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 − 4·5-s + 9·9-s + 26·11-s − 2·13-s + 12·15-s + 36·17-s − 76·19-s + 114·23-s − 109·25-s − 27·27-s + 6·29-s − 256·31-s − 78·33-s − 86·37-s + 6·39-s − 160·41-s + 220·43-s − 36·45-s + 308·47-s − 108·51-s + 258·53-s − 104·55-s + 228·57-s + 264·59-s − 606·61-s + 8·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.357·5-s + 1/3·9-s + 0.712·11-s − 0.0426·13-s + 0.206·15-s + 0.513·17-s − 0.917·19-s + 1.03·23-s − 0.871·25-s − 0.192·27-s + 0.0384·29-s − 1.48·31-s − 0.411·33-s − 0.382·37-s + 0.0246·39-s − 0.609·41-s + 0.780·43-s − 0.119·45-s + 0.955·47-s − 0.296·51-s + 0.668·53-s − 0.254·55-s + 0.529·57-s + 0.582·59-s − 1.27·61-s + 0.0152·65-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 \( 1 + 4 T + p^{3} T^{2} \)
11 \( 1 - 26 T + p^{3} T^{2} \)
13 \( 1 + 2 T + p^{3} T^{2} \)
17 \( 1 - 36 T + p^{3} T^{2} \)
19 \( 1 + 4 p T + p^{3} T^{2} \)
23 \( 1 - 114 T + p^{3} T^{2} \)
29 \( 1 - 6 T + p^{3} T^{2} \)
31 \( 1 + 256 T + p^{3} T^{2} \)
37 \( 1 + 86 T + p^{3} T^{2} \)
41 \( 1 + 160 T + p^{3} T^{2} \)
43 \( 1 - 220 T + p^{3} T^{2} \)
47 \( 1 - 308 T + p^{3} T^{2} \)
53 \( 1 - 258 T + p^{3} T^{2} \)
59 \( 1 - 264 T + p^{3} T^{2} \)
61 \( 1 + 606 T + p^{3} T^{2} \)
67 \( 1 - 520 T + p^{3} T^{2} \)
71 \( 1 - 286 T + p^{3} T^{2} \)
73 \( 1 - 530 T + p^{3} T^{2} \)
79 \( 1 - 44 T + p^{3} T^{2} \)
83 \( 1 - 1012 T + p^{3} T^{2} \)
89 \( 1 + 768 T + p^{3} T^{2} \)
97 \( 1 + 222 T + p^{3} T^{2} \)
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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

−8.233189527945701230680574967528, −7.34625463998216708225115176322, −6.75025046582949378189545643050, −5.87026022849043240716165359829, −5.15247336607215317330370575493, −4.15462631442213561591336148099, −3.52709486463188557989681977937, −2.19444978242734083122256032300, −1.10576222542035645039706575422, 0, 1.10576222542035645039706575422, 2.19444978242734083122256032300, 3.52709486463188557989681977937, 4.15462631442213561591336148099, 5.15247336607215317330370575493, 5.87026022849043240716165359829, 6.75025046582949378189545643050, 7.34625463998216708225115176322, 8.233189527945701230680574967528

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