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 + 6.95·5-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: \(1\)
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 \( 1 - 6.95T + 125T^{2} \)
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

−8.296415394716699407146706429337, −7.68485070059839949233223790658, −6.63447782544841362007988187425, −5.88417439212052389323744446981, −5.21277158964052701423457356170, −4.01224083911137744005391565121, −3.32446503282435461266908869954, −2.21890908793913834133819503056, −1.49978067305721876004197705330, 0, 1.49978067305721876004197705330, 2.21890908793913834133819503056, 3.32446503282435461266908869954, 4.01224083911137744005391565121, 5.21277158964052701423457356170, 5.88417439212052389323744446981, 6.63447782544841362007988187425, 7.68485070059839949233223790658, 8.296415394716699407146706429337

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