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 − 8.16·5-s + 9·9-s − 37.8·11-s − 39.9·13-s − 24.5·15-s − 9.93·17-s − 90.4·19-s − 118.·23-s − 58.2·25-s + 27·27-s − 78.4·29-s + 92.0·31-s − 113.·33-s + 332.·37-s − 119.·39-s − 71.7·41-s + 115.·43-s − 73.5·45-s + 307.·47-s − 29.8·51-s − 403.·53-s + 308.·55-s − 271.·57-s + 593.·59-s − 333.·61-s + 326.·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.730·5-s + 0.333·9-s − 1.03·11-s − 0.851·13-s − 0.421·15-s − 0.141·17-s − 1.09·19-s − 1.07·23-s − 0.466·25-s + 0.192·27-s − 0.502·29-s + 0.533·31-s − 0.598·33-s + 1.47·37-s − 0.491·39-s − 0.273·41-s + 0.411·43-s − 0.243·45-s + 0.955·47-s − 0.0818·51-s − 1.04·53-s + 0.757·55-s − 0.630·57-s + 1.31·59-s − 0.699·61-s + 0.622·65-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

\(L(2)\) \(\approx\) \(1.224924646\)
\(L(\frac12)\) \(\approx\) \(1.224924646\)
\(L(\frac{5}{2})\) 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 \( 1 + 8.16T + 125T^{2} \)
11 \( 1 + 37.8T + 1.33e3T^{2} \)
13 \( 1 + 39.9T + 2.19e3T^{2} \)
17 \( 1 + 9.93T + 4.91e3T^{2} \)
19 \( 1 + 90.4T + 6.85e3T^{2} \)
23 \( 1 + 118.T + 1.21e4T^{2} \)
29 \( 1 + 78.4T + 2.43e4T^{2} \)
31 \( 1 - 92.0T + 2.97e4T^{2} \)
37 \( 1 - 332.T + 5.06e4T^{2} \)
41 \( 1 + 71.7T + 6.89e4T^{2} \)
43 \( 1 - 115.T + 7.95e4T^{2} \)
47 \( 1 - 307.T + 1.03e5T^{2} \)
53 \( 1 + 403.T + 1.48e5T^{2} \)
59 \( 1 - 593.T + 2.05e5T^{2} \)
61 \( 1 + 333.T + 2.26e5T^{2} \)
67 \( 1 - 743.T + 3.00e5T^{2} \)
71 \( 1 - 728.T + 3.57e5T^{2} \)
73 \( 1 - 801.T + 3.89e5T^{2} \)
79 \( 1 + 1.06e3T + 4.93e5T^{2} \)
83 \( 1 - 906.T + 5.71e5T^{2} \)
89 \( 1 + 1.11e3T + 7.04e5T^{2} \)
97 \( 1 + 1.48e3T + 9.12e5T^{2} \)
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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

−8.338941638912948123355467752026, −7.985953179768969249606102164046, −7.33804015967586631224285422015, −6.39559029664777746351088489161, −5.41794736094286262928782943348, −4.44224226278406675913669249475, −3.85764998581098195098715083384, −2.69675777371874876366061734684, −2.07243133821239135972075228488, −0.45039596435440305849176762267, 0.45039596435440305849176762267, 2.07243133821239135972075228488, 2.69675777371874876366061734684, 3.85764998581098195098715083384, 4.44224226278406675913669249475, 5.41794736094286262928782943348, 6.39559029664777746351088489161, 7.33804015967586631224285422015, 7.985953179768969249606102164046, 8.338941638912948123355467752026

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