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 − 0.585·5-s + 9-s + 0.828·11-s + 1.41·13-s + 0.585·15-s + 2.24·17-s − 6.82·19-s + 4.82·23-s − 4.65·25-s − 27-s − 8.48·29-s + 5.17·31-s − 0.828·33-s − 1.65·37-s − 1.41·39-s − 0.585·41-s + 8·43-s − 0.585·45-s + 6.82·47-s − 2.24·51-s + 13.3·53-s − 0.485·55-s + 6.82·57-s − 5.17·59-s − 13.8·61-s − 0.828·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.261·5-s + 0.333·9-s + 0.249·11-s + 0.392·13-s + 0.151·15-s + 0.543·17-s − 1.56·19-s + 1.00·23-s − 0.931·25-s − 0.192·27-s − 1.57·29-s + 0.928·31-s − 0.144·33-s − 0.272·37-s − 0.226·39-s − 0.0914·41-s + 1.21·43-s − 0.0873·45-s + 0.996·47-s − 0.314·51-s + 1.82·53-s − 0.0654·55-s + 0.904·57-s − 0.673·59-s − 1.77·61-s − 0.102·65-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 \( 1 + 0.585T + 5T^{2} \)
11 \( 1 - 0.828T + 11T^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
17 \( 1 - 2.24T + 17T^{2} \)
19 \( 1 + 6.82T + 19T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 + 8.48T + 29T^{2} \)
31 \( 1 - 5.17T + 31T^{2} \)
37 \( 1 + 1.65T + 37T^{2} \)
41 \( 1 + 0.585T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 6.82T + 47T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 + 5.17T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 0.828T + 71T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 + 2.34T + 79T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + 7.75T + 97T^{2} \)
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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

−7.29637436743534577713554355704, −6.69737356723734448609093185346, −5.84420023150915375503525269754, −5.55119412452877128167365991855, −4.35854454827406274967142386740, −4.09720697182944309693238985361, −3.08851288744705720909537427889, −2.09096610714618596634542791377, −1.13383660485897597425103463939, 0, 1.13383660485897597425103463939, 2.09096610714618596634542791377, 3.08851288744705720909537427889, 4.09720697182944309693238985361, 4.35854454827406274967142386740, 5.55119412452877128167365991855, 5.84420023150915375503525269754, 6.69737356723734448609093185346, 7.29637436743534577713554355704

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