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Properties

Label 2-164-164.163-c0-0-2
Degree $2$
Conductor $164$
Sign $1$
Analytic cond. $0.0818466$
Root an. cond. $0.286088$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 2·5-s + 8-s − 9-s − 2·10-s + 16-s − 18-s − 2·20-s + 3·25-s + 32-s − 36-s − 2·37-s − 2·40-s + 41-s + 2·45-s − 49-s + 3·50-s + 2·61-s + 64-s − 72-s − 2·73-s − 2·74-s − 2·80-s + 81-s + 82-s + 2·90-s + ⋯
L(s)  = 1  + 2-s + 4-s − 2·5-s + 8-s − 9-s − 2·10-s + 16-s − 18-s − 2·20-s + 3·25-s + 32-s − 36-s − 2·37-s − 2·40-s + 41-s + 2·45-s − 49-s + 3·50-s + 2·61-s + 64-s − 72-s − 2·73-s − 2·74-s − 2·80-s + 81-s + 82-s + 2·90-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $1$
Analytic conductor: \(0.0818466\)
Root analytic conductor: \(0.286088\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{164} (163, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8709512319\)
\(L(\frac12)\) \(\approx\) \(0.8709512319\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
41 \( 1 - T \)
good3 \( 1 + T^{2} \)
5 \( ( 1 + T )^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 + T )^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( ( 1 + T )^{2} \)
79 \( 1 + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−12.90632094096469162594744104233, −11.99334529142724562601063672904, −11.46813589545324245719361295830, −10.63217591090368908602780976149, −8.636716592973613372975798469663, −7.75750761223569401880141234036, −6.76516852373750768608997934158, −5.24663724267846224107369018513, −4.05018568360181817306673501450, −3.06556400547627146487640594369, 3.06556400547627146487640594369, 4.05018568360181817306673501450, 5.24663724267846224107369018513, 6.76516852373750768608997934158, 7.75750761223569401880141234036, 8.636716592973613372975798469663, 10.63217591090368908602780976149, 11.46813589545324245719361295830, 11.99334529142724562601063672904, 12.90632094096469162594744104233

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