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Properties

Label 2-164-164.163-c0-0-0
Degree $2$
Conductor $164$
Sign $1$
Analytic cond. $0.0818466$
Root an. cond. $0.286088$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.41·3-s + 4-s + 1.41·6-s + 1.41·7-s − 8-s + 1.00·9-s + 1.41·11-s − 1.41·12-s − 1.41·14-s + 16-s − 1.00·18-s − 1.41·19-s − 2.00·21-s − 1.41·22-s + 1.41·24-s − 25-s + 1.41·28-s − 32-s − 2.00·33-s + 1.00·36-s + 1.41·38-s + 41-s + 2.00·42-s + 1.41·44-s − 1.41·47-s − 1.41·48-s + ⋯
L(s)  = 1  − 2-s − 1.41·3-s + 4-s + 1.41·6-s + 1.41·7-s − 8-s + 1.00·9-s + 1.41·11-s − 1.41·12-s − 1.41·14-s + 16-s − 1.00·18-s − 1.41·19-s − 2.00·21-s − 1.41·22-s + 1.41·24-s − 25-s + 1.41·28-s − 32-s − 2.00·33-s + 1.00·36-s + 1.41·38-s + 41-s + 2.00·42-s + 1.41·44-s − 1.41·47-s − 1.41·48-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: no
Arithmetic: yes
Character: $\chi_{164} (163, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3481268795\)
\(L(\frac12)\) \(\approx\) \(0.3481268795\)
\(L(1)\) 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 \( 1 + 1.41T + T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 - 1.41T + T^{2} \)
11 \( 1 - 1.41T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.41T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 2T + T^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 1.41T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−12.51833864361520564569950340750, −11.59109040205698406505356516171, −11.25219882261673457397142875486, −10.30712458699540265229593530298, −9.024266535468335828680798604443, −7.958130159799043255838239927925, −6.69921581546472461321751958786, −5.83779551634800273249445863948, −4.40272053803656526702806000032, −1.61066445217681730691510386296, 1.61066445217681730691510386296, 4.40272053803656526702806000032, 5.83779551634800273249445863948, 6.69921581546472461321751958786, 7.958130159799043255838239927925, 9.024266535468335828680798604443, 10.30712458699540265229593530298, 11.25219882261673457397142875486, 11.59109040205698406505356516171, 12.51833864361520564569950340750

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