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 + 14.5·5-s + 9·9-s − 10.1·11-s − 0.623·13-s − 43.5·15-s − 16.8·17-s − 75.7·19-s − 46.7·23-s + 85.5·25-s − 27·27-s + 73.5·29-s + 135.·31-s + 30.3·33-s + 133.·37-s + 1.87·39-s − 69.3·41-s − 279.·43-s + 130.·45-s − 372.·47-s + 50.4·51-s + 656.·53-s − 146.·55-s + 227.·57-s + 730.·59-s + 39.5·61-s − 9.05·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.29·5-s + 0.333·9-s − 0.277·11-s − 0.0133·13-s − 0.749·15-s − 0.239·17-s − 0.915·19-s − 0.423·23-s + 0.684·25-s − 0.192·27-s + 0.470·29-s + 0.784·31-s + 0.160·33-s + 0.591·37-s + 0.00768·39-s − 0.263·41-s − 0.990·43-s + 0.432·45-s − 1.15·47-s + 0.138·51-s + 1.70·53-s − 0.359·55-s + 0.528·57-s + 1.61·59-s + 0.0829·61-s − 0.0172·65-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

\(L(2)\) \(\approx\) \(2.130222616\)
\(L(\frac12)\) \(\approx\) \(2.130222616\)
\(L(\frac{5}{2})\) 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 \( 1 - 14.5T + 125T^{2} \)
11 \( 1 + 10.1T + 1.33e3T^{2} \)
13 \( 1 + 0.623T + 2.19e3T^{2} \)
17 \( 1 + 16.8T + 4.91e3T^{2} \)
19 \( 1 + 75.7T + 6.85e3T^{2} \)
23 \( 1 + 46.7T + 1.21e4T^{2} \)
29 \( 1 - 73.5T + 2.43e4T^{2} \)
31 \( 1 - 135.T + 2.97e4T^{2} \)
37 \( 1 - 133.T + 5.06e4T^{2} \)
41 \( 1 + 69.3T + 6.89e4T^{2} \)
43 \( 1 + 279.T + 7.95e4T^{2} \)
47 \( 1 + 372.T + 1.03e5T^{2} \)
53 \( 1 - 656.T + 1.48e5T^{2} \)
59 \( 1 - 730.T + 2.05e5T^{2} \)
61 \( 1 - 39.5T + 2.26e5T^{2} \)
67 \( 1 + 417.T + 3.00e5T^{2} \)
71 \( 1 - 831.T + 3.57e5T^{2} \)
73 \( 1 - 1.19e3T + 3.89e5T^{2} \)
79 \( 1 - 1.03e3T + 4.93e5T^{2} \)
83 \( 1 + 117.T + 5.71e5T^{2} \)
89 \( 1 + 678.T + 7.04e5T^{2} \)
97 \( 1 - 1.65e3T + 9.12e5T^{2} \)
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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

−8.652942027554562718687106140973, −7.943300427266794056006878061027, −6.69259955664938555633765276700, −6.40014893433255897060794146361, −5.49439480836448642090401600645, −4.89103495386709809954805240351, −3.86720714161151397618292153121, −2.54767705438025116152259258321, −1.84374970390552561806724164713, −0.66504521738311640956374860662, 0.66504521738311640956374860662, 1.84374970390552561806724164713, 2.54767705438025116152259258321, 3.86720714161151397618292153121, 4.89103495386709809954805240351, 5.49439480836448642090401600645, 6.40014893433255897060794146361, 6.69259955664938555633765276700, 7.943300427266794056006878061027, 8.652942027554562718687106140973

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