Properties

Label 2-96e2-1.1-c1-0-58
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  + 3.46·5-s + 1.41·7-s − 4.89·11-s − 1.41·13-s − 4.89·17-s + 6·19-s + 6.92·23-s + 6.99·25-s − 3.46·29-s − 1.41·31-s + 4.89·35-s + 9.89·37-s − 4.89·41-s + 6·43-s + 6.92·47-s − 5·49-s + 3.46·53-s − 16.9·55-s − 9.79·59-s + 7.07·61-s − 4.89·65-s − 8·67-s + 13.8·71-s − 12·73-s − 6.92·77-s + 15.5·79-s + 14.6·83-s + ⋯
L(s)  = 1  + 1.54·5-s + 0.534·7-s − 1.47·11-s − 0.392·13-s − 1.18·17-s + 1.37·19-s + 1.44·23-s + 1.39·25-s − 0.643·29-s − 0.254·31-s + 0.828·35-s + 1.62·37-s − 0.765·41-s + 0.914·43-s + 1.01·47-s − 0.714·49-s + 0.475·53-s − 2.28·55-s − 1.27·59-s + 0.905·61-s − 0.607·65-s − 0.977·67-s + 1.64·71-s − 1.40·73-s − 0.789·77-s + 1.75·79-s + 1.61·83-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

\(L(1)\) \(\approx\) \(2.852554293\)
\(L(\frac12)\) \(\approx\) \(2.852554293\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.46T + 5T^{2} \)
7 \( 1 - 1.41T + 7T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 - 6.92T + 23T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
31 \( 1 + 1.41T + 31T^{2} \)
37 \( 1 - 9.89T + 37T^{2} \)
41 \( 1 + 4.89T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 - 3.46T + 53T^{2} \)
59 \( 1 + 9.79T + 59T^{2} \)
61 \( 1 - 7.07T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + 12T + 73T^{2} \)
79 \( 1 - 15.5T + 79T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 + 9.79T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−7.59492244242467116794767097861, −7.12951411940704568799165331067, −6.22836724972442483671143706290, −5.54643703755115722252275865552, −5.10237112928620599902096303998, −4.52320016458681835185982467920, −3.15756537599390890860562648810, −2.49690078150222062003343782749, −1.90676579022112223803440717071, −0.811667604645753709043682570700, 0.811667604645753709043682570700, 1.90676579022112223803440717071, 2.49690078150222062003343782749, 3.15756537599390890860562648810, 4.52320016458681835185982467920, 5.10237112928620599902096303998, 5.54643703755115722252275865552, 6.22836724972442483671143706290, 7.12951411940704568799165331067, 7.59492244242467116794767097861

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