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 + 9-s − 2·11-s − 13-s − 2·17-s + 5·19-s − 6·23-s − 5·25-s − 27-s + 8·29-s + 3·31-s + 2·33-s + 9·37-s + 39-s + 2·41-s + 43-s − 8·47-s + 2·51-s − 6·53-s − 5·57-s + 6·59-s + 2·61-s − 5·67-s + 6·69-s − 4·71-s − 11·73-s + 5·75-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 0.485·17-s + 1.14·19-s − 1.25·23-s − 25-s − 0.192·27-s + 1.48·29-s + 0.538·31-s + 0.348·33-s + 1.47·37-s + 0.160·39-s + 0.312·41-s + 0.152·43-s − 1.16·47-s + 0.280·51-s − 0.824·53-s − 0.662·57-s + 0.781·59-s + 0.256·61-s − 0.610·67-s + 0.722·69-s − 0.474·71-s − 1.28·73-s + 0.577·75-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 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−7.58375107691877218755030320416, −6.43182411277543087472216858535, −6.14236856942922112406158983668, −5.25699149983290473950556496989, −4.67971751550780395389568173757, −3.96187662121981911608523900945, −2.96627158213516769548423321206, −2.20372611588543136801007546913, −1.10379900433015833862739804009, 0, 1.10379900433015833862739804009, 2.20372611588543136801007546913, 2.96627158213516769548423321206, 3.96187662121981911608523900945, 4.67971751550780395389568173757, 5.25699149983290473950556496989, 6.14236856942922112406158983668, 6.43182411277543087472216858535, 7.58375107691877218755030320416

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