Properties

Label 2-3564-396.263-c0-0-0
Degree $2$
Conductor $3564$
Sign $-0.642 + 0.766i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 17-s − 2·19-s + (0.499 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + 0.999·28-s + (0.5 − 0.866i)29-s + (−0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 17-s − 2·19-s + (0.499 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + 0.999·28-s + (0.5 − 0.866i)29-s + (−0.499 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3564\)    =    \(2^{2} \cdot 3^{4} \cdot 11\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(1.77866\)
Root analytic conductor: \(1.33366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3564} (2375, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3564,\ (\ :0),\ -0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04353077375\)
\(L(\frac12)\) \(\approx\) \(0.04353077375\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
11 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 + 2T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + 2T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.967132347421788892073667049175, −8.534624625904106299465286447631, −7.929555447329775812759920320191, −6.76816349997819403265102159720, −6.24585507867256146063172032112, −5.90109163867976834950857776994, −4.71928605413644973607799073780, −4.07189271107812219881674394447, −2.76232506792537979809814918890, −1.72288800520640974622082928013, 0.02842847575095271850409578002, 1.65891832304948384759162438358, 2.37502415471120635942855888509, 3.66841486213620620685021198556, 4.12876320244910594769997603844, 4.86538076322577173850613884241, 6.27008775833454374760507993378, 6.95609358637747146699893211448, 7.52250402813517444638418819208, 8.639279730557338929301860622533

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