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 − 6.95·5-s + 9·9-s − 43.9·11-s − 83.5·13-s + 20.8·15-s + 10.4·17-s − 4.27·19-s − 160.·23-s − 76.5·25-s − 27·27-s + 9.93·29-s + 133.·31-s + 131.·33-s − 357.·37-s + 250.·39-s − 127.·41-s − 343.·43-s − 62.6·45-s − 77.4·47-s − 31.3·51-s − 460.·53-s + 305.·55-s + 12.8·57-s − 272.·59-s − 51.3·61-s + 581.·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.622·5-s + 0.333·9-s − 1.20·11-s − 1.78·13-s + 0.359·15-s + 0.148·17-s − 0.0516·19-s − 1.45·23-s − 0.612·25-s − 0.192·27-s + 0.0635·29-s + 0.774·31-s + 0.694·33-s − 1.58·37-s + 1.02·39-s − 0.484·41-s − 1.21·43-s − 0.207·45-s − 0.240·47-s − 0.0859·51-s − 1.19·53-s + 0.749·55-s + 0.0298·57-s − 0.601·59-s − 0.107·61-s + 1.10·65-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

\(L(2)\) \(\approx\) \(0.05055365546\)
\(L(\frac12)\) \(\approx\) \(0.05055365546\)
\(L(\frac{5}{2})\) 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 \( 1 + 6.95T + 125T^{2} \)
11 \( 1 + 43.9T + 1.33e3T^{2} \)
13 \( 1 + 83.5T + 2.19e3T^{2} \)
17 \( 1 - 10.4T + 4.91e3T^{2} \)
19 \( 1 + 4.27T + 6.85e3T^{2} \)
23 \( 1 + 160.T + 1.21e4T^{2} \)
29 \( 1 - 9.93T + 2.43e4T^{2} \)
31 \( 1 - 133.T + 2.97e4T^{2} \)
37 \( 1 + 357.T + 5.06e4T^{2} \)
41 \( 1 + 127.T + 6.89e4T^{2} \)
43 \( 1 + 343.T + 7.95e4T^{2} \)
47 \( 1 + 77.4T + 1.03e5T^{2} \)
53 \( 1 + 460.T + 1.48e5T^{2} \)
59 \( 1 + 272.T + 2.05e5T^{2} \)
61 \( 1 + 51.3T + 2.26e5T^{2} \)
67 \( 1 - 327.T + 3.00e5T^{2} \)
71 \( 1 - 571.T + 3.57e5T^{2} \)
73 \( 1 - 206.T + 3.89e5T^{2} \)
79 \( 1 + 923.T + 4.93e5T^{2} \)
83 \( 1 - 1.10e3T + 5.71e5T^{2} \)
89 \( 1 + 1.53e3T + 7.04e5T^{2} \)
97 \( 1 - 97.6T + 9.12e5T^{2} \)
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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

−8.351903738115710470042413731629, −7.84601205953829740160641485568, −7.18629901226496990623456575113, −6.32931996060875069449410273864, −5.27395102459675750887708131261, −4.85959379643837231747259212359, −3.85905180844342084976918265061, −2.78025405556676087704015210639, −1.81882822535050180288754829137, −0.097302972617646981771182896733, 0.097302972617646981771182896733, 1.81882822535050180288754829137, 2.78025405556676087704015210639, 3.85905180844342084976918265061, 4.85959379643837231747259212359, 5.27395102459675750887708131261, 6.32931996060875069449410273864, 7.18629901226496990623456575113, 7.84601205953829740160641485568, 8.351903738115710470042413731629

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