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Properties

Label 2-164-164.23-c0-0-0
Degree $2$
Conductor $164$
Sign $0.430 - 0.902i$
Analytic cond. $0.0818466$
Root an. cond. $0.286088$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.5 − 0.363i)5-s + (−0.809 − 0.587i)8-s − 9-s + (0.5 + 0.363i)10-s + (−1.11 + 0.363i)13-s + (0.309 − 0.951i)16-s + (1.11 − 1.53i)17-s + (−0.309 − 0.951i)18-s + (−0.190 + 0.587i)20-s + (−0.190 + 0.587i)25-s + (−0.690 − 0.951i)26-s + 32-s + (1.80 + 0.587i)34-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.5 − 0.363i)5-s + (−0.809 − 0.587i)8-s − 9-s + (0.5 + 0.363i)10-s + (−1.11 + 0.363i)13-s + (0.309 − 0.951i)16-s + (1.11 − 1.53i)17-s + (−0.309 − 0.951i)18-s + (−0.190 + 0.587i)20-s + (−0.190 + 0.587i)25-s + (−0.690 − 0.951i)26-s + 32-s + (1.80 + 0.587i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $0.430 - 0.902i$
Analytic conductor: \(0.0818466\)
Root analytic conductor: \(0.286088\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :0),\ 0.430 - 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7028397928\)
\(L(\frac12)\) \(\approx\) \(0.7028397928\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + T^{2} \)
5 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.809 - 0.587i)T^{2} \)
11 \( 1 + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (0.809 - 0.587i)T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.309 - 0.951i)T^{2} \)
71 \( 1 + (0.309 + 0.951i)T^{2} \)
73 \( 1 - 1.61T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \)
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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

−13.59412316652849152304994509105, −12.37269724649348323876085268275, −11.61977360039862515155857433548, −9.829941880548668674480452489535, −9.144277440944001181458604538181, −7.965951516206819742473072934641, −6.93886035143021211153624870729, −5.60628249265597878547519197620, −4.88951706528056482401978081920, −3.03848108752394923544774568067, 2.26742282274066328026911668242, 3.57109128524881941414714243813, 5.23869840015675314642057463457, 6.12510857055227658078927412167, 7.932204997056532795828957939533, 9.101205702288621514215501238496, 10.19815625914200241977185593840, 10.78712556184697619327846629630, 12.09604094449971856836424148047, 12.61692017991889847957373106400

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