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Properties

Label 2-164-164.163-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(0.5752967736\)
\(L(\frac12)\) \(\approx\) \(0.5752967736\)
\(L(1)\) 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

−13.18763867837712168221678951560, −12.16056558853819033271782428387, −10.64515854163204002054397390352, −9.694866627355286219347224837793, −9.187105519113440642461821141702, −7.955392826552363289980170655329, −7.33632915986505954677530283474, −5.85609162058118679786749317933, −3.38262324183870129332137983046, −2.53413337667629057470742593665, 2.53413337667629057470742593665, 3.38262324183870129332137983046, 5.85609162058118679786749317933, 7.33632915986505954677530283474, 7.955392826552363289980170655329, 9.187105519113440642461821141702, 9.694866627355286219347224837793, 10.64515854163204002054397390352, 12.16056558853819033271782428387, 13.18763867837712168221678951560

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