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Properties

Label 2-253-253.164-c0-0-0
Degree 22
Conductor 253253
Sign 0.381+0.924i0.381 + 0.924i
Analytic cond. 0.1262630.126263
Root an. cond. 0.3553350.355335
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.698 − 1.53i)3-s + (−0.959 + 0.281i)4-s + (0.186 − 0.215i)5-s + (−1.19 − 1.38i)9-s + (−0.142 + 0.989i)11-s + (−0.239 + 1.66i)12-s + (−0.198 − 0.435i)15-s + (0.841 − 0.540i)16-s + (−0.118 + 0.258i)20-s + (−0.142 + 0.989i)23-s + (0.130 + 0.909i)25-s + (−1.34 + 0.393i)27-s + (−0.544 − 1.19i)31-s + (1.41 + 0.909i)33-s + (1.54 + 0.989i)36-s + (1.25 + 1.45i)37-s + ⋯
L(s)  = 1  + (0.698 − 1.53i)3-s + (−0.959 + 0.281i)4-s + (0.186 − 0.215i)5-s + (−1.19 − 1.38i)9-s + (−0.142 + 0.989i)11-s + (−0.239 + 1.66i)12-s + (−0.198 − 0.435i)15-s + (0.841 − 0.540i)16-s + (−0.118 + 0.258i)20-s + (−0.142 + 0.989i)23-s + (0.130 + 0.909i)25-s + (−1.34 + 0.393i)27-s + (−0.544 − 1.19i)31-s + (1.41 + 0.909i)33-s + (1.54 + 0.989i)36-s + (1.25 + 1.45i)37-s + ⋯

Functional equation

Λ(s)=(253s/2 ΓC(s) L(s)=((0.381+0.924i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(253s/2 ΓC(s) L(s)=((0.381+0.924i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 253253    =    11⋅2311 \cdot 23
Sign: 0.381+0.924i0.381 + 0.924i
Analytic conductor: 0.1262630.126263
Root analytic conductor: 0.3553350.355335
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ253(164,⋅)\chi_{253} (164, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 253, ( :0), 0.381+0.924i)(2,\ 253,\ (\ :0),\ 0.381 + 0.924i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.76225890190.7622589019
L(12)L(\frac12) ≈\approx 0.76225890190.7622589019
L(1)L(1) 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)
bad11 1+(0.142−0.989i)T 1 + (0.142 - 0.989i)T
23 1+(0.142−0.989i)T 1 + (0.142 - 0.989i)T
good2 1+(0.959−0.281i)T2 1 + (0.959 - 0.281i)T^{2}
3 1+(−0.698+1.53i)T+(−0.654−0.755i)T2 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2}
5 1+(−0.186+0.215i)T+(−0.142−0.989i)T2 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2}
7 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
13 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{2}
17 1+(−0.841−0.540i)T2 1 + (-0.841 - 0.540i)T^{2}
19 1+(−0.841+0.540i)T2 1 + (-0.841 + 0.540i)T^{2}
29 1+(−0.841−0.540i)T2 1 + (-0.841 - 0.540i)T^{2}
31 1+(0.544+1.19i)T+(−0.654+0.755i)T2 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2}
37 1+(−1.25−1.45i)T+(−0.142+0.989i)T2 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2}
41 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
43 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
47 1+1.30T+T2 1 + 1.30T + T^{2}
53 1+(1.61−1.03i)T+(0.415−0.909i)T2 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2}
59 1+(1.61+1.03i)T+(0.415+0.909i)T2 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2}
61 1+(0.654−0.755i)T2 1 + (0.654 - 0.755i)T^{2}
67 1+(0.239+1.66i)T+(−0.959+0.281i)T2 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2}
71 1+(−0.0405−0.281i)T+(−0.959+0.281i)T2 1 + (-0.0405 - 0.281i)T + (-0.959 + 0.281i)T^{2}
73 1+(−0.841+0.540i)T2 1 + (-0.841 + 0.540i)T^{2}
79 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{2}
83 1+(0.142−0.989i)T2 1 + (0.142 - 0.989i)T^{2}
89 1+(−0.698+1.53i)T+(−0.654−0.755i)T2 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2}
97 1+(−0.857+0.989i)T+(−0.142−0.989i)T2 1 + (-0.857 + 0.989i)T + (-0.142 - 0.989i)T^{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

−12.49154366927145877277450247798, −11.51513193698249743325494190188, −9.748377989077920958412778338852, −9.117518297748097578961038873262, −7.947310992096860865403685019615, −7.51511919653137768636864798925, −6.20838038062075400816381662075, −4.80806915706543327818837999032, −3.25297195246930043817121171869, −1.68272420117990294505165198837, 2.96486559833008781205357508615, 4.06621496211129415832596826603, 4.96332786324097638901111599888, 6.08064293609029139551270269205, 8.074580207792284169757914988639, 8.818363654283799960990648776082, 9.522607472550855290831919765184, 10.42578848606262017158353264009, 10.98211323671080688402176227543, 12.61192772817713725273934437109

Graph of the ZZ-function along the critical line