Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 6·5-s + 9·9-s − 36·11-s − 62·13-s + 18·15-s − 114·17-s − 76·19-s + 24·23-s − 89·25-s − 27·27-s + 54·29-s − 112·31-s + 108·33-s − 178·37-s + 186·39-s − 378·41-s + 172·43-s − 54·45-s − 192·47-s + 342·51-s − 402·53-s + 216·55-s + 228·57-s + 396·59-s − 254·61-s + 372·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.536·5-s + 1/3·9-s − 0.986·11-s − 1.32·13-s + 0.309·15-s − 1.62·17-s − 0.917·19-s + 0.217·23-s − 0.711·25-s − 0.192·27-s + 0.345·29-s − 0.648·31-s + 0.569·33-s − 0.790·37-s + 0.763·39-s − 1.43·41-s + 0.609·43-s − 0.178·45-s − 0.595·47-s + 0.939·51-s − 1.04·53-s + 0.529·55-s + 0.529·57-s + 0.873·59-s − 0.533·61-s + 0.709·65-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: \(2\)
Selberg data: \((2,\ 2352,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
7 \( 1 \)
good5 \( 1 + 6 T + p^{3} T^{2} \)
11 \( 1 + 36 T + p^{3} T^{2} \)
13 \( 1 + 62 T + p^{3} T^{2} \)
17 \( 1 + 114 T + p^{3} T^{2} \)
19 \( 1 + 4 p T + p^{3} T^{2} \)
23 \( 1 - 24 T + p^{3} T^{2} \)
29 \( 1 - 54 T + p^{3} T^{2} \)
31 \( 1 + 112 T + p^{3} T^{2} \)
37 \( 1 + 178 T + p^{3} T^{2} \)
41 \( 1 + 378 T + p^{3} T^{2} \)
43 \( 1 - 4 p T + p^{3} T^{2} \)
47 \( 1 + 192 T + p^{3} T^{2} \)
53 \( 1 + 402 T + p^{3} T^{2} \)
59 \( 1 - 396 T + p^{3} T^{2} \)
61 \( 1 + 254 T + p^{3} T^{2} \)
67 \( 1 - 1012 T + p^{3} T^{2} \)
71 \( 1 + 840 T + p^{3} T^{2} \)
73 \( 1 + 890 T + p^{3} T^{2} \)
79 \( 1 + 80 T + p^{3} T^{2} \)
83 \( 1 + 108 T + p^{3} T^{2} \)
89 \( 1 - 1638 T + p^{3} T^{2} \)
97 \( 1 + 1010 T + p^{3} T^{2} \)
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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

−7.80452394903879155305266210951, −7.05181899270936034958892534206, −6.42801129436450314274599407367, −5.31465867550635048839253610822, −4.74918057678642258547637444342, −3.95275110609942894412594355155, −2.70244599289275974427492577321, −1.86508995635353036071437105148, 0, 0, 1.86508995635353036071437105148, 2.70244599289275974427492577321, 3.95275110609942894412594355155, 4.74918057678642258547637444342, 5.31465867550635048839253610822, 6.42801129436450314274599407367, 7.05181899270936034958892534206, 7.80452394903879155305266210951

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