Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3.41·5-s + 9-s − 0.828·11-s − 4.24·13-s − 3.41·15-s − 7.41·17-s + 6.82·19-s − 4.82·23-s + 6.65·25-s − 27-s − 2.82·29-s + 2.82·31-s + 0.828·33-s + 1.65·37-s + 4.24·39-s − 10.2·41-s + 11.3·43-s + 3.41·45-s + 4.48·47-s + 7.41·51-s + 2·53-s − 2.82·55-s − 6.82·57-s + 8.48·59-s + 11.0·61-s − 14.4·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.52·5-s + 0.333·9-s − 0.249·11-s − 1.17·13-s − 0.881·15-s − 1.79·17-s + 1.56·19-s − 1.00·23-s + 1.33·25-s − 0.192·27-s − 0.525·29-s + 0.508·31-s + 0.144·33-s + 0.272·37-s + 0.679·39-s − 1.59·41-s + 1.72·43-s + 0.508·45-s + 0.654·47-s + 1.03·51-s + 0.274·53-s − 0.381·55-s − 0.904·57-s + 1.10·59-s + 1.41·61-s − 1.79·65-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: \(1\)
Selberg data: \((2,\ 9408,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.41T + 5T^{2} \)
11 \( 1 + 0.828T + 11T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + 7.41T + 17T^{2} \)
19 \( 1 - 6.82T + 19T^{2} \)
23 \( 1 + 4.82T + 23T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 - 2.82T + 31T^{2} \)
37 \( 1 - 1.65T + 37T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 - 4.48T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 8.48T + 59T^{2} \)
61 \( 1 - 11.0T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 + 7.75T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + 5.75T + 89T^{2} \)
97 \( 1 - 0.242T + 97T^{2} \)
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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

−7.13375207033334933103282385731, −6.68396337751472781930153468536, −5.83911055540854050457606803128, −5.44384045371663765909036901586, −4.78503278027433362046481685777, −4.00348074781870620256710503447, −2.67266292166849134808137184604, −2.25064078013074864030346091015, −1.30759791282426558021598316197, 0, 1.30759791282426558021598316197, 2.25064078013074864030346091015, 2.67266292166849134808137184604, 4.00348074781870620256710503447, 4.78503278027433362046481685777, 5.44384045371663765909036901586, 5.83911055540854050457606803128, 6.68396337751472781930153468536, 7.13375207033334933103282385731

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