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Properties

Label 2-252-1.1-c7-0-8
Degree 22
Conductor 252252
Sign 11
Analytic cond. 78.721078.7210
Root an. cond. 8.872488.87248
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 373.·5-s + 343·7-s + 4.60e3·11-s − 5.28e3·13-s + 1.60e4·17-s + 2.83e4·19-s + 3.90e4·23-s + 6.14e4·25-s + 9.73e4·29-s − 2.41e5·31-s + 1.28e5·35-s + 2.49e5·37-s + 1.08e5·41-s − 2.28e5·43-s − 3.52e5·47-s + 1.17e5·49-s − 7.78e5·53-s + 1.72e6·55-s − 1.76e6·59-s + 2.60e5·61-s − 1.97e6·65-s − 1.24e6·67-s + 1.36e6·71-s + 1.75e6·73-s + 1.57e6·77-s + 6.81e6·79-s + 7.92e6·83-s + ⋯
L(s)  = 1  + 1.33·5-s + 0.377·7-s + 1.04·11-s − 0.667·13-s + 0.793·17-s + 0.949·19-s + 0.668·23-s + 0.786·25-s + 0.741·29-s − 1.45·31-s + 0.505·35-s + 0.808·37-s + 0.245·41-s − 0.438·43-s − 0.495·47-s + 0.142·49-s − 0.718·53-s + 1.39·55-s − 1.12·59-s + 0.146·61-s − 0.892·65-s − 0.504·67-s + 0.453·71-s + 0.527·73-s + 0.394·77-s + 1.55·79-s + 1.52·83-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+7/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: 11
Analytic conductor: 78.721078.7210
Root analytic conductor: 8.872488.87248
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 252, ( :7/2), 1)(2,\ 252,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 3.4396110973.439611097
L(12)L(\frac12) ≈\approx 3.4396110973.439611097
L(92)L(\frac{9}{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)
bad2 1 1
3 1 1
7 1−343T 1 - 343T
good5 1−373.T+7.81e4T2 1 - 373.T + 7.81e4T^{2}
11 1−4.60e3T+1.94e7T2 1 - 4.60e3T + 1.94e7T^{2}
13 1+5.28e3T+6.27e7T2 1 + 5.28e3T + 6.27e7T^{2}
17 1−1.60e4T+4.10e8T2 1 - 1.60e4T + 4.10e8T^{2}
19 1−2.83e4T+8.93e8T2 1 - 2.83e4T + 8.93e8T^{2}
23 1−3.90e4T+3.40e9T2 1 - 3.90e4T + 3.40e9T^{2}
29 1−9.73e4T+1.72e10T2 1 - 9.73e4T + 1.72e10T^{2}
31 1+2.41e5T+2.75e10T2 1 + 2.41e5T + 2.75e10T^{2}
37 1−2.49e5T+9.49e10T2 1 - 2.49e5T + 9.49e10T^{2}
41 1−1.08e5T+1.94e11T2 1 - 1.08e5T + 1.94e11T^{2}
43 1+2.28e5T+2.71e11T2 1 + 2.28e5T + 2.71e11T^{2}
47 1+3.52e5T+5.06e11T2 1 + 3.52e5T + 5.06e11T^{2}
53 1+7.78e5T+1.17e12T2 1 + 7.78e5T + 1.17e12T^{2}
59 1+1.76e6T+2.48e12T2 1 + 1.76e6T + 2.48e12T^{2}
61 1−2.60e5T+3.14e12T2 1 - 2.60e5T + 3.14e12T^{2}
67 1+1.24e6T+6.06e12T2 1 + 1.24e6T + 6.06e12T^{2}
71 1−1.36e6T+9.09e12T2 1 - 1.36e6T + 9.09e12T^{2}
73 1−1.75e6T+1.10e13T2 1 - 1.75e6T + 1.10e13T^{2}
79 1−6.81e6T+1.92e13T2 1 - 6.81e6T + 1.92e13T^{2}
83 1−7.92e6T+2.71e13T2 1 - 7.92e6T + 2.71e13T^{2}
89 1−2.49e6T+4.42e13T2 1 - 2.49e6T + 4.42e13T^{2}
97 1−6.93e5T+8.07e13T2 1 - 6.93e5T + 8.07e13T^{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

−10.70666662874727957510885609623, −9.624094750852025448855275546814, −9.229853571291429333450961154620, −7.81583727520378254295417038126, −6.72024297712191847758432074450, −5.71648795586946143076185612135, −4.82477475196548627588423593600, −3.27918655609939795175772085419, −1.96497674369795894388626784417, −1.01785267369795611692021190370, 1.01785267369795611692021190370, 1.96497674369795894388626784417, 3.27918655609939795175772085419, 4.82477475196548627588423593600, 5.71648795586946143076185612135, 6.72024297712191847758432074450, 7.81583727520378254295417038126, 9.229853571291429333450961154620, 9.624094750852025448855275546814, 10.70666662874727957510885609623

Graph of the ZZ-function along the critical line