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 − 7.81·5-s + 9·9-s − 42.7·11-s − 10.9·13-s + 23.4·15-s + 80.2·17-s − 112.·19-s + 70.1·23-s − 63.9·25-s − 27·27-s − 147.·29-s + 144.·31-s + 128.·33-s − 0.292·37-s + 32.8·39-s − 294.·41-s − 337.·43-s − 70.3·45-s − 104.·47-s − 240.·51-s − 143.·53-s + 334.·55-s + 338.·57-s − 520.·59-s + 399.·61-s + 85.5·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.698·5-s + 0.333·9-s − 1.17·11-s − 0.233·13-s + 0.403·15-s + 1.14·17-s − 1.36·19-s + 0.636·23-s − 0.511·25-s − 0.192·27-s − 0.941·29-s + 0.837·31-s + 0.676·33-s − 0.00130·37-s + 0.134·39-s − 1.12·41-s − 1.19·43-s − 0.232·45-s − 0.324·47-s − 0.661·51-s − 0.370·53-s + 0.819·55-s + 0.785·57-s − 1.14·59-s + 0.839·61-s + 0.163·65-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

\(L(2)\) \(\approx\) \(0.5845242731\)
\(L(\frac12)\) \(\approx\) \(0.5845242731\)
\(L(\frac{5}{2})\) 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 \( 1 + 7.81T + 125T^{2} \)
11 \( 1 + 42.7T + 1.33e3T^{2} \)
13 \( 1 + 10.9T + 2.19e3T^{2} \)
17 \( 1 - 80.2T + 4.91e3T^{2} \)
19 \( 1 + 112.T + 6.85e3T^{2} \)
23 \( 1 - 70.1T + 1.21e4T^{2} \)
29 \( 1 + 147.T + 2.43e4T^{2} \)
31 \( 1 - 144.T + 2.97e4T^{2} \)
37 \( 1 + 0.292T + 5.06e4T^{2} \)
41 \( 1 + 294.T + 6.89e4T^{2} \)
43 \( 1 + 337.T + 7.95e4T^{2} \)
47 \( 1 + 104.T + 1.03e5T^{2} \)
53 \( 1 + 143.T + 1.48e5T^{2} \)
59 \( 1 + 520.T + 2.05e5T^{2} \)
61 \( 1 - 399.T + 2.26e5T^{2} \)
67 \( 1 + 137.T + 3.00e5T^{2} \)
71 \( 1 + 266.T + 3.57e5T^{2} \)
73 \( 1 - 524.T + 3.89e5T^{2} \)
79 \( 1 + 433.T + 4.93e5T^{2} \)
83 \( 1 + 664.T + 5.71e5T^{2} \)
89 \( 1 - 803.T + 7.04e5T^{2} \)
97 \( 1 - 445.T + 9.12e5T^{2} \)
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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

−8.370811700281012585394647739001, −7.910436990734369506700315184778, −7.14434848461213598237958924269, −6.29256963112627649988660523419, −5.39442777695429088296655400439, −4.77236397189150343050974935587, −3.81340249208850988286401426088, −2.89473863210373399408862699394, −1.70224764335885133634673116433, −0.34413226614665664911416399725, 0.34413226614665664911416399725, 1.70224764335885133634673116433, 2.89473863210373399408862699394, 3.81340249208850988286401426088, 4.77236397189150343050974935587, 5.39442777695429088296655400439, 6.29256963112627649988660523419, 7.14434848461213598237958924269, 7.910436990734369506700315184778, 8.370811700281012585394647739001

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