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 + 2·5-s + 9-s − 2·11-s − 4·13-s + 2·15-s − 6·17-s + 8·19-s − 6·23-s − 25-s + 27-s + 10·29-s − 4·31-s − 2·33-s − 6·37-s − 4·39-s + 6·41-s − 4·43-s + 2·45-s − 8·47-s − 6·51-s − 2·53-s − 4·55-s + 8·57-s − 4·59-s − 8·61-s − 8·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 0.516·15-s − 1.45·17-s + 1.83·19-s − 1.25·23-s − 1/5·25-s + 0.192·27-s + 1.85·29-s − 0.718·31-s − 0.348·33-s − 0.986·37-s − 0.640·39-s + 0.937·41-s − 0.609·43-s + 0.298·45-s − 1.16·47-s − 0.840·51-s − 0.274·53-s − 0.539·55-s + 1.05·57-s − 0.520·59-s − 1.02·61-s − 0.992·65-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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−7.50054219932255182031639197944, −6.66646405864906805591665057994, −6.07661650808441808640686227898, −5.11648933272940261648105167766, −4.79314821415251420536899717164, −3.73563079980436139104373699234, −2.81614860024748248378813859786, −2.29214130621326749347781840762, −1.48049640836312376553144082550, 0, 1.48049640836312376553144082550, 2.29214130621326749347781840762, 2.81614860024748248378813859786, 3.73563079980436139104373699234, 4.79314821415251420536899717164, 5.11648933272940261648105167766, 6.07661650808441808640686227898, 6.66646405864906805591665057994, 7.50054219932255182031639197944

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