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 − 11.4·5-s + 9·9-s + 52.5·11-s + 5.48·13-s − 34.3·15-s − 85.3·17-s − 110.·19-s + 209.·23-s + 5.99·25-s + 27·27-s − 132.·29-s − 49.3·31-s + 157.·33-s + 160.·37-s + 16.4·39-s − 138.·41-s + 365.·43-s − 103.·45-s + 131.·47-s − 256.·51-s − 561.·53-s − 601.·55-s − 331.·57-s − 436.·59-s + 291.·61-s − 62.7·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.02·5-s + 0.333·9-s + 1.44·11-s + 0.116·13-s − 0.591·15-s − 1.21·17-s − 1.33·19-s + 1.89·23-s + 0.0479·25-s + 0.192·27-s − 0.850·29-s − 0.285·31-s + 0.832·33-s + 0.712·37-s + 0.0675·39-s − 0.526·41-s + 1.29·43-s − 0.341·45-s + 0.407·47-s − 0.703·51-s − 1.45·53-s − 1.47·55-s − 0.771·57-s − 0.962·59-s + 0.612·61-s − 0.119·65-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 \( 1 - 3T \)
7 \( 1 \)
good5 \( 1 + 11.4T + 125T^{2} \)
11 \( 1 - 52.5T + 1.33e3T^{2} \)
13 \( 1 - 5.48T + 2.19e3T^{2} \)
17 \( 1 + 85.3T + 4.91e3T^{2} \)
19 \( 1 + 110.T + 6.85e3T^{2} \)
23 \( 1 - 209.T + 1.21e4T^{2} \)
29 \( 1 + 132.T + 2.43e4T^{2} \)
31 \( 1 + 49.3T + 2.97e4T^{2} \)
37 \( 1 - 160.T + 5.06e4T^{2} \)
41 \( 1 + 138.T + 6.89e4T^{2} \)
43 \( 1 - 365.T + 7.95e4T^{2} \)
47 \( 1 - 131.T + 1.03e5T^{2} \)
53 \( 1 + 561.T + 1.48e5T^{2} \)
59 \( 1 + 436.T + 2.05e5T^{2} \)
61 \( 1 - 291.T + 2.26e5T^{2} \)
67 \( 1 - 593.T + 3.00e5T^{2} \)
71 \( 1 + 775.T + 3.57e5T^{2} \)
73 \( 1 + 330.T + 3.89e5T^{2} \)
79 \( 1 + 243.T + 4.93e5T^{2} \)
83 \( 1 - 332.T + 5.71e5T^{2} \)
89 \( 1 - 979.T + 7.04e5T^{2} \)
97 \( 1 + 466.T + 9.12e5T^{2} \)
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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

−8.357290667340199722417074989507, −7.46769957752197581983024311630, −6.83512853265035739111984730046, −6.13125032868207808789385002721, −4.71576735326991678127664231498, −4.12810189244668091782891846759, −3.46861034537734163450884498036, −2.35894350103838170068575492100, −1.25212773238234968325918313147, 0, 1.25212773238234968325918313147, 2.35894350103838170068575492100, 3.46861034537734163450884498036, 4.12810189244668091782891846759, 4.71576735326991678127664231498, 6.13125032868207808789385002721, 6.83512853265035739111984730046, 7.46769957752197581983024311630, 8.357290667340199722417074989507

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