Properties

Label 2-96e2-1.1-c1-0-12
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.79·5-s − 0.158·7-s − 5.37·11-s − 5.95·13-s + 3.05·17-s − 3.05·19-s + 2.82·23-s − 1.76·25-s + 2.96·29-s + 4.15·31-s + 0.285·35-s − 8.46·37-s − 2.60·41-s − 8.13·43-s − 2.82·47-s − 6.97·49-s + 5.03·53-s + 9.65·55-s − 5.65·59-s − 5.18·61-s + 10.7·65-s − 1.08·67-s − 0.317·71-s + 1.33·73-s + 0.853·77-s + 9.69·79-s − 0.163·83-s + ⋯
L(s)  = 1  − 0.804·5-s − 0.0600·7-s − 1.61·11-s − 1.65·13-s + 0.740·17-s − 0.700·19-s + 0.589·23-s − 0.353·25-s + 0.551·29-s + 0.746·31-s + 0.0483·35-s − 1.39·37-s − 0.406·41-s − 1.24·43-s − 0.412·47-s − 0.996·49-s + 0.690·53-s + 1.30·55-s − 0.736·59-s − 0.664·61-s + 1.32·65-s − 0.132·67-s − 0.0377·71-s + 0.156·73-s + 0.0972·77-s + 1.09·79-s − 0.0179·83-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: \(0\)
Selberg data: \((2,\ 9216,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5036208204\)
\(L(\frac12)\) \(\approx\) \(0.5036208204\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.79T + 5T^{2} \)
7 \( 1 + 0.158T + 7T^{2} \)
11 \( 1 + 5.37T + 11T^{2} \)
13 \( 1 + 5.95T + 13T^{2} \)
17 \( 1 - 3.05T + 17T^{2} \)
19 \( 1 + 3.05T + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 - 2.96T + 29T^{2} \)
31 \( 1 - 4.15T + 31T^{2} \)
37 \( 1 + 8.46T + 37T^{2} \)
41 \( 1 + 2.60T + 41T^{2} \)
43 \( 1 + 8.13T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 5.03T + 53T^{2} \)
59 \( 1 + 5.65T + 59T^{2} \)
61 \( 1 + 5.18T + 61T^{2} \)
67 \( 1 + 1.08T + 67T^{2} \)
71 \( 1 + 0.317T + 71T^{2} \)
73 \( 1 - 1.33T + 73T^{2} \)
79 \( 1 - 9.69T + 79T^{2} \)
83 \( 1 + 0.163T + 83T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + 0.571T + 97T^{2} \)
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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

−7.85290218150113295648272926386, −7.14260519689702900339450513185, −6.51878072633939075946485028254, −5.39243283953204492780182737324, −5.01011804036323856153603482135, −4.35111986509268644094144722114, −3.27363032643052051363211392454, −2.77420661818527796200533177698, −1.83795707156280769139226829736, −0.32141600898816960907333942012, 0.32141600898816960907333942012, 1.83795707156280769139226829736, 2.77420661818527796200533177698, 3.27363032643052051363211392454, 4.35111986509268644094144722114, 5.01011804036323856153603482135, 5.39243283953204492780182737324, 6.51878072633939075946485028254, 7.14260519689702900339450513185, 7.85290218150113295648272926386

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