Properties

Label 2-9408-1.1-c1-0-55
Degree $2$
Conductor $9408$
Sign $1$
Analytic cond. $75.1232$
Root an. cond. $8.66736$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3.27·5-s + 9-s − 3.27·11-s + 6.27·13-s − 3.27·15-s + 4·17-s − 6.27·19-s + 4·23-s + 5.72·25-s − 27-s − 5.27·29-s + 31-s + 3.27·33-s + 2.27·37-s − 6.27·39-s + 4.54·41-s − 0.274·43-s + 3.27·45-s + 6·47-s − 4·51-s − 9.27·53-s − 10.7·55-s + 6.27·57-s − 1.27·59-s + 10·61-s + 20.5·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.46·5-s + 0.333·9-s − 0.987·11-s + 1.74·13-s − 0.845·15-s + 0.970·17-s − 1.43·19-s + 0.834·23-s + 1.14·25-s − 0.192·27-s − 0.979·29-s + 0.179·31-s + 0.570·33-s + 0.373·37-s − 1.00·39-s + 0.710·41-s − 0.0419·43-s + 0.488·45-s + 0.875·47-s − 0.560·51-s − 1.27·53-s − 1.44·55-s + 0.831·57-s − 0.165·59-s + 1.28·61-s + 2.54·65-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: $1$
Analytic conductor: \(75.1232\)
Root analytic conductor: \(8.66736\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9408,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.467045613\)
\(L(\frac12)\) \(\approx\) \(2.467045613\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.27T + 5T^{2} \)
11 \( 1 + 3.27T + 11T^{2} \)
13 \( 1 - 6.27T + 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + 6.27T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 5.27T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 - 2.27T + 37T^{2} \)
41 \( 1 - 4.54T + 41T^{2} \)
43 \( 1 + 0.274T + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + 9.27T + 53T^{2} \)
59 \( 1 + 1.27T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 0.274T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 4.27T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 - 7.27T + 83T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 + 8.72T + 97T^{2} \)
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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

−7.76119634281237893591342427928, −6.70969853403009036705803696575, −6.26546714427569278029189793645, −5.62288555779463556181298404206, −5.28523032267949041717496278463, −4.29097249145847133527920600111, −3.41851975336195283532751203654, −2.45649385784649586130640903739, −1.69292327475941469974604554495, −0.805992375472753294090837094381, 0.805992375472753294090837094381, 1.69292327475941469974604554495, 2.45649385784649586130640903739, 3.41851975336195283532751203654, 4.29097249145847133527920600111, 5.28523032267949041717496278463, 5.62288555779463556181298404206, 6.26546714427569278029189793645, 6.70969853403009036705803696575, 7.76119634281237893591342427928

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