Properties

Degree $2$
Conductor $9408$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3.27·5-s + 9-s − 3.27·11-s − 6.27·13-s − 3.27·15-s − 4·17-s + 6.27·19-s + 4·23-s + 5.72·25-s + 27-s − 5.27·29-s − 31-s − 3.27·33-s + 2.27·37-s − 6.27·39-s − 4.54·41-s − 0.274·43-s − 3.27·45-s − 6·47-s − 4·51-s − 9.27·53-s + 10.7·55-s + 6.27·57-s + 1.27·59-s − 10·61-s + 20.5·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.46·5-s + 0.333·9-s − 0.987·11-s − 1.74·13-s − 0.845·15-s − 0.970·17-s + 1.43·19-s + 0.834·23-s + 1.14·25-s + 0.192·27-s − 0.979·29-s − 0.179·31-s − 0.570·33-s + 0.373·37-s − 1.00·39-s − 0.710·41-s − 0.0419·43-s − 0.488·45-s − 0.875·47-s − 0.560·51-s − 1.27·53-s + 1.44·55-s + 0.831·57-s + 0.165·59-s − 1.28·61-s + 2.54·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9408\)    =    \(2^{6} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{9408} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9408,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8129602825\)
\(L(\frac12)\) \(\approx\) \(0.8129602825\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
good5 \( 1 + 3.27T + 5T^{2} \)
11 \( 1 + 3.27T + 11T^{2} \)
13 \( 1 + 6.27T + 13T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 - 6.27T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 5.27T + 29T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 - 2.27T + 37T^{2} \)
41 \( 1 + 4.54T + 41T^{2} \)
43 \( 1 + 0.274T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 9.27T + 53T^{2} \)
59 \( 1 - 1.27T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 0.274T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 4.27T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 + 7.27T + 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 - 8.72T + 97T^{2} \)
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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

−7.66328531982910988548536257136, −7.33126789711657316884512611365, −6.65380468460929706355078888468, −5.34953985618329060208957704906, −4.86044661479823303219741803988, −4.24471567835145342075536636457, −3.24488238917503924636695537243, −2.88047509841753621674424224774, −1.86465137484388749313564147516, −0.39781015255148828026636074956, 0.39781015255148828026636074956, 1.86465137484388749313564147516, 2.88047509841753621674424224774, 3.24488238917503924636695537243, 4.24471567835145342075536636457, 4.86044661479823303219741803988, 5.34953985618329060208957704906, 6.65380468460929706355078888468, 7.33126789711657316884512611365, 7.66328531982910988548536257136

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