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 + 14.6·5-s + 9·9-s + 11.4·11-s + 20.8·13-s − 43.9·15-s − 18.8·17-s + 91.0·19-s − 28.9·23-s + 89.3·25-s − 27·27-s + 281.·29-s − 276.·31-s − 34.3·33-s + 250.·37-s − 62.4·39-s − 12.1·41-s − 65.2·43-s + 131.·45-s + 199.·47-s + 56.4·51-s − 64.1·53-s + 167.·55-s − 273.·57-s − 69.0·59-s + 656.·61-s + 304.·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.30·5-s + 0.333·9-s + 0.313·11-s + 0.443·13-s − 0.756·15-s − 0.268·17-s + 1.09·19-s − 0.262·23-s + 0.715·25-s − 0.192·27-s + 1.80·29-s − 1.59·31-s − 0.181·33-s + 1.11·37-s − 0.256·39-s − 0.0462·41-s − 0.231·43-s + 0.436·45-s + 0.620·47-s + 0.155·51-s − 0.166·53-s + 0.410·55-s − 0.634·57-s − 0.152·59-s + 1.37·61-s + 0.581·65-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

\(L(2)\) \(\approx\) \(2.739938897\)
\(L(\frac12)\) \(\approx\) \(2.739938897\)
\(L(\frac{5}{2})\) 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 \( 1 - 14.6T + 125T^{2} \)
11 \( 1 - 11.4T + 1.33e3T^{2} \)
13 \( 1 - 20.8T + 2.19e3T^{2} \)
17 \( 1 + 18.8T + 4.91e3T^{2} \)
19 \( 1 - 91.0T + 6.85e3T^{2} \)
23 \( 1 + 28.9T + 1.21e4T^{2} \)
29 \( 1 - 281.T + 2.43e4T^{2} \)
31 \( 1 + 276.T + 2.97e4T^{2} \)
37 \( 1 - 250.T + 5.06e4T^{2} \)
41 \( 1 + 12.1T + 6.89e4T^{2} \)
43 \( 1 + 65.2T + 7.95e4T^{2} \)
47 \( 1 - 199.T + 1.03e5T^{2} \)
53 \( 1 + 64.1T + 1.48e5T^{2} \)
59 \( 1 + 69.0T + 2.05e5T^{2} \)
61 \( 1 - 656.T + 2.26e5T^{2} \)
67 \( 1 - 419.T + 3.00e5T^{2} \)
71 \( 1 + 1.02e3T + 3.57e5T^{2} \)
73 \( 1 - 103.T + 3.89e5T^{2} \)
79 \( 1 + 260.T + 4.93e5T^{2} \)
83 \( 1 - 879.T + 5.71e5T^{2} \)
89 \( 1 - 1.48e3T + 7.04e5T^{2} \)
97 \( 1 - 177.T + 9.12e5T^{2} \)
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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

−8.834439404436465735621568066058, −7.80369787697610890467946316732, −6.86917658769974071454426939715, −6.21006689578783724856102304888, −5.59240629846379069101675569230, −4.87554662771421914713538876669, −3.81501351750869924358437545032, −2.67330448953823047186293382137, −1.67312514306597313840634644695, −0.800498491916692887339731902263, 0.800498491916692887339731902263, 1.67312514306597313840634644695, 2.67330448953823047186293382137, 3.81501351750869924358437545032, 4.87554662771421914713538876669, 5.59240629846379069101675569230, 6.21006689578783724856102304888, 6.86917658769974071454426939715, 7.80369787697610890467946316732, 8.834439404436465735621568066058

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