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 + 3.04·5-s + 9-s + 3.93·11-s + 4.88·13-s + 3.04·15-s + 5.34·17-s − 2.30·19-s + 7.93·23-s + 4.25·25-s + 27-s − 5.55·29-s + 0.645·31-s + 3.93·33-s − 5.65·37-s + 4.88·39-s − 10.0·41-s + 8.91·43-s + 3.04·45-s + 6.61·47-s + 5.34·51-s − 1.25·53-s + 11.9·55-s − 2.30·57-s + 3.04·59-s − 2.97·61-s + 14.8·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.36·5-s + 0.333·9-s + 1.18·11-s + 1.35·13-s + 0.785·15-s + 1.29·17-s − 0.528·19-s + 1.65·23-s + 0.851·25-s + 0.192·27-s − 1.03·29-s + 0.115·31-s + 0.684·33-s − 0.929·37-s + 0.782·39-s − 1.57·41-s + 1.35·43-s + 0.453·45-s + 0.964·47-s + 0.748·51-s − 0.172·53-s + 1.61·55-s − 0.304·57-s + 0.396·59-s − 0.380·61-s + 1.84·65-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

\(L(1)\) \(\approx\) \(4.634610556\)
\(L(\frac12)\) \(\approx\) \(4.634610556\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.04T + 5T^{2} \)
11 \( 1 - 3.93T + 11T^{2} \)
13 \( 1 - 4.88T + 13T^{2} \)
17 \( 1 - 5.34T + 17T^{2} \)
19 \( 1 + 2.30T + 19T^{2} \)
23 \( 1 - 7.93T + 23T^{2} \)
29 \( 1 + 5.55T + 29T^{2} \)
31 \( 1 - 0.645T + 31T^{2} \)
37 \( 1 + 5.65T + 37T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 - 8.91T + 43T^{2} \)
47 \( 1 - 6.61T + 47T^{2} \)
53 \( 1 + 1.25T + 53T^{2} \)
59 \( 1 - 3.04T + 59T^{2} \)
61 \( 1 + 2.97T + 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + 4.67T + 73T^{2} \)
79 \( 1 - 1.05T + 79T^{2} \)
83 \( 1 - 8.60T + 83T^{2} \)
89 \( 1 + 4.85T + 89T^{2} \)
97 \( 1 + 18.3T + 97T^{2} \)
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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

−7.69786840031974159598767997332, −6.89851653984685002716988899000, −6.36025258291813769711978918023, −5.69681940671574214756329395579, −5.11137462637700636458642321367, −3.97688846144305640247017059653, −3.46412126946019179475609358687, −2.59605252897496069167722099109, −1.55537560428189180123287616461, −1.19578799521226615499487068169, 1.19578799521226615499487068169, 1.55537560428189180123287616461, 2.59605252897496069167722099109, 3.46412126946019179475609358687, 3.97688846144305640247017059653, 5.11137462637700636458642321367, 5.69681940671574214756329395579, 6.36025258291813769711978918023, 6.89851653984685002716988899000, 7.69786840031974159598767997332

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