Properties

Label 2-96e2-1.1-c1-0-125
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  − 0.585·5-s + 3.69·7-s − 4.14·11-s + 3.41·13-s − 2.82·17-s − 6.30·19-s + 6.75·23-s − 4.65·25-s − 7.41·29-s + 3.06·31-s − 2.16·35-s + 9.07·37-s + 4·41-s − 1.08·43-s − 3.06·47-s + 6.65·49-s − 4.58·53-s + 2.42·55-s − 1.08·59-s − 1.07·61-s − 2·65-s + 1.97·67-s + 8.02·71-s − 6.48·73-s − 15.3·77-s − 14.7·79-s − 13.6·83-s + ⋯
L(s)  = 1  − 0.261·5-s + 1.39·7-s − 1.24·11-s + 0.946·13-s − 0.685·17-s − 1.44·19-s + 1.40·23-s − 0.931·25-s − 1.37·29-s + 0.549·31-s − 0.365·35-s + 1.49·37-s + 0.624·41-s − 0.165·43-s − 0.446·47-s + 0.950·49-s − 0.629·53-s + 0.327·55-s − 0.140·59-s − 0.137·61-s − 0.248·65-s + 0.241·67-s + 0.952·71-s − 0.759·73-s − 1.74·77-s − 1.66·79-s − 1.50·83-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: \(1\)
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 \( 1 + 0.585T + 5T^{2} \)
7 \( 1 - 3.69T + 7T^{2} \)
11 \( 1 + 4.14T + 11T^{2} \)
13 \( 1 - 3.41T + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 + 6.30T + 19T^{2} \)
23 \( 1 - 6.75T + 23T^{2} \)
29 \( 1 + 7.41T + 29T^{2} \)
31 \( 1 - 3.06T + 31T^{2} \)
37 \( 1 - 9.07T + 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 + 1.08T + 43T^{2} \)
47 \( 1 + 3.06T + 47T^{2} \)
53 \( 1 + 4.58T + 53T^{2} \)
59 \( 1 + 1.08T + 59T^{2} \)
61 \( 1 + 1.07T + 61T^{2} \)
67 \( 1 - 1.97T + 67T^{2} \)
71 \( 1 - 8.02T + 71T^{2} \)
73 \( 1 + 6.48T + 73T^{2} \)
79 \( 1 + 14.7T + 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 + 4.82T + 89T^{2} \)
97 \( 1 - 5.17T + 97T^{2} \)
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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

−7.57215006728486142387231879555, −6.74296565437651285290087324360, −5.90165196871777992755349370870, −5.30831334700932836176600640147, −4.47805994756070011413855338964, −4.10764818238172151577803702751, −2.92469802377438730878936135794, −2.15789399517423854871645815589, −1.32222178361332820850877716905, 0, 1.32222178361332820850877716905, 2.15789399517423854871645815589, 2.92469802377438730878936135794, 4.10764818238172151577803702751, 4.47805994756070011413855338964, 5.30831334700932836176600640147, 5.90165196871777992755349370870, 6.74296565437651285290087324360, 7.57215006728486142387231879555

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