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Properties

Label 2-253-253.142-c0-0-0
Degree $2$
Conductor $253$
Sign $0.854 - 0.519i$
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  + (0.273 − 0.0801i)3-s + (−0.654 + 0.755i)4-s + (0.698 + 0.449i)5-s + (−0.773 + 0.496i)9-s + (0.415 − 0.909i)11-s + (−0.118 + 0.258i)12-s + (0.226 + 0.0666i)15-s + (−0.142 − 0.989i)16-s + (−0.797 + 0.234i)20-s + (0.415 − 0.909i)23-s + (−0.128 − 0.281i)25-s + (−0.357 + 0.412i)27-s + (−1.61 − 0.474i)31-s + (0.0405 − 0.281i)33-s + (0.130 − 0.909i)36-s + (−1.10 + 0.708i)37-s + ⋯
L(s)  = 1  + (0.273 − 0.0801i)3-s + (−0.654 + 0.755i)4-s + (0.698 + 0.449i)5-s + (−0.773 + 0.496i)9-s + (0.415 − 0.909i)11-s + (−0.118 + 0.258i)12-s + (0.226 + 0.0666i)15-s + (−0.142 − 0.989i)16-s + (−0.797 + 0.234i)20-s + (0.415 − 0.909i)23-s + (−0.128 − 0.281i)25-s + (−0.357 + 0.412i)27-s + (−1.61 − 0.474i)31-s + (0.0405 − 0.281i)33-s + (0.130 − 0.909i)36-s + (−1.10 + 0.708i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $0.854 - 0.519i$
Analytic conductor: \(0.126263\)
Root analytic conductor: \(0.355335\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{253} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 253,\ (\ :0),\ 0.854 - 0.519i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7531075655\)
\(L(\frac12)\) \(\approx\) \(0.7531075655\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (-0.415 + 0.909i)T \)
good2 \( 1 + (0.654 - 0.755i)T^{2} \)
3 \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \)
5 \( 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2} \)
7 \( 1 + (0.959 - 0.281i)T^{2} \)
13 \( 1 + (0.959 + 0.281i)T^{2} \)
17 \( 1 + (0.142 - 0.989i)T^{2} \)
19 \( 1 + (0.142 + 0.989i)T^{2} \)
29 \( 1 + (0.142 - 0.989i)T^{2} \)
31 \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \)
37 \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \)
41 \( 1 + (-0.415 - 0.909i)T^{2} \)
43 \( 1 + (-0.841 + 0.540i)T^{2} \)
47 \( 1 - 1.68T + T^{2} \)
53 \( 1 + (-0.186 - 1.29i)T + (-0.959 + 0.281i)T^{2} \)
59 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
61 \( 1 + (-0.841 - 0.540i)T^{2} \)
67 \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \)
71 \( 1 + (-0.345 - 0.755i)T + (-0.654 + 0.755i)T^{2} \)
73 \( 1 + (0.142 + 0.989i)T^{2} \)
79 \( 1 + (0.959 + 0.281i)T^{2} \)
83 \( 1 + (-0.415 + 0.909i)T^{2} \)
89 \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \)
97 \( 1 + (-1.41 - 0.909i)T + (0.415 + 0.909i)T^{2} \)
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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

−12.47210654363265279190258569691, −11.38302731627264562742466384245, −10.48319473389806095820384317070, −9.207524629169377354398411676708, −8.603102279585484001284824124959, −7.60154651504119032577021266338, −6.30308306723527688568184656695, −5.18108674211041084179501382885, −3.66906143112531900718428331295, −2.54329993479200130555116758660, 1.76931037847701579835770158532, 3.73837341827522809199844503466, 5.13061017494103047681522785563, 5.84870719200424728815925701926, 7.17914306159097609389332492981, 8.770634713219353808441745851333, 9.247802481790663834214128852846, 10.00892184959326235700756560231, 11.14228407888340720476565015099, 12.32741771006146296281028344554

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