Properties

Label 2-96e2-1.1-c1-0-140
Degree $2$
Conductor $9216$
Sign $-1$
Analytic cond. $73.5901$
Root an. cond. $8.57846$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.27·7-s − 6.31·13-s − 3.46·19-s − 5·25-s + 4.52·31-s − 11.9·37-s + 10.3·43-s + 20.8·49-s − 5.00·61-s − 16·67-s − 13.8·73-s − 15.0·79-s − 33.3·91-s − 13.8·97-s + 16.5·103-s − 13.2·109-s + ⋯
L(s)  = 1  + 1.99·7-s − 1.75·13-s − 0.794·19-s − 25-s + 0.811·31-s − 1.96·37-s + 1.58·43-s + 2.97·49-s − 0.640·61-s − 1.95·67-s − 1.62·73-s − 1.69·79-s − 3.49·91-s − 1.40·97-s + 1.63·103-s − 1.27·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9216\)    =    \(2^{10} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(73.5901\)
Root analytic conductor: \(8.57846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{9216} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9216,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good5 \( 1 + 5T^{2} \)
7 \( 1 - 5.27T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 6.31T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4.52T + 31T^{2} \)
37 \( 1 + 11.9T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 5.00T + 61T^{2} \)
67 \( 1 + 16T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−7.54819145834121486924122582681, −6.89905346564336736789714322690, −5.80364309809751450505556660585, −5.26227966166048988244772824239, −4.50558026123186279143574713940, −4.21742800480488082328991502184, −2.83198476199136641689326549927, −2.09787078974469372903157306254, −1.43425819682647374049095168760, 0, 1.43425819682647374049095168760, 2.09787078974469372903157306254, 2.83198476199136641689326549927, 4.21742800480488082328991502184, 4.50558026123186279143574713940, 5.26227966166048988244772824239, 5.80364309809751450505556660585, 6.89905346564336736789714322690, 7.54819145834121486924122582681

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