Properties

Label 2-96e2-1.1-c1-0-129
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  + 0.378·7-s + 3.48·13-s + 3.46·19-s − 5·25-s − 10.1·31-s − 2.17·37-s − 10.3·43-s − 6.85·49-s − 14.7·61-s − 16·67-s + 13.8·73-s + 9.41·79-s + 1.32·91-s + 13.8·97-s + 11.6·103-s + 16.1·109-s + ⋯
L(s)  = 1  + 0.143·7-s + 0.966·13-s + 0.794·19-s − 25-s − 1.82·31-s − 0.357·37-s − 1.58·43-s − 0.979·49-s − 1.89·61-s − 1.95·67-s + 1.62·73-s + 1.05·79-s + 0.138·91-s + 1.40·97-s + 1.15·103-s + 1.54·109-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 \( 1 - 0.378T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 3.48T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + 2.17T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 14.7T + 61T^{2} \)
67 \( 1 + 16T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 - 9.41T + 79T^{2} \)
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

−7.55503101313494924666217368445, −6.62618106138811124186833099516, −6.02785302153785115512040732434, −5.33346530256710295936851411287, −4.65148409012182754932447750642, −3.63510243770382464811985115218, −3.29487586134192794883138674384, −2.02828459907767621748096648731, −1.34925923964010256885618909609, 0, 1.34925923964010256885618909609, 2.02828459907767621748096648731, 3.29487586134192794883138674384, 3.63510243770382464811985115218, 4.65148409012182754932447750642, 5.33346530256710295936851411287, 6.02785302153785115512040732434, 6.62618106138811124186833099516, 7.55503101313494924666217368445

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