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 − 19.1·5-s + 9·9-s − 40.5·11-s + 50.4·13-s − 57.4·15-s − 51.9·17-s + 33.1·19-s − 62.8·23-s + 241.·25-s + 27·27-s + 129.·29-s − 242.·31-s − 121.·33-s − 389.·37-s + 151.·39-s − 470.·41-s + 125.·43-s − 172.·45-s + 386.·47-s − 155.·51-s − 611.·53-s + 777.·55-s + 99.3·57-s − 226.·59-s + 725.·61-s − 966.·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.71·5-s + 0.333·9-s − 1.11·11-s + 1.07·13-s − 0.988·15-s − 0.740·17-s + 0.400·19-s − 0.569·23-s + 1.93·25-s + 0.192·27-s + 0.831·29-s − 1.40·31-s − 0.642·33-s − 1.73·37-s + 0.621·39-s − 1.79·41-s + 0.443·43-s − 0.570·45-s + 1.19·47-s − 0.427·51-s − 1.58·53-s + 1.90·55-s + 0.230·57-s − 0.499·59-s + 1.52·61-s − 1.84·65-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

\(L(2)\) \(\approx\) \(1.089570633\)
\(L(\frac12)\) \(\approx\) \(1.089570633\)
\(L(\frac{5}{2})\) 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 \( 1 + 19.1T + 125T^{2} \)
11 \( 1 + 40.5T + 1.33e3T^{2} \)
13 \( 1 - 50.4T + 2.19e3T^{2} \)
17 \( 1 + 51.9T + 4.91e3T^{2} \)
19 \( 1 - 33.1T + 6.85e3T^{2} \)
23 \( 1 + 62.8T + 1.21e4T^{2} \)
29 \( 1 - 129.T + 2.43e4T^{2} \)
31 \( 1 + 242.T + 2.97e4T^{2} \)
37 \( 1 + 389.T + 5.06e4T^{2} \)
41 \( 1 + 470.T + 6.89e4T^{2} \)
43 \( 1 - 125.T + 7.95e4T^{2} \)
47 \( 1 - 386.T + 1.03e5T^{2} \)
53 \( 1 + 611.T + 1.48e5T^{2} \)
59 \( 1 + 226.T + 2.05e5T^{2} \)
61 \( 1 - 725.T + 2.26e5T^{2} \)
67 \( 1 + 1.04e3T + 3.00e5T^{2} \)
71 \( 1 + 169.T + 3.57e5T^{2} \)
73 \( 1 + 381.T + 3.89e5T^{2} \)
79 \( 1 - 1.16e3T + 4.93e5T^{2} \)
83 \( 1 - 808.T + 5.71e5T^{2} \)
89 \( 1 + 319.T + 7.04e5T^{2} \)
97 \( 1 - 1.13e3T + 9.12e5T^{2} \)
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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

−8.569302137646119533619975233665, −7.88308719331413103876208745406, −7.35681243522572896085222994328, −6.52702930523488048932359887899, −5.30975653108228084796515366446, −4.47149218209711935123478066904, −3.62803347856914513679570152679, −3.11981175771736267175130935061, −1.83703662488179994590676692329, −0.43956662763254163552126478447, 0.43956662763254163552126478447, 1.83703662488179994590676692329, 3.11981175771736267175130935061, 3.62803347856914513679570152679, 4.47149218209711935123478066904, 5.30975653108228084796515366446, 6.52702930523488048932359887899, 7.35681243522572896085222994328, 7.88308719331413103876208745406, 8.569302137646119533619975233665

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