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Properties

Label 2-3564-891.274-c0-0-1
Degree $2$
Conductor $3564$
Sign $-0.981 + 0.192i$
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  + (0.396 − 0.918i)3-s + (0.0798 − 1.37i)5-s + (−0.686 − 0.727i)9-s + (−0.835 + 0.549i)11-s + (−1.22 − 0.615i)15-s + (−1.82 − 0.433i)23-s + (−0.877 − 0.102i)25-s + (−0.939 + 0.342i)27-s + (−0.227 − 0.758i)31-s + (0.173 + 0.984i)33-s + (0.337 − 1.91i)37-s + (−1.05 + 0.882i)45-s + (0.0333 − 0.111i)47-s + (0.893 − 0.448i)49-s + (−0.597 + 1.03i)53-s + ⋯
L(s)  = 1  + (0.396 − 0.918i)3-s + (0.0798 − 1.37i)5-s + (−0.686 − 0.727i)9-s + (−0.835 + 0.549i)11-s + (−1.22 − 0.615i)15-s + (−1.82 − 0.433i)23-s + (−0.877 − 0.102i)25-s + (−0.939 + 0.342i)27-s + (−0.227 − 0.758i)31-s + (0.173 + 0.984i)33-s + (0.337 − 1.91i)37-s + (−1.05 + 0.882i)45-s + (0.0333 − 0.111i)47-s + (0.893 − 0.448i)49-s + (−0.597 + 1.03i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3564\)    =    \(2^{2} \cdot 3^{4} \cdot 11\)
Sign: $-0.981 + 0.192i$
Analytic conductor: \(1.77866\)
Root analytic conductor: \(1.33366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3564} (1165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3564,\ (\ :0),\ -0.981 + 0.192i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.016032071\)
\(L(\frac12)\) \(\approx\) \(1.016032071\)
\(L(1)\) 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 \( 1 + (-0.396 + 0.918i)T \)
11 \( 1 + (0.835 - 0.549i)T \)
good5 \( 1 + (-0.0798 + 1.37i)T + (-0.993 - 0.116i)T^{2} \)
7 \( 1 + (-0.893 + 0.448i)T^{2} \)
13 \( 1 + (-0.973 + 0.230i)T^{2} \)
17 \( 1 + (-0.173 - 0.984i)T^{2} \)
19 \( 1 + (-0.173 + 0.984i)T^{2} \)
23 \( 1 + (1.82 + 0.433i)T + (0.893 + 0.448i)T^{2} \)
29 \( 1 + (0.286 + 0.957i)T^{2} \)
31 \( 1 + (0.227 + 0.758i)T + (-0.835 + 0.549i)T^{2} \)
37 \( 1 + (-0.337 + 1.91i)T + (-0.939 - 0.342i)T^{2} \)
41 \( 1 + (0.686 + 0.727i)T^{2} \)
43 \( 1 + (-0.597 - 0.802i)T^{2} \)
47 \( 1 + (-0.0333 + 0.111i)T + (-0.835 - 0.549i)T^{2} \)
53 \( 1 + (0.597 - 1.03i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.997 + 0.656i)T + (0.396 + 0.918i)T^{2} \)
61 \( 1 + (0.0581 - 0.998i)T^{2} \)
67 \( 1 + (-0.473 - 0.635i)T + (-0.286 + 0.957i)T^{2} \)
71 \( 1 + (-1.57 + 0.571i)T + (0.766 - 0.642i)T^{2} \)
73 \( 1 + (-0.766 - 0.642i)T^{2} \)
79 \( 1 + (0.686 - 0.727i)T^{2} \)
83 \( 1 + (0.686 - 0.727i)T^{2} \)
89 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (-0.00676 - 0.116i)T + (-0.993 + 0.116i)T^{2} \)
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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

−8.174398062994023097628432996150, −7.88837185964028182910819558598, −7.09245779745475141891879836624, −6.02571009992282231935160410988, −5.52829642191348903640211851496, −4.55823706945418930583730429393, −3.78326023249971786369810410687, −2.42917079852136461471858241793, −1.82212512154975846599608397921, −0.51875516575688316270491929821, 2.06596383602908879014916518440, 2.94473183742976062656139228826, 3.44652721593062732120140699520, 4.38798449813506657898303420353, 5.35206570083809055671551921958, 6.06221670541581673553901854141, 6.82934429318568971869993709959, 7.86911060705684390468273916416, 8.172704693657813576030860376695, 9.183737989251347464887043992376

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