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 + 2.82·5-s + 9-s − 2.82·11-s − 2.82·15-s + 2.82·17-s − 4·19-s − 8.48·23-s + 3.00·25-s − 27-s − 2·29-s + 2.82·33-s + 6·37-s + 8.48·41-s + 11.3·43-s + 2.82·45-s + 8·47-s − 2.82·51-s − 6·53-s − 8.00·55-s + 4·57-s − 12·59-s − 5.65·61-s − 5.65·67-s + 8.48·69-s − 2.82·71-s − 5.65·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.26·5-s + 0.333·9-s − 0.852·11-s − 0.730·15-s + 0.685·17-s − 0.917·19-s − 1.76·23-s + 0.600·25-s − 0.192·27-s − 0.371·29-s + 0.492·33-s + 0.986·37-s + 1.32·41-s + 1.72·43-s + 0.421·45-s + 1.16·47-s − 0.396·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s − 1.56·59-s − 0.724·61-s − 0.691·67-s + 1.02·69-s − 0.335·71-s − 0.662·73-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 \( 1 - 2.82T + 5T^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 8.48T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 8.48T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 5.65T + 61T^{2} \)
67 \( 1 + 5.65T + 67T^{2} \)
71 \( 1 + 2.82T + 71T^{2} \)
73 \( 1 + 5.65T + 73T^{2} \)
79 \( 1 + 5.65T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 2.82T + 89T^{2} \)
97 \( 1 + 16.9T + 97T^{2} \)
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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

−7.51770160447679466405554167716, −6.34935909758512881236318372356, −5.86716003977435541880860976679, −5.66467579258691980208546520947, −4.60721313110744683055976063586, −4.04028781612530651836348719963, −2.77489630792132571537210518484, −2.18230947785858181304749472356, −1.29007633253592092844614777481, 0, 1.29007633253592092844614777481, 2.18230947785858181304749472356, 2.77489630792132571537210518484, 4.04028781612530651836348719963, 4.60721313110744683055976063586, 5.66467579258691980208546520947, 5.86716003977435541880860976679, 6.34935909758512881236318372356, 7.51770160447679466405554167716

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