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 − 16.6·5-s + 9·9-s + 49.2·11-s + 64.4·13-s + 49.8·15-s + 132.·17-s + 82.0·19-s − 82.0·23-s + 150.·25-s − 27·27-s + 157.·29-s + 185.·31-s − 147.·33-s − 51.9·37-s − 193.·39-s − 49.4·41-s − 313.·43-s − 149.·45-s + 553.·47-s − 397.·51-s − 619.·53-s − 817.·55-s − 246.·57-s + 712.·59-s − 287.·61-s − 1.07e3·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.48·5-s + 0.333·9-s + 1.34·11-s + 1.37·13-s + 0.857·15-s + 1.88·17-s + 0.990·19-s − 0.743·23-s + 1.20·25-s − 0.192·27-s + 1.00·29-s + 1.07·31-s − 0.779·33-s − 0.230·37-s − 0.794·39-s − 0.188·41-s − 1.11·43-s − 0.494·45-s + 1.71·47-s − 1.09·51-s − 1.60·53-s − 2.00·55-s − 0.572·57-s + 1.57·59-s − 0.603·61-s − 2.04·65-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

\(L(2)\) \(\approx\) \(1.896267483\)
\(L(\frac12)\) \(\approx\) \(1.896267483\)
\(L(\frac{5}{2})\) 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 \( 1 + 16.6T + 125T^{2} \)
11 \( 1 - 49.2T + 1.33e3T^{2} \)
13 \( 1 - 64.4T + 2.19e3T^{2} \)
17 \( 1 - 132.T + 4.91e3T^{2} \)
19 \( 1 - 82.0T + 6.85e3T^{2} \)
23 \( 1 + 82.0T + 1.21e4T^{2} \)
29 \( 1 - 157.T + 2.43e4T^{2} \)
31 \( 1 - 185.T + 2.97e4T^{2} \)
37 \( 1 + 51.9T + 5.06e4T^{2} \)
41 \( 1 + 49.4T + 6.89e4T^{2} \)
43 \( 1 + 313.T + 7.95e4T^{2} \)
47 \( 1 - 553.T + 1.03e5T^{2} \)
53 \( 1 + 619.T + 1.48e5T^{2} \)
59 \( 1 - 712.T + 2.05e5T^{2} \)
61 \( 1 + 287.T + 2.26e5T^{2} \)
67 \( 1 + 226.T + 3.00e5T^{2} \)
71 \( 1 + 55.3T + 3.57e5T^{2} \)
73 \( 1 - 799.T + 3.89e5T^{2} \)
79 \( 1 - 120.T + 4.93e5T^{2} \)
83 \( 1 - 857.T + 5.71e5T^{2} \)
89 \( 1 + 377.T + 7.04e5T^{2} \)
97 \( 1 + 1.26e3T + 9.12e5T^{2} \)
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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

−8.363863861257719684149544711567, −7.978885185184447007321274487170, −7.07129346958763367118365657848, −6.34978905499968776991628594544, −5.54934240287517458452306505654, −4.49729600594054760345473952202, −3.72751067918108962341354364575, −3.25781663591593930184682379372, −1.30351176812353816387036055896, −0.75758484406899958127357751025, 0.75758484406899958127357751025, 1.30351176812353816387036055896, 3.25781663591593930184682379372, 3.72751067918108962341354364575, 4.49729600594054760345473952202, 5.54934240287517458452306505654, 6.34978905499968776991628594544, 7.07129346958763367118365657848, 7.978885185184447007321274487170, 8.363863861257719684149544711567

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