Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 6·5-s + 9·9-s + 30·11-s + 53·13-s + 18·15-s − 84·17-s + 97·19-s − 84·23-s − 89·25-s − 27·27-s − 180·29-s − 179·31-s − 90·33-s − 145·37-s − 159·39-s + 126·41-s + 325·43-s − 54·45-s + 366·47-s + 252·51-s − 768·53-s − 180·55-s − 291·57-s + 264·59-s + 818·61-s − 318·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.536·5-s + 1/3·9-s + 0.822·11-s + 1.13·13-s + 0.309·15-s − 1.19·17-s + 1.17·19-s − 0.761·23-s − 0.711·25-s − 0.192·27-s − 1.15·29-s − 1.03·31-s − 0.474·33-s − 0.644·37-s − 0.652·39-s + 0.479·41-s + 1.15·43-s − 0.178·45-s + 1.13·47-s + 0.691·51-s − 1.99·53-s − 0.441·55-s − 0.676·57-s + 0.582·59-s + 1.71·61-s − 0.606·65-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: \(1\)
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 \( 1 - 30 T + p^{3} T^{2} \)
13 \( 1 - 53 T + p^{3} T^{2} \)
17 \( 1 + 84 T + p^{3} T^{2} \)
19 \( 1 - 97 T + p^{3} T^{2} \)
23 \( 1 + 84 T + p^{3} T^{2} \)
29 \( 1 + 180 T + p^{3} T^{2} \)
31 \( 1 + 179 T + p^{3} T^{2} \)
37 \( 1 + 145 T + p^{3} T^{2} \)
41 \( 1 - 126 T + p^{3} T^{2} \)
43 \( 1 - 325 T + p^{3} T^{2} \)
47 \( 1 - 366 T + p^{3} T^{2} \)
53 \( 1 + 768 T + p^{3} T^{2} \)
59 \( 1 - 264 T + p^{3} T^{2} \)
61 \( 1 - 818 T + p^{3} T^{2} \)
67 \( 1 - 523 T + p^{3} T^{2} \)
71 \( 1 - 342 T + p^{3} T^{2} \)
73 \( 1 + 43 T + p^{3} T^{2} \)
79 \( 1 - 1171 T + p^{3} T^{2} \)
83 \( 1 - 810 T + p^{3} T^{2} \)
89 \( 1 + 600 T + p^{3} T^{2} \)
97 \( 1 - 386 T + p^{3} T^{2} \)
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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

−8.180157180188761159910189100917, −7.43956913901578222832544317799, −6.65366274290268190844818240804, −5.93275271951102379654144354565, −5.16958801012272596769146089255, −3.88946257906629496342719239274, −3.79022034828191887907232180700, −2.15026721211002765939059628758, −1.11133623147651298710133847655, 0, 1.11133623147651298710133847655, 2.15026721211002765939059628758, 3.79022034828191887907232180700, 3.88946257906629496342719239274, 5.16958801012272596769146089255, 5.93275271951102379654144354565, 6.65366274290268190844818240804, 7.43956913901578222832544317799, 8.180157180188761159910189100917

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