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 + 6.30·5-s + 9·9-s − 48.9·11-s + 2.60·13-s + 18.9·15-s − 136.·17-s + 45.2·19-s + 38.1·23-s − 85.2·25-s + 27·27-s + 52.7·29-s − 14.7·31-s − 146.·33-s + 333.·37-s + 7.82·39-s − 227.·41-s + 398.·43-s + 56.7·45-s − 184.·47-s − 410.·51-s + 359.·53-s − 308.·55-s + 135.·57-s + 99.9·59-s + 674.·61-s + 16.4·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.563·5-s + 0.333·9-s − 1.34·11-s + 0.0556·13-s + 0.325·15-s − 1.95·17-s + 0.545·19-s + 0.345·23-s − 0.682·25-s + 0.192·27-s + 0.337·29-s − 0.0856·31-s − 0.774·33-s + 1.48·37-s + 0.0321·39-s − 0.865·41-s + 1.41·43-s + 0.187·45-s − 0.572·47-s − 1.12·51-s + 0.932·53-s − 0.755·55-s + 0.315·57-s + 0.220·59-s + 1.41·61-s + 0.0313·65-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

\(L(2)\) \(\approx\) \(2.561143045\)
\(L(\frac12)\) \(\approx\) \(2.561143045\)
\(L(\frac{5}{2})\) 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 \( 1 - 6.30T + 125T^{2} \)
11 \( 1 + 48.9T + 1.33e3T^{2} \)
13 \( 1 - 2.60T + 2.19e3T^{2} \)
17 \( 1 + 136.T + 4.91e3T^{2} \)
19 \( 1 - 45.2T + 6.85e3T^{2} \)
23 \( 1 - 38.1T + 1.21e4T^{2} \)
29 \( 1 - 52.7T + 2.43e4T^{2} \)
31 \( 1 + 14.7T + 2.97e4T^{2} \)
37 \( 1 - 333.T + 5.06e4T^{2} \)
41 \( 1 + 227.T + 6.89e4T^{2} \)
43 \( 1 - 398.T + 7.95e4T^{2} \)
47 \( 1 + 184.T + 1.03e5T^{2} \)
53 \( 1 - 359.T + 1.48e5T^{2} \)
59 \( 1 - 99.9T + 2.05e5T^{2} \)
61 \( 1 - 674.T + 2.26e5T^{2} \)
67 \( 1 - 376.T + 3.00e5T^{2} \)
71 \( 1 - 1.18e3T + 3.57e5T^{2} \)
73 \( 1 - 735.T + 3.89e5T^{2} \)
79 \( 1 - 836.T + 4.93e5T^{2} \)
83 \( 1 - 293.T + 5.71e5T^{2} \)
89 \( 1 + 1.29e3T + 7.04e5T^{2} \)
97 \( 1 - 201.T + 9.12e5T^{2} \)
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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

−8.575538226071418621300594096551, −7.987728412473259123450125143397, −7.13301480205609200915360921693, −6.36756352232415575144649635388, −5.42609436563531896250073384390, −4.68700379603590535584730063871, −3.71445667165639460204541437344, −2.52315834012646762498768932446, −2.15858186618317234552972511798, −0.66564183129699699194764421106, 0.66564183129699699194764421106, 2.15858186618317234552972511798, 2.52315834012646762498768932446, 3.71445667165639460204541437344, 4.68700379603590535584730063871, 5.42609436563531896250073384390, 6.36756352232415575144649635388, 7.13301480205609200915360921693, 7.987728412473259123450125143397, 8.575538226071418621300594096551

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