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Properties

Label 2-3267-297.290-c0-0-0
Degree $2$
Conductor $3267$
Sign $0.625 + 0.780i$
Analytic cond. $1.63044$
Root an. cond. $1.27688$
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  + (0.615 − 0.788i)3-s + (0.848 + 0.529i)4-s + (0.893 − 0.924i)5-s + (−0.241 − 0.970i)9-s + (0.939 − 0.342i)12-s + (−0.178 − 1.27i)15-s + (0.438 + 0.898i)16-s + (1.24 − 0.311i)20-s + (1.11 + 1.32i)23-s + (−0.0227 − 0.652i)25-s + (−0.913 − 0.406i)27-s + (−1.87 + 0.131i)31-s + (0.309 − 0.951i)36-s + (−1.02 + 1.13i)37-s + (−1.11 − 0.642i)45-s + ⋯
L(s)  = 1  + (0.615 − 0.788i)3-s + (0.848 + 0.529i)4-s + (0.893 − 0.924i)5-s + (−0.241 − 0.970i)9-s + (0.939 − 0.342i)12-s + (−0.178 − 1.27i)15-s + (0.438 + 0.898i)16-s + (1.24 − 0.311i)20-s + (1.11 + 1.32i)23-s + (−0.0227 − 0.652i)25-s + (−0.913 − 0.406i)27-s + (−1.87 + 0.131i)31-s + (0.309 − 0.951i)36-s + (−1.02 + 1.13i)37-s + (−1.11 − 0.642i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.625 + 0.780i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (1775, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.625 + 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.199330915\)
\(L(\frac12)\) \(\approx\) \(2.199330915\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.615 + 0.788i)T \)
11 \( 1 \)
good2 \( 1 + (-0.848 - 0.529i)T^{2} \)
5 \( 1 + (-0.893 + 0.924i)T + (-0.0348 - 0.999i)T^{2} \)
7 \( 1 + (-0.719 - 0.694i)T^{2} \)
13 \( 1 + (-0.374 + 0.927i)T^{2} \)
17 \( 1 + (-0.669 - 0.743i)T^{2} \)
19 \( 1 + (-0.104 + 0.994i)T^{2} \)
23 \( 1 + (-1.11 - 1.32i)T + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (0.241 - 0.970i)T^{2} \)
31 \( 1 + (1.87 - 0.131i)T + (0.990 - 0.139i)T^{2} \)
37 \( 1 + (1.02 - 1.13i)T + (-0.104 - 0.994i)T^{2} \)
41 \( 1 + (0.241 + 0.970i)T^{2} \)
43 \( 1 + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (1.04 + 1.67i)T + (-0.438 + 0.898i)T^{2} \)
53 \( 1 + (-0.402 + 0.553i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.321 - 0.603i)T + (-0.559 + 0.829i)T^{2} \)
61 \( 1 + (0.990 + 0.139i)T^{2} \)
67 \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.801 + 1.79i)T + (-0.669 - 0.743i)T^{2} \)
73 \( 1 + (0.913 + 0.406i)T^{2} \)
79 \( 1 + (0.848 + 0.529i)T^{2} \)
83 \( 1 + (0.374 + 0.927i)T^{2} \)
89 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.249 - 0.241i)T + (0.0348 - 0.999i)T^{2} \)
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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

−8.811571150569839372923550699156, −7.916425027549621932970832500979, −7.24122903127454778885106112968, −6.66701798595500699876806295497, −5.74931483869707306727271614993, −5.12345771744203805805062705191, −3.71174777594770913522595220722, −3.03602366620213668387788919251, −1.91050168910141843602122568696, −1.43708086815946960770463360257, 1.73029750469992573987548461156, 2.54503739463375259760213341868, 3.10173019682214830690256068898, 4.20860297969104187158885032631, 5.34404990034968275656173051254, 5.79037361204289476849971649785, 6.84531076267642127780004506138, 7.18062771173498819076120597638, 8.277268301569945820397678054093, 9.153689256151153156793864475808

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