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 + 16.6·5-s + 9·9-s + 71.7·11-s + 65.3·13-s + 49.9·15-s − 90.4·17-s + 163.·19-s − 79.2·23-s + 152.·25-s + 27·27-s − 43.2·29-s + 135.·31-s + 215.·33-s + 270.·37-s + 196.·39-s − 152.·41-s + 177.·43-s + 149.·45-s − 45.6·47-s − 271.·51-s − 158.·53-s + 1.19e3·55-s + 491.·57-s − 391.·59-s − 551.·61-s + 1.08e3·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.48·5-s + 0.333·9-s + 1.96·11-s + 1.39·13-s + 0.860·15-s − 1.29·17-s + 1.97·19-s − 0.718·23-s + 1.21·25-s + 0.192·27-s − 0.277·29-s + 0.785·31-s + 1.13·33-s + 1.20·37-s + 0.805·39-s − 0.579·41-s + 0.630·43-s + 0.496·45-s − 0.141·47-s − 0.744·51-s − 0.410·53-s + 2.93·55-s + 1.14·57-s − 0.864·59-s − 1.15·61-s + 2.07·65-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

\(L(2)\) \(\approx\) \(5.328369992\)
\(L(\frac12)\) \(\approx\) \(5.328369992\)
\(L(\frac{5}{2})\) 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 \( 1 - 16.6T + 125T^{2} \)
11 \( 1 - 71.7T + 1.33e3T^{2} \)
13 \( 1 - 65.3T + 2.19e3T^{2} \)
17 \( 1 + 90.4T + 4.91e3T^{2} \)
19 \( 1 - 163.T + 6.85e3T^{2} \)
23 \( 1 + 79.2T + 1.21e4T^{2} \)
29 \( 1 + 43.2T + 2.43e4T^{2} \)
31 \( 1 - 135.T + 2.97e4T^{2} \)
37 \( 1 - 270.T + 5.06e4T^{2} \)
41 \( 1 + 152.T + 6.89e4T^{2} \)
43 \( 1 - 177.T + 7.95e4T^{2} \)
47 \( 1 + 45.6T + 1.03e5T^{2} \)
53 \( 1 + 158.T + 1.48e5T^{2} \)
59 \( 1 + 391.T + 2.05e5T^{2} \)
61 \( 1 + 551.T + 2.26e5T^{2} \)
67 \( 1 + 458.T + 3.00e5T^{2} \)
71 \( 1 - 486.T + 3.57e5T^{2} \)
73 \( 1 + 574.T + 3.89e5T^{2} \)
79 \( 1 - 668.T + 4.93e5T^{2} \)
83 \( 1 - 76.2T + 5.71e5T^{2} \)
89 \( 1 + 1.36e3T + 7.04e5T^{2} \)
97 \( 1 + 242.T + 9.12e5T^{2} \)
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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

−8.927954958875309952957026433376, −8.000520049231780603615460759143, −6.87555460371582779561365871624, −6.28880199858472558214832587842, −5.75576136128451806053761298531, −4.52515383051941230917677821815, −3.71907606166266782625378880684, −2.76073500848258481946540788565, −1.61247356156542201866858968265, −1.18177444876055460447460858930, 1.18177444876055460447460858930, 1.61247356156542201866858968265, 2.76073500848258481946540788565, 3.71907606166266782625378880684, 4.52515383051941230917677821815, 5.75576136128451806053761298531, 6.28880199858472558214832587842, 6.87555460371582779561365871624, 8.000520049231780603615460759143, 8.927954958875309952957026433376

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