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Properties

Label 2-253-253.164-c0-0-0
Degree $2$
Conductor $253$
Sign $0.381 + 0.924i$
Analytic cond. $0.126263$
Root an. cond. $0.355335$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $0.381 + 0.924i$
Analytic conductor: \(0.126263\)
Root analytic conductor: \(0.355335\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{253} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 253,\ (\ :0),\ 0.381 + 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7622589019\)
\(L(\frac12)\) \(\approx\) \(0.7622589019\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.142 - 0.989i)T \)
23 \( 1 + (0.142 - 0.989i)T \)
good2 \( 1 + (0.959 - 0.281i)T^{2} \)
3 \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \)
5 \( 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2} \)
7 \( 1 + (-0.415 + 0.909i)T^{2} \)
13 \( 1 + (-0.415 - 0.909i)T^{2} \)
17 \( 1 + (-0.841 - 0.540i)T^{2} \)
19 \( 1 + (-0.841 + 0.540i)T^{2} \)
29 \( 1 + (-0.841 - 0.540i)T^{2} \)
31 \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \)
37 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
41 \( 1 + (0.142 + 0.989i)T^{2} \)
43 \( 1 + (0.654 + 0.755i)T^{2} \)
47 \( 1 + 1.30T + T^{2} \)
53 \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \)
59 \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \)
61 \( 1 + (0.654 - 0.755i)T^{2} \)
67 \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \)
71 \( 1 + (-0.0405 - 0.281i)T + (-0.959 + 0.281i)T^{2} \)
73 \( 1 + (-0.841 + 0.540i)T^{2} \)
79 \( 1 + (-0.415 - 0.909i)T^{2} \)
83 \( 1 + (0.142 - 0.989i)T^{2} \)
89 \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \)
97 \( 1 + (-0.857 + 0.989i)T + (-0.142 - 0.989i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line