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 + 4.13·5-s + 9-s + 2.29·11-s − 2.43·13-s − 4.13·15-s − 3.71·17-s − 7.84·19-s − 1.70·23-s + 12.0·25-s − 27-s − 3.25·29-s − 1.80·31-s − 2.29·33-s + 5.65·37-s + 2.43·39-s + 6.81·41-s − 5.44·43-s + 4.13·45-s − 13.2·47-s + 3.71·51-s − 9.09·53-s + 9.50·55-s + 7.84·57-s + 14.9·59-s + 2.16·61-s − 10.0·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.84·5-s + 0.333·9-s + 0.693·11-s − 0.674·13-s − 1.06·15-s − 0.900·17-s − 1.80·19-s − 0.354·23-s + 2.41·25-s − 0.192·27-s − 0.603·29-s − 0.324·31-s − 0.400·33-s + 0.929·37-s + 0.389·39-s + 1.06·41-s − 0.829·43-s + 0.616·45-s − 1.93·47-s + 0.519·51-s − 1.24·53-s + 1.28·55-s + 1.03·57-s + 1.94·59-s + 0.277·61-s − 1.24·65-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 \( 1 - 4.13T + 5T^{2} \)
11 \( 1 - 2.29T + 11T^{2} \)
13 \( 1 + 2.43T + 13T^{2} \)
17 \( 1 + 3.71T + 17T^{2} \)
19 \( 1 + 7.84T + 19T^{2} \)
23 \( 1 + 1.70T + 23T^{2} \)
29 \( 1 + 3.25T + 29T^{2} \)
31 \( 1 + 1.80T + 31T^{2} \)
37 \( 1 - 5.65T + 37T^{2} \)
41 \( 1 - 6.81T + 41T^{2} \)
43 \( 1 + 5.44T + 43T^{2} \)
47 \( 1 + 13.2T + 47T^{2} \)
53 \( 1 + 9.09T + 53T^{2} \)
59 \( 1 - 14.9T + 59T^{2} \)
61 \( 1 - 2.16T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 - 3.95T + 71T^{2} \)
73 \( 1 + 9.68T + 73T^{2} \)
79 \( 1 + 10.0T + 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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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

−7.00108494286456351253541727330, −6.51109054045034091050652935717, −6.10830919919443025978611546850, −5.41608629921436858138843734300, −4.68417843644144082043038189682, −4.06208818729085408413804552982, −2.74392897487219044537449184496, −2.04921832936424813394860869877, −1.45412699182750151296424549210, 0, 1.45412699182750151296424549210, 2.04921832936424813394860869877, 2.74392897487219044537449184496, 4.06208818729085408413804552982, 4.68417843644144082043038189682, 5.41608629921436858138843734300, 6.10830919919443025978611546850, 6.51109054045034091050652935717, 7.00108494286456351253541727330

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