Properties

Label 2-96e2-1.1-c1-0-106
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  − 3.41·5-s + 1.53·7-s + 4.77·11-s + 0.585·13-s + 2.82·17-s − 0.448·19-s − 5.86·23-s + 6.65·25-s − 4.58·29-s − 7.39·31-s − 5.22·35-s − 5.07·37-s + 4·41-s − 2.61·43-s + 7.39·47-s − 4.65·49-s − 7.41·53-s − 16.3·55-s − 2.61·59-s + 13.0·61-s − 2·65-s − 10.0·67-s + 11.9·71-s + 10.4·73-s + 7.31·77-s − 6.12·79-s − 3.50·83-s + ⋯
L(s)  = 1  − 1.52·5-s + 0.578·7-s + 1.44·11-s + 0.162·13-s + 0.685·17-s − 0.102·19-s − 1.22·23-s + 1.33·25-s − 0.851·29-s − 1.32·31-s − 0.883·35-s − 0.833·37-s + 0.624·41-s − 0.398·43-s + 1.07·47-s − 0.665·49-s − 1.01·53-s − 2.19·55-s − 0.340·59-s + 1.67·61-s − 0.248·65-s − 1.22·67-s + 1.42·71-s + 1.22·73-s + 0.833·77-s − 0.688·79-s − 0.385·83-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 \( 1 + 3.41T + 5T^{2} \)
7 \( 1 - 1.53T + 7T^{2} \)
11 \( 1 - 4.77T + 11T^{2} \)
13 \( 1 - 0.585T + 13T^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
19 \( 1 + 0.448T + 19T^{2} \)
23 \( 1 + 5.86T + 23T^{2} \)
29 \( 1 + 4.58T + 29T^{2} \)
31 \( 1 + 7.39T + 31T^{2} \)
37 \( 1 + 5.07T + 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 + 2.61T + 43T^{2} \)
47 \( 1 - 7.39T + 47T^{2} \)
53 \( 1 + 7.41T + 53T^{2} \)
59 \( 1 + 2.61T + 59T^{2} \)
61 \( 1 - 13.0T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + 6.12T + 79T^{2} \)
83 \( 1 + 3.50T + 83T^{2} \)
89 \( 1 - 0.828T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−7.48787713878485507105136508729, −6.85663056754080157692536146738, −6.04593072551731099995337686584, −5.24584881398966307612519510281, −4.37566759450764295616120560281, −3.77193287918665618326138021370, −3.46329845077836699306993877127, −2.05539576898860840693665372343, −1.18085164726689439997569162348, 0, 1.18085164726689439997569162348, 2.05539576898860840693665372343, 3.46329845077836699306993877127, 3.77193287918665618326138021370, 4.37566759450764295616120560281, 5.24584881398966307612519510281, 6.04593072551731099995337686584, 6.85663056754080157692536146738, 7.48787713878485507105136508729

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