Properties

Label 2-2352-3.2-c0-0-0
Degree $2$
Conductor $2352$
Sign $1$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
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  − 3-s + 9-s − 13-s + 19-s + 25-s − 27-s + 31-s − 37-s + 39-s + 43-s − 57-s + 2·61-s + 67-s − 73-s − 75-s + 79-s + 81-s − 93-s + 2·97-s + 103-s − 109-s + 111-s − 117-s + ⋯
L(s)  = 1  − 3-s + 9-s − 13-s + 19-s + 25-s − 27-s + 31-s − 37-s + 39-s + 43-s − 57-s + 2·61-s + 67-s − 73-s − 75-s + 79-s + 81-s − 93-s + 2·97-s + 103-s − 109-s + 111-s − 117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2352} (785, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8477909302\)
\(L(\frac12)\) \(\approx\) \(0.8477909302\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
7 \( 1 \)
good5 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 - T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 - T + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 - T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )^{2} \)
67 \( 1 - T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
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

−9.359234384190361867969580729156, −8.360086758631098365062223874895, −7.37353349778628892824086506281, −6.92296483761230805782514019751, −6.00230167626282075679164904954, −5.16552387184826496020098368778, −4.65190882045998153680741138615, −3.52210941996624872160860458130, −2.33376119657853550905392114686, −0.943826337148702968312648393411, 0.943826337148702968312648393411, 2.33376119657853550905392114686, 3.52210941996624872160860458130, 4.65190882045998153680741138615, 5.16552387184826496020098368778, 6.00230167626282075679164904954, 6.92296483761230805782514019751, 7.37353349778628892824086506281, 8.360086758631098365062223874895, 9.359234384190361867969580729156

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