Properties

Label 2-96e2-1.1-c1-0-32
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  − 0.841·5-s − 1.64·7-s − 4.75·11-s + 3.74·13-s + 5.53·17-s + 5.15·19-s + 4.33·23-s − 4.29·25-s − 8.66·29-s + 5.64·31-s + 1.38·35-s − 0.913·37-s − 7.91·41-s − 0.500·43-s + 9.10·47-s − 4.29·49-s − 6.97·53-s + 3.99·55-s + 6.13·59-s + 0.913·61-s − 3.14·65-s + 5.65·67-s − 13.4·71-s − 3.29·73-s + 7.82·77-s + 9.64·79-s − 4.75·83-s + ⋯
L(s)  = 1  − 0.376·5-s − 0.622·7-s − 1.43·11-s + 1.03·13-s + 1.34·17-s + 1.18·19-s + 0.904·23-s − 0.858·25-s − 1.60·29-s + 1.01·31-s + 0.234·35-s − 0.150·37-s − 1.23·41-s − 0.0763·43-s + 1.32·47-s − 0.613·49-s − 0.958·53-s + 0.539·55-s + 0.799·59-s + 0.116·61-s − 0.390·65-s + 0.691·67-s − 1.59·71-s − 0.385·73-s + 0.891·77-s + 1.08·79-s − 0.521·83-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9216,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.488865097\)
\(L(\frac12)\) \(\approx\) \(1.488865097\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.841T + 5T^{2} \)
7 \( 1 + 1.64T + 7T^{2} \)
11 \( 1 + 4.75T + 11T^{2} \)
13 \( 1 - 3.74T + 13T^{2} \)
17 \( 1 - 5.53T + 17T^{2} \)
19 \( 1 - 5.15T + 19T^{2} \)
23 \( 1 - 4.33T + 23T^{2} \)
29 \( 1 + 8.66T + 29T^{2} \)
31 \( 1 - 5.64T + 31T^{2} \)
37 \( 1 + 0.913T + 37T^{2} \)
41 \( 1 + 7.91T + 41T^{2} \)
43 \( 1 + 0.500T + 43T^{2} \)
47 \( 1 - 9.10T + 47T^{2} \)
53 \( 1 + 6.97T + 53T^{2} \)
59 \( 1 - 6.13T + 59T^{2} \)
61 \( 1 - 0.913T + 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 + 3.29T + 73T^{2} \)
79 \( 1 - 9.64T + 79T^{2} \)
83 \( 1 + 4.75T + 83T^{2} \)
89 \( 1 - 2.38T + 89T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−7.72545761853854312008284171429, −7.20487807320839184085925525128, −6.30552231354422231781685610943, −5.50267940144337104148466545582, −5.23276446550456190649043132134, −4.06679356479533735338904227738, −3.30890727682589875548415769953, −2.92191405297734064502918328095, −1.65525907361738288423917510910, −0.59306958411625150675633905494, 0.59306958411625150675633905494, 1.65525907361738288423917510910, 2.92191405297734064502918328095, 3.30890727682589875548415769953, 4.06679356479533735338904227738, 5.23276446550456190649043132134, 5.50267940144337104148466545582, 6.30552231354422231781685610943, 7.20487807320839184085925525128, 7.72545761853854312008284171429

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