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 + 10.6·5-s + 9·9-s + 6.65·11-s − 75.9·13-s + 31.9·15-s + 104.·17-s + 85.4·19-s + 68.6·23-s − 11.4·25-s + 27·27-s + 87.7·29-s + 62.7·31-s + 19.9·33-s + 42.2·37-s − 227.·39-s − 313.·41-s − 306.·43-s + 95.8·45-s + 215.·47-s + 312.·51-s + 525.·53-s + 70.8·55-s + 256.·57-s + 360.·59-s − 800.·61-s − 809.·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.953·5-s + 0.333·9-s + 0.182·11-s − 1.62·13-s + 0.550·15-s + 1.48·17-s + 1.03·19-s + 0.622·23-s − 0.0917·25-s + 0.192·27-s + 0.562·29-s + 0.363·31-s + 0.105·33-s + 0.187·37-s − 0.935·39-s − 1.19·41-s − 1.08·43-s + 0.317·45-s + 0.667·47-s + 0.859·51-s + 1.36·53-s + 0.173·55-s + 0.595·57-s + 0.795·59-s − 1.68·61-s − 1.54·65-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

\(L(2)\) \(\approx\) \(3.705105929\)
\(L(\frac12)\) \(\approx\) \(3.705105929\)
\(L(\frac{5}{2})\) 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 \( 1 - 10.6T + 125T^{2} \)
11 \( 1 - 6.65T + 1.33e3T^{2} \)
13 \( 1 + 75.9T + 2.19e3T^{2} \)
17 \( 1 - 104.T + 4.91e3T^{2} \)
19 \( 1 - 85.4T + 6.85e3T^{2} \)
23 \( 1 - 68.6T + 1.21e4T^{2} \)
29 \( 1 - 87.7T + 2.43e4T^{2} \)
31 \( 1 - 62.7T + 2.97e4T^{2} \)
37 \( 1 - 42.2T + 5.06e4T^{2} \)
41 \( 1 + 313.T + 6.89e4T^{2} \)
43 \( 1 + 306.T + 7.95e4T^{2} \)
47 \( 1 - 215.T + 1.03e5T^{2} \)
53 \( 1 - 525.T + 1.48e5T^{2} \)
59 \( 1 - 360.T + 2.05e5T^{2} \)
61 \( 1 + 800.T + 2.26e5T^{2} \)
67 \( 1 - 40.2T + 3.00e5T^{2} \)
71 \( 1 - 298.T + 3.57e5T^{2} \)
73 \( 1 + 517.T + 3.89e5T^{2} \)
79 \( 1 - 1.22e3T + 4.93e5T^{2} \)
83 \( 1 - 1.32e3T + 5.71e5T^{2} \)
89 \( 1 - 639.T + 7.04e5T^{2} \)
97 \( 1 + 1.42e3T + 9.12e5T^{2} \)
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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

−8.731984558431045861405934615024, −7.75367671630912342921493856329, −7.25802872577714727631928653194, −6.32850620208133741237664013587, −5.33892012148367356742345086038, −4.87256475181815114387385599764, −3.53629202192852517029617313025, −2.77577043148608344370699657733, −1.89657453017150708096786866809, −0.852218576134019872757778431108, 0.852218576134019872757778431108, 1.89657453017150708096786866809, 2.77577043148608344370699657733, 3.53629202192852517029617313025, 4.87256475181815114387385599764, 5.33892012148367356742345086038, 6.32850620208133741237664013587, 7.25802872577714727631928653194, 7.75367671630912342921493856329, 8.731984558431045861405934615024

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