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Properties

Label 2-2624-164.163-c0-0-1
Degree $2$
Conductor $2624$
Sign $1$
Analytic cond. $1.30954$
Root an. cond. $1.14435$
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  − 1.41·3-s − 1.41·7-s + 1.00·9-s + 1.41·11-s − 1.41·19-s + 2.00·21-s − 25-s − 2.00·33-s + 41-s + 1.41·47-s + 1.00·49-s + 2.00·57-s + 2·61-s − 1.41·63-s + 1.41·67-s + 1.41·71-s + 1.41·75-s − 2.00·77-s + 1.41·79-s − 0.999·81-s + 1.41·99-s + ⋯
L(s)  = 1  − 1.41·3-s − 1.41·7-s + 1.00·9-s + 1.41·11-s − 1.41·19-s + 2.00·21-s − 25-s − 2.00·33-s + 41-s + 1.41·47-s + 1.00·49-s + 2.00·57-s + 2·61-s − 1.41·63-s + 1.41·67-s + 1.41·71-s + 1.41·75-s − 2.00·77-s + 1.41·79-s − 0.999·81-s + 1.41·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $1$
Analytic conductor: \(1.30954\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (2623, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2624,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5692844002\)
\(L(\frac12)\) \(\approx\) \(0.5692844002\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
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

−9.280907958022290083985316068125, −8.375614469347444373140465859671, −7.13526760690638206462475322907, −6.48914747921063413796135061090, −6.17861051868916698730733355412, −5.38050581067249907645391772889, −4.21247143803608509772161973167, −3.69504420127389968519596264318, −2.26655906144415210864048456258, −0.73713795966271918933991058572, 0.73713795966271918933991058572, 2.26655906144415210864048456258, 3.69504420127389968519596264318, 4.21247143803608509772161973167, 5.38050581067249907645391772889, 6.17861051868916698730733355412, 6.48914747921063413796135061090, 7.13526760690638206462475322907, 8.375614469347444373140465859671, 9.280907958022290083985316068125

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