Properties

Label 2-96e2-1.1-c1-0-62
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.473·5-s + 4.55·7-s + 3.49·11-s + 0.0840·13-s + 3.61·17-s − 3.61·19-s − 2.82·23-s − 4.77·25-s + 7.30·29-s − 0.557·31-s − 2.15·35-s − 6.20·37-s + 9.27·41-s − 2.27·43-s + 2.82·47-s + 13.7·49-s + 0.697·53-s − 1.65·55-s + 5.65·59-s + 3.85·61-s − 0.0397·65-s + 5.33·67-s + 9.11·71-s − 0.541·73-s + 15.9·77-s + 10.9·79-s − 15.0·83-s + ⋯
L(s)  = 1  − 0.211·5-s + 1.72·7-s + 1.05·11-s + 0.0233·13-s + 0.877·17-s − 0.829·19-s − 0.589·23-s − 0.955·25-s + 1.35·29-s − 0.100·31-s − 0.364·35-s − 1.01·37-s + 1.44·41-s − 0.347·43-s + 0.412·47-s + 1.96·49-s + 0.0958·53-s − 0.223·55-s + 0.736·59-s + 0.494·61-s − 0.00493·65-s + 0.652·67-s + 1.08·71-s − 0.0633·73-s + 1.81·77-s + 1.23·79-s − 1.65·83-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

\(L(1)\) \(\approx\) \(2.871194665\)
\(L(\frac12)\) \(\approx\) \(2.871194665\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.473T + 5T^{2} \)
7 \( 1 - 4.55T + 7T^{2} \)
11 \( 1 - 3.49T + 11T^{2} \)
13 \( 1 - 0.0840T + 13T^{2} \)
17 \( 1 - 3.61T + 17T^{2} \)
19 \( 1 + 3.61T + 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 - 7.30T + 29T^{2} \)
31 \( 1 + 0.557T + 31T^{2} \)
37 \( 1 + 6.20T + 37T^{2} \)
41 \( 1 - 9.27T + 41T^{2} \)
43 \( 1 + 2.27T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 0.697T + 53T^{2} \)
59 \( 1 - 5.65T + 59T^{2} \)
61 \( 1 - 3.85T + 61T^{2} \)
67 \( 1 - 5.33T + 67T^{2} \)
71 \( 1 - 9.11T + 71T^{2} \)
73 \( 1 + 0.541T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + 15.0T + 83T^{2} \)
89 \( 1 - 14.6T + 89T^{2} \)
97 \( 1 - 4.31T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.896602411965699926671345771584, −7.10762485280107739201189350129, −6.35930821114949051138955105975, −5.59719294586141430542835224665, −4.91877062813087041637966123047, −4.16909348135625314875352381348, −3.71718489341468301002549159158, −2.42992764600780078171011300336, −1.69145531827896506059156077095, −0.874041656161496072794554406245, 0.874041656161496072794554406245, 1.69145531827896506059156077095, 2.42992764600780078171011300336, 3.71718489341468301002549159158, 4.16909348135625314875352381348, 4.91877062813087041637966123047, 5.59719294586141430542835224665, 6.35930821114949051138955105975, 7.10762485280107739201189350129, 7.896602411965699926671345771584

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