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Properties

Label 2-3564-99.86-c0-0-0
Degree $2$
Conductor $3564$
Sign $-0.677 - 0.735i$
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  + (−0.336 + 1.58i)5-s + (−0.104 + 0.994i)7-s + (0.207 + 0.978i)11-s + (1.08 + 1.20i)13-s + (−0.587 + 0.190i)17-s + (0.809 − 0.587i)19-s + (−0.535 − 0.309i)23-s + (−1.47 − 0.658i)25-s + (0.994 + 0.104i)29-s + (−0.669 − 0.743i)31-s + (−1.53 − 0.5i)35-s + (−0.994 + 0.104i)41-s + (0.658 − 1.47i)47-s + (−0.951 − 0.309i)53-s − 1.61·55-s + ⋯
L(s)  = 1  + (−0.336 + 1.58i)5-s + (−0.104 + 0.994i)7-s + (0.207 + 0.978i)11-s + (1.08 + 1.20i)13-s + (−0.587 + 0.190i)17-s + (0.809 − 0.587i)19-s + (−0.535 − 0.309i)23-s + (−1.47 − 0.658i)25-s + (0.994 + 0.104i)29-s + (−0.669 − 0.743i)31-s + (−1.53 − 0.5i)35-s + (−0.994 + 0.104i)41-s + (0.658 − 1.47i)47-s + (−0.951 − 0.309i)53-s − 1.61·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3564\)    =    \(2^{2} \cdot 3^{4} \cdot 11\)
Sign: $-0.677 - 0.735i$
Analytic conductor: \(1.77866\)
Root analytic conductor: \(1.33366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3564} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3564,\ (\ :0),\ -0.677 - 0.735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.157672725\)
\(L(\frac12)\) \(\approx\) \(1.157672725\)
\(L(1)\) 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 \( 1 + (-0.207 - 0.978i)T \)
good5 \( 1 + (0.336 - 1.58i)T + (-0.913 - 0.406i)T^{2} \)
7 \( 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2} \)
13 \( 1 + (-1.08 - 1.20i)T + (-0.104 + 0.994i)T^{2} \)
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 \( 1 + (-0.994 - 0.104i)T + (0.978 + 0.207i)T^{2} \)
31 \( 1 + (0.669 + 0.743i)T + (-0.104 + 0.994i)T^{2} \)
37 \( 1 + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.994 - 0.104i)T + (0.978 - 0.207i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.658 + 1.47i)T + (-0.669 - 0.743i)T^{2} \)
53 \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.406 - 0.913i)T + (-0.669 + 0.743i)T^{2} \)
61 \( 1 + (-1.08 + 1.20i)T + (-0.104 - 0.994i)T^{2} \)
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 \( 1 + (0.913 - 0.406i)T^{2} \)
83 \( 1 + (0.743 + 0.669i)T + (0.104 + 0.994i)T^{2} \)
89 \( 1 - 1.61iT - T^{2} \)
97 \( 1 + (0.913 - 0.406i)T^{2} \)
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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

−9.020587073861702411580571617031, −8.311304807448088059469409202999, −7.34029726451999780511214654605, −6.68300273661487003767281784373, −6.36415839771713009332288850051, −5.32133229982064303017163130284, −4.23939642298579004431693883669, −3.54564738644207972702086845957, −2.55079609191567882982391131410, −1.90246269574288130598806183072, 0.73680054677766818153763417167, 1.38662072809457344517243270113, 3.18812532623789225330781950376, 3.81895973060177700226668648544, 4.60247234452994849650107255935, 5.47249818358501842681240543045, 6.03029690386118359734322116753, 7.11053841649717128368783839529, 7.990491195066915256694324408313, 8.400026947582216252546884280586

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