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Properties

Label 2-3564-99.20-c0-0-1
Degree $2$
Conductor $3564$
Sign $-0.607 + 0.794i$
Analytic cond. $1.77866$
Root an. cond. $1.33366$
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  + (−1.20 − 1.08i)5-s + (0.913 − 0.406i)7-s + (0.743 − 0.669i)11-s + (−1.58 + 0.336i)13-s + (−0.587 + 0.190i)17-s + (0.809 − 0.587i)19-s + (0.535 − 0.309i)23-s + (0.169 + 1.60i)25-s + (−0.406 − 0.913i)29-s + (0.978 − 0.207i)31-s + (−1.53 − 0.5i)35-s + (0.406 − 0.913i)41-s + (−1.60 + 0.169i)47-s + (−0.951 − 0.309i)53-s − 1.61·55-s + ⋯
L(s)  = 1  + (−1.20 − 1.08i)5-s + (0.913 − 0.406i)7-s + (0.743 − 0.669i)11-s + (−1.58 + 0.336i)13-s + (−0.587 + 0.190i)17-s + (0.809 − 0.587i)19-s + (0.535 − 0.309i)23-s + (0.169 + 1.60i)25-s + (−0.406 − 0.913i)29-s + (0.978 − 0.207i)31-s + (−1.53 − 0.5i)35-s + (0.406 − 0.913i)41-s + (−1.60 + 0.169i)47-s + (−0.951 − 0.309i)53-s − 1.61·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3564\)    =    \(2^{2} \cdot 3^{4} \cdot 11\)
Sign: $-0.607 + 0.794i$
Analytic conductor: \(1.77866\)
Root analytic conductor: \(1.33366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3564} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3564,\ (\ :0),\ -0.607 + 0.794i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8460295483\)
\(L(\frac12)\) \(\approx\) \(0.8460295483\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + (-0.743 + 0.669i)T \)
good5 \( 1 + (1.20 + 1.08i)T + (0.104 + 0.994i)T^{2} \)
7 \( 1 + (-0.913 + 0.406i)T + (0.669 - 0.743i)T^{2} \)
13 \( 1 + (1.58 - 0.336i)T + (0.913 - 0.406i)T^{2} \)
17 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.535 + 0.309i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.406 + 0.913i)T + (-0.669 + 0.743i)T^{2} \)
31 \( 1 + (-0.978 + 0.207i)T + (0.913 - 0.406i)T^{2} \)
37 \( 1 + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.406 + 0.913i)T + (-0.669 - 0.743i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (1.60 - 0.169i)T + (0.978 - 0.207i)T^{2} \)
53 \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.994 + 0.104i)T + (0.978 + 0.207i)T^{2} \)
61 \( 1 + (1.58 + 0.336i)T + (0.913 + 0.406i)T^{2} \)
67 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.104 + 0.994i)T^{2} \)
83 \( 1 + (0.207 - 0.978i)T + (-0.913 - 0.406i)T^{2} \)
89 \( 1 - 1.61iT - T^{2} \)
97 \( 1 + (-0.104 + 0.994i)T^{2} \)
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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

−8.317093741595574282471577103935, −7.84858462537402727035151731275, −7.23242941637222329943902286027, −6.32639002506444586864539984799, −5.05178361013644638261636240437, −4.67902102146248858416597645485, −4.08096706273983942236265404542, −3.00172953993049612693622912838, −1.65848736701930271620406465412, −0.50541519917335363860328526507, 1.62235351879194360245541349273, 2.77170555244268682007789398858, 3.41514667000033797358829684803, 4.61980480293545876850704208760, 4.87727029729359238426008925430, 6.17119642329828736420453330526, 7.04302753779029770336881144321, 7.50316964616306676810637622898, 8.002419588164279594462078598667, 8.955787158256886410096698052094

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