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 + 3.20·5-s + 9-s − 4.24·11-s + 3.15·13-s − 3.20·15-s − 4.40·17-s − 3.15·19-s + 4.40·23-s + 5.24·25-s − 27-s − 7.20·29-s + 2.04·31-s + 4.24·33-s + 9.65·37-s − 3.15·39-s − 10.4·41-s − 0.750·43-s + 3.20·45-s − 2.40·47-s + 4.40·51-s + 3.29·53-s − 13.6·55-s + 3.15·57-s − 8.24·59-s + 8.09·61-s + 10.0·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.43·5-s + 0.333·9-s − 1.28·11-s + 0.874·13-s − 0.826·15-s − 1.06·17-s − 0.723·19-s + 0.918·23-s + 1.04·25-s − 0.192·27-s − 1.33·29-s + 0.367·31-s + 0.739·33-s + 1.58·37-s − 0.504·39-s − 1.63·41-s − 0.114·43-s + 0.477·45-s − 0.350·47-s + 0.616·51-s + 0.452·53-s − 1.83·55-s + 0.417·57-s − 1.07·59-s + 1.03·61-s + 1.25·65-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 \( 1 - 3.20T + 5T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 - 3.15T + 13T^{2} \)
17 \( 1 + 4.40T + 17T^{2} \)
19 \( 1 + 3.15T + 19T^{2} \)
23 \( 1 - 4.40T + 23T^{2} \)
29 \( 1 + 7.20T + 29T^{2} \)
31 \( 1 - 2.04T + 31T^{2} \)
37 \( 1 - 9.65T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 + 0.750T + 43T^{2} \)
47 \( 1 + 2.40T + 47T^{2} \)
53 \( 1 - 3.29T + 53T^{2} \)
59 \( 1 + 8.24T + 59T^{2} \)
61 \( 1 - 8.09T + 61T^{2} \)
67 \( 1 + 5.15T + 67T^{2} \)
71 \( 1 - 6.40T + 71T^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 + 16.4T + 79T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 - 2.40T + 89T^{2} \)
97 \( 1 - 4.24T + 97T^{2} \)
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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

−7.16994662981887554743564024795, −6.49729110467948366785857685470, −6.00964920184741095216465889215, −5.35521093745068329349696699794, −4.84321290027232170982453048277, −3.92472690580912264738824293409, −2.80303143413862762195587299371, −2.14930174716000056105808740955, −1.32039026572221818321214680682, 0, 1.32039026572221818321214680682, 2.14930174716000056105808740955, 2.80303143413862762195587299371, 3.92472690580912264738824293409, 4.84321290027232170982453048277, 5.35521093745068329349696699794, 6.00964920184741095216465889215, 6.49729110467948366785857685470, 7.16994662981887554743564024795

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