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 − 6.73·5-s + 9·9-s + 53.6·11-s + 1.73·13-s − 20.2·15-s − 46.9·17-s + 20.1·19-s − 118.·23-s − 79.6·25-s + 27·27-s − 103.·29-s + 157.·31-s + 161.·33-s − 37.7·37-s + 5.20·39-s − 287.·41-s − 504.·43-s − 60.6·45-s + 220.·47-s − 140.·51-s + 292.·53-s − 361.·55-s + 60.4·57-s + 595.·59-s − 265.·61-s − 11.6·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.602·5-s + 0.333·9-s + 1.47·11-s + 0.0370·13-s − 0.347·15-s − 0.669·17-s + 0.243·19-s − 1.07·23-s − 0.636·25-s + 0.192·27-s − 0.663·29-s + 0.912·31-s + 0.849·33-s − 0.167·37-s + 0.0213·39-s − 1.09·41-s − 1.79·43-s − 0.200·45-s + 0.684·47-s − 0.386·51-s + 0.757·53-s − 0.886·55-s + 0.140·57-s + 1.31·59-s − 0.557·61-s − 0.0223·65-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 \( 1 + 6.73T + 125T^{2} \)
11 \( 1 - 53.6T + 1.33e3T^{2} \)
13 \( 1 - 1.73T + 2.19e3T^{2} \)
17 \( 1 + 46.9T + 4.91e3T^{2} \)
19 \( 1 - 20.1T + 6.85e3T^{2} \)
23 \( 1 + 118.T + 1.21e4T^{2} \)
29 \( 1 + 103.T + 2.43e4T^{2} \)
31 \( 1 - 157.T + 2.97e4T^{2} \)
37 \( 1 + 37.7T + 5.06e4T^{2} \)
41 \( 1 + 287.T + 6.89e4T^{2} \)
43 \( 1 + 504.T + 7.95e4T^{2} \)
47 \( 1 - 220.T + 1.03e5T^{2} \)
53 \( 1 - 292.T + 1.48e5T^{2} \)
59 \( 1 - 595.T + 2.05e5T^{2} \)
61 \( 1 + 265.T + 2.26e5T^{2} \)
67 \( 1 + 936.T + 3.00e5T^{2} \)
71 \( 1 - 545.T + 3.57e5T^{2} \)
73 \( 1 - 299.T + 3.89e5T^{2} \)
79 \( 1 - 940.T + 4.93e5T^{2} \)
83 \( 1 + 611.T + 5.71e5T^{2} \)
89 \( 1 + 1.17e3T + 7.04e5T^{2} \)
97 \( 1 - 1.48e3T + 9.12e5T^{2} \)
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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

−8.361026282241550955767731889188, −7.53097614491260272531074617683, −6.77031794688048113931296091401, −6.08598502198048380891107114012, −4.88101163399044358715771657050, −3.95124931337291476430299127191, −3.56296518763115254373628572880, −2.27052620838613659077811634949, −1.33907039418938657415600937837, 0, 1.33907039418938657415600937837, 2.27052620838613659077811634949, 3.56296518763115254373628572880, 3.95124931337291476430299127191, 4.88101163399044358715771657050, 6.08598502198048380891107114012, 6.77031794688048113931296091401, 7.53097614491260272531074617683, 8.361026282241550955767731889188

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