Properties

Label 2-2352-1.1-c3-0-64
Degree $2$
Conductor $2352$
Sign $1$
Analytic cond. $138.772$
Root an. cond. $11.7801$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 15.8·5-s + 9·9-s + 30.2·11-s + 61.6·13-s + 47.4·15-s − 56.1·17-s − 139.·19-s + 8.14·23-s + 125.·25-s + 27·27-s − 0.217·29-s + 176.·31-s + 90.7·33-s + 211.·37-s + 184.·39-s + 293.·41-s − 434.·43-s + 142.·45-s + 483.·47-s − 168.·51-s + 20.5·53-s + 478.·55-s − 418.·57-s − 231.·59-s + 838.·61-s + 974.·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.41·5-s + 0.333·9-s + 0.829·11-s + 1.31·13-s + 0.816·15-s − 0.801·17-s − 1.68·19-s + 0.0738·23-s + 1.00·25-s + 0.192·27-s − 0.00139·29-s + 1.02·31-s + 0.478·33-s + 0.937·37-s + 0.759·39-s + 1.11·41-s − 1.54·43-s + 0.471·45-s + 1.50·47-s − 0.462·51-s + 0.0531·53-s + 1.17·55-s − 0.971·57-s − 0.510·59-s + 1.76·61-s + 1.85·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(138.772\)
Root analytic conductor: \(11.7801\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.483179334\)
\(L(\frac12)\) \(\approx\) \(4.483179334\)
\(L(\frac{5}{2})\) 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 \( 1 - 3T \)
7 \( 1 \)
good5 \( 1 - 15.8T + 125T^{2} \)
11 \( 1 - 30.2T + 1.33e3T^{2} \)
13 \( 1 - 61.6T + 2.19e3T^{2} \)
17 \( 1 + 56.1T + 4.91e3T^{2} \)
19 \( 1 + 139.T + 6.85e3T^{2} \)
23 \( 1 - 8.14T + 1.21e4T^{2} \)
29 \( 1 + 0.217T + 2.43e4T^{2} \)
31 \( 1 - 176.T + 2.97e4T^{2} \)
37 \( 1 - 211.T + 5.06e4T^{2} \)
41 \( 1 - 293.T + 6.89e4T^{2} \)
43 \( 1 + 434.T + 7.95e4T^{2} \)
47 \( 1 - 483.T + 1.03e5T^{2} \)
53 \( 1 - 20.5T + 1.48e5T^{2} \)
59 \( 1 + 231.T + 2.05e5T^{2} \)
61 \( 1 - 838.T + 2.26e5T^{2} \)
67 \( 1 - 624.T + 3.00e5T^{2} \)
71 \( 1 + 227.T + 3.57e5T^{2} \)
73 \( 1 + 43.0T + 3.89e5T^{2} \)
79 \( 1 - 309.T + 4.93e5T^{2} \)
83 \( 1 - 1.23e3T + 5.71e5T^{2} \)
89 \( 1 + 1.14e3T + 7.04e5T^{2} \)
97 \( 1 + 1.68e3T + 9.12e5T^{2} \)
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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

−8.713789776824128299514122604084, −8.166105048587492002749114079284, −6.77418323494350141671193733453, −6.39416426659826242869782038442, −5.70802579721869781049949036092, −4.49377037612110718924872850548, −3.81642616167790977906240562002, −2.57522119540594118702356112993, −1.91038626894690978397886560573, −0.963753306978551847029839855192, 0.963753306978551847029839855192, 1.91038626894690978397886560573, 2.57522119540594118702356112993, 3.81642616167790977906240562002, 4.49377037612110718924872850548, 5.70802579721869781049949036092, 6.39416426659826242869782038442, 6.77418323494350141671193733453, 8.166105048587492002749114079284, 8.713789776824128299514122604084

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