Properties

Degree $2$
Conductor $9408$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s + 9-s − 6·11-s − 3·13-s + 2·15-s − 4·17-s + 5·19-s + 4·23-s − 25-s + 27-s + 4·29-s + 7·31-s − 6·33-s + 9·37-s − 3·39-s + 2·41-s − 43-s + 2·45-s + 2·47-s − 4·51-s − 8·53-s − 12·55-s + 5·57-s + 10·61-s − 6·65-s − 15·67-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.80·11-s − 0.832·13-s + 0.516·15-s − 0.970·17-s + 1.14·19-s + 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.742·29-s + 1.25·31-s − 1.04·33-s + 1.47·37-s − 0.480·39-s + 0.312·41-s − 0.152·43-s + 0.298·45-s + 0.291·47-s − 0.560·51-s − 1.09·53-s − 1.61·55-s + 0.662·57-s + 1.28·61-s − 0.744·65-s − 1.83·67-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: \(0\)
Selberg data: \((2,\ 9408,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.651024325\)
\(L(\frac12)\) \(\approx\) \(2.651024325\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 15 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−7.81353925695778742696645984148, −7.10670595704600128790727444218, −6.36345186673824788020069626177, −5.55411270528915953242583826978, −4.95179140945389843725361192855, −4.39549585954368139301756377391, −3.03355910817079593112688407362, −2.67283925129501260972827793377, −1.99619127590767462191308397743, −0.73522365973514908809032779124, 0.73522365973514908809032779124, 1.99619127590767462191308397743, 2.67283925129501260972827793377, 3.03355910817079593112688407362, 4.39549585954368139301756377391, 4.95179140945389843725361192855, 5.55411270528915953242583826978, 6.36345186673824788020069626177, 7.10670595704600128790727444218, 7.81353925695778742696645984148

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