Properties

Degree $2$
Conductor $2352$
Sign $1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 9.82·5-s + 9·9-s + 14.1·11-s + 26.1·13-s + 29.4·15-s + 78.5·17-s + 73.1·19-s + 96·23-s − 28.4·25-s − 27·27-s + 173.·29-s + 67.2·31-s − 42.5·33-s − 301.·37-s − 78.3·39-s − 472.·41-s + 463.·43-s − 88.4·45-s − 91.1·47-s − 235.·51-s − 163.·53-s − 139.·55-s − 219.·57-s − 600.·59-s − 571.·61-s − 256.·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.878·5-s + 0.333·9-s + 0.388·11-s + 0.557·13-s + 0.507·15-s + 1.12·17-s + 0.883·19-s + 0.870·23-s − 0.227·25-s − 0.192·27-s + 1.10·29-s + 0.389·31-s − 0.224·33-s − 1.34·37-s − 0.321·39-s − 1.79·41-s + 1.64·43-s − 0.292·45-s − 0.283·47-s − 0.647·51-s − 0.423·53-s − 0.341·55-s − 0.510·57-s − 1.32·59-s − 1.19·61-s − 0.489·65-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: $1$
Motivic weight: \(3\)
Character: $\chi_{2352} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.604181548\)
\(L(\frac12)\) \(\approx\) \(1.604181548\)
\(L(\frac{5}{2})\) 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 \( 1 + 9.82T + 125T^{2} \)
11 \( 1 - 14.1T + 1.33e3T^{2} \)
13 \( 1 - 26.1T + 2.19e3T^{2} \)
17 \( 1 - 78.5T + 4.91e3T^{2} \)
19 \( 1 - 73.1T + 6.85e3T^{2} \)
23 \( 1 - 96T + 1.21e4T^{2} \)
29 \( 1 - 173.T + 2.43e4T^{2} \)
31 \( 1 - 67.2T + 2.97e4T^{2} \)
37 \( 1 + 301.T + 5.06e4T^{2} \)
41 \( 1 + 472.T + 6.89e4T^{2} \)
43 \( 1 - 463.T + 7.95e4T^{2} \)
47 \( 1 + 91.1T + 1.03e5T^{2} \)
53 \( 1 + 163.T + 1.48e5T^{2} \)
59 \( 1 + 600.T + 2.05e5T^{2} \)
61 \( 1 + 571.T + 2.26e5T^{2} \)
67 \( 1 - 539.T + 3.00e5T^{2} \)
71 \( 1 - 1.06e3T + 3.57e5T^{2} \)
73 \( 1 - 442.T + 3.89e5T^{2} \)
79 \( 1 - 45.7T + 4.93e5T^{2} \)
83 \( 1 + 686.T + 5.71e5T^{2} \)
89 \( 1 + 660.T + 7.04e5T^{2} \)
97 \( 1 + 658.T + 9.12e5T^{2} \)
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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

−8.515776966569671419793689465071, −7.83108889368526994990910407245, −7.09598964342515668281207126588, −6.35297706838601530476770412716, −5.42860290427695971804113790803, −4.71698154816232514611088165826, −3.71352353418520223317620954081, −3.09425955349054042103898303139, −1.47748345174797004057789524296, −0.63255798946810150526164685050, 0.63255798946810150526164685050, 1.47748345174797004057789524296, 3.09425955349054042103898303139, 3.71352353418520223317620954081, 4.71698154816232514611088165826, 5.42860290427695971804113790803, 6.35297706838601530476770412716, 7.09598964342515668281207126588, 7.83108889368526994990910407245, 8.515776966569671419793689465071

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