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Properties

Label 2-54e2-108.79-c0-0-7
Degree $2$
Conductor $2916$
Sign $-0.893 + 0.448i$
Analytic cond. $1.45527$
Root an. cond. $1.20634$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (1.17 − 0.984i)5-s + (−0.5 + 0.866i)8-s + (−0.766 − 1.32i)10-s + (−0.326 − 1.85i)13-s + (0.766 + 0.642i)16-s + (−0.173 − 0.300i)17-s + (−1.43 + 0.524i)20-s + (0.233 − 1.32i)25-s − 1.87·26-s + (−0.326 + 1.85i)29-s + (0.766 − 0.642i)32-s + (−0.326 + 0.118i)34-s + (−0.173 − 0.300i)37-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (1.17 − 0.984i)5-s + (−0.5 + 0.866i)8-s + (−0.766 − 1.32i)10-s + (−0.326 − 1.85i)13-s + (0.766 + 0.642i)16-s + (−0.173 − 0.300i)17-s + (−1.43 + 0.524i)20-s + (0.233 − 1.32i)25-s − 1.87·26-s + (−0.326 + 1.85i)29-s + (0.766 − 0.642i)32-s + (−0.326 + 0.118i)34-s + (−0.173 − 0.300i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2916\)    =    \(2^{2} \cdot 3^{6}\)
Sign: $-0.893 + 0.448i$
Analytic conductor: \(1.45527\)
Root analytic conductor: \(1.20634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2916} (2107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2916,\ (\ :0),\ -0.893 + 0.448i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.370992357\)
\(L(\frac12)\) \(\approx\) \(1.370992357\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 \)
good5 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
7 \( 1 + (-0.766 + 0.642i)T^{2} \)
11 \( 1 + (-0.173 - 0.984i)T^{2} \)
13 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
31 \( 1 + (-0.766 - 0.642i)T^{2} \)
37 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.939 + 0.342i)T^{2} \)
89 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
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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

−8.800910460929156454493611103559, −8.254787513473872486839074230396, −7.14955240220238183344692076603, −5.89078757509523033832727238151, −5.30793695540953230247256314005, −4.94810610466266011073301723656, −3.69980927059747300265465460985, −2.78985233730874395320066317602, −1.86461517595307305650890250739, −0.827707415859005414518448527889, 1.81985502454879515883742146540, 2.76548861828931479714133682718, 3.96582018262559757042141847575, 4.67430508265983183386205318041, 5.74325351903277886686492728423, 6.32262677142072812877967235766, 6.80017397082443854438384157191, 7.52984750408915900047355367515, 8.461385479717817499785954584348, 9.383520436106035668111057151158

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