Properties

Label 2-96e2-1.1-c1-0-14
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  − 1.41·5-s − 2.82·7-s − 4.24·13-s − 4·17-s + 8·19-s − 5.65·23-s − 2.99·25-s + 1.41·29-s + 2.82·31-s + 4.00·35-s − 4.24·37-s + 4·41-s − 11.3·47-s + 1.00·49-s − 7.07·53-s − 12·59-s − 1.41·61-s + 6·65-s − 4·67-s + 11.3·71-s + 14·73-s + 8.48·79-s − 16·83-s + 5.65·85-s − 6·89-s + 12·91-s − 11.3·95-s + ⋯
L(s)  = 1  − 0.632·5-s − 1.06·7-s − 1.17·13-s − 0.970·17-s + 1.83·19-s − 1.17·23-s − 0.599·25-s + 0.262·29-s + 0.508·31-s + 0.676·35-s − 0.697·37-s + 0.624·41-s − 1.65·47-s + 0.142·49-s − 0.971·53-s − 1.56·59-s − 0.181·61-s + 0.744·65-s − 0.488·67-s + 1.34·71-s + 1.63·73-s + 0.954·79-s − 1.75·83-s + 0.613·85-s − 0.635·89-s + 1.25·91-s − 1.16·95-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

\(L(1)\) \(\approx\) \(0.6314553219\)
\(L(\frac12)\) \(\approx\) \(0.6314553219\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.41T + 5T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 - 2.82T + 31T^{2} \)
37 \( 1 + 4.24T + 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 + 7.07T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 1.41T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 - 8.48T + 79T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 16T + 97T^{2} \)
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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

−7.85051135596474282411277184443, −6.95863552327635429807377373134, −6.52555476265611447481759362629, −5.67147594439776898682701723090, −4.90135339268055628654269880337, −4.19616079737161093746243679368, −3.36271488682540483112672085126, −2.81381040486688729354367446694, −1.78058121932092624245849088188, −0.36513715477833542505483706902, 0.36513715477833542505483706902, 1.78058121932092624245849088188, 2.81381040486688729354367446694, 3.36271488682540483112672085126, 4.19616079737161093746243679368, 4.90135339268055628654269880337, 5.67147594439776898682701723090, 6.52555476265611447481759362629, 6.95863552327635429807377373134, 7.85051135596474282411277184443

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