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 + 15.7·5-s + 9·9-s − 58.6·11-s − 20.7·13-s + 47.2·15-s + 42.9·17-s − 137.·19-s − 29.0·23-s + 122.·25-s + 27·27-s + 8.68·29-s + 202.·31-s − 176.·33-s − 15.2·37-s − 62.2·39-s + 117.·41-s + 101.·43-s + 141.·45-s − 588.·47-s + 128.·51-s + 404.·53-s − 923.·55-s − 411.·57-s − 10.8·59-s − 894.·61-s − 326.·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.40·5-s + 0.333·9-s − 1.60·11-s − 0.442·13-s + 0.812·15-s + 0.612·17-s − 1.65·19-s − 0.263·23-s + 0.980·25-s + 0.192·27-s + 0.0555·29-s + 1.17·31-s − 0.928·33-s − 0.0678·37-s − 0.255·39-s + 0.446·41-s + 0.361·43-s + 0.469·45-s − 1.82·47-s + 0.353·51-s + 1.04·53-s − 2.26·55-s − 0.956·57-s − 0.0239·59-s − 1.87·61-s − 0.622·65-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 \( 1 - 15.7T + 125T^{2} \)
11 \( 1 + 58.6T + 1.33e3T^{2} \)
13 \( 1 + 20.7T + 2.19e3T^{2} \)
17 \( 1 - 42.9T + 4.91e3T^{2} \)
19 \( 1 + 137.T + 6.85e3T^{2} \)
23 \( 1 + 29.0T + 1.21e4T^{2} \)
29 \( 1 - 8.68T + 2.43e4T^{2} \)
31 \( 1 - 202.T + 2.97e4T^{2} \)
37 \( 1 + 15.2T + 5.06e4T^{2} \)
41 \( 1 - 117.T + 6.89e4T^{2} \)
43 \( 1 - 101.T + 7.95e4T^{2} \)
47 \( 1 + 588.T + 1.03e5T^{2} \)
53 \( 1 - 404.T + 1.48e5T^{2} \)
59 \( 1 + 10.8T + 2.05e5T^{2} \)
61 \( 1 + 894.T + 2.26e5T^{2} \)
67 \( 1 - 703.T + 3.00e5T^{2} \)
71 \( 1 + 1.16e3T + 3.57e5T^{2} \)
73 \( 1 + 1.11e3T + 3.89e5T^{2} \)
79 \( 1 + 138.T + 4.93e5T^{2} \)
83 \( 1 - 894.T + 5.71e5T^{2} \)
89 \( 1 + 681.T + 7.04e5T^{2} \)
97 \( 1 - 246.T + 9.12e5T^{2} \)
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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

−8.255237690608461120930876860555, −7.64573270883740923009935760262, −6.60510476145190263366415038430, −5.89799710238499290538794189969, −5.13505169401371739889768265095, −4.31717376280979203520723533958, −2.90269520919566328380165373338, −2.41956805084769586497092803025, −1.52779039632218138644907005679, 0, 1.52779039632218138644907005679, 2.41956805084769586497092803025, 2.90269520919566328380165373338, 4.31717376280979203520723533958, 5.13505169401371739889768265095, 5.89799710238499290538794189969, 6.60510476145190263366415038430, 7.64573270883740923009935760262, 8.255237690608461120930876860555

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