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Properties

Label 2-3564-99.76-c0-0-0
Degree $2$
Conductor $3564$
Sign $-0.173 - 0.984i$
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)5-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)31-s − 37-s + (1 + 1.73i)47-s + (−0.5 + 0.866i)49-s − 2·53-s − 0.999·55-s + (−0.5 + 0.866i)59-s + (0.5 − 0.866i)67-s + 71-s + 89-s + (0.5 + 0.866i)97-s + (−1 + 1.73i)103-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)31-s − 37-s + (1 + 1.73i)47-s + (−0.5 + 0.866i)49-s − 2·53-s − 0.999·55-s + (−0.5 + 0.866i)59-s + (0.5 − 0.866i)67-s + 71-s + 89-s + (0.5 + 0.866i)97-s + (−1 + 1.73i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3564\)    =    \(2^{2} \cdot 3^{4} \cdot 11\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(1.77866\)
Root analytic conductor: \(1.33366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3564} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3564,\ (\ :0),\ -0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.002584639\)
\(L(\frac12)\) \(\approx\) \(1.002584639\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + 2T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T + T^{2} \)
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

−9.151740430141561575778290799274, −7.80096398420849722257704126524, −7.68343250780441453838327237863, −6.69323250348870990701161590583, −6.18589601749676901396088439726, −5.10225801549432100411753064226, −4.23991885382064208723350049729, −3.50720792175616483717968063769, −2.61359261150623324909969206787, −1.51462559573905334687584014688, 0.60285134557988965737172579772, 1.82220665370982325137675746722, 3.12146058141741375600072761703, 3.90894522683410197302525395647, 4.72046645966272003925420229798, 5.41953319676735788975284956762, 6.36703710371480099058705666104, 6.97629090249182380522490676531, 8.120187366538958578417502747502, 8.442368667786624040725882294375

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