832366251081019899665

Graph of the ZZ-function along the critical line