Properties

Label 2-96e2-1.1-c1-0-33
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  + 2.82·5-s − 4.24·7-s + 4·11-s − 4.24·13-s − 6·17-s − 2·19-s + 2.82·23-s + 3.00·25-s − 5.65·29-s + 4.24·31-s − 12·35-s − 4.24·37-s + 10·41-s + 6·43-s − 2.82·47-s + 10.9·49-s − 5.65·53-s + 11.3·55-s + 4.24·61-s − 12·65-s − 4·67-s + 2.82·71-s + 16·73-s − 16.9·77-s + 4.24·79-s + 16·83-s − 16.9·85-s + ⋯
L(s)  = 1  + 1.26·5-s − 1.60·7-s + 1.20·11-s − 1.17·13-s − 1.45·17-s − 0.458·19-s + 0.589·23-s + 0.600·25-s − 1.05·29-s + 0.762·31-s − 2.02·35-s − 0.697·37-s + 1.56·41-s + 0.914·43-s − 0.412·47-s + 1.57·49-s − 0.777·53-s + 1.52·55-s + 0.543·61-s − 1.48·65-s − 0.488·67-s + 0.335·71-s + 1.87·73-s − 1.93·77-s + 0.477·79-s + 1.75·83-s − 1.84·85-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

\(L(1)\) \(\approx\) \(1.760393218\)
\(L(\frac12)\) \(\approx\) \(1.760393218\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.82T + 5T^{2} \)
7 \( 1 + 4.24T + 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + 5.65T + 29T^{2} \)
31 \( 1 - 4.24T + 31T^{2} \)
37 \( 1 + 4.24T + 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + 5.65T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 4.24T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 2.82T + 71T^{2} \)
73 \( 1 - 16T + 73T^{2} \)
79 \( 1 - 4.24T + 79T^{2} \)
83 \( 1 - 16T + 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−7.46711368586190404627363655533, −6.81426977650685007789003506336, −6.35792083247936148795549021974, −5.93586110990468018698790064105, −4.98263281409415164383855533722, −4.21048939951121476702749181852, −3.38112227141562985392063126173, −2.47457607562477944304603797251, −1.96995799776714901236618145174, −0.60954835033349355277274915156, 0.60954835033349355277274915156, 1.96995799776714901236618145174, 2.47457607562477944304603797251, 3.38112227141562985392063126173, 4.21048939951121476702749181852, 4.98263281409415164383855533722, 5.93586110990468018698790064105, 6.35792083247936148795549021974, 6.81426977650685007789003506336, 7.46711368586190404627363655533

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