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Properties

Label 2-3267-297.158-c0-0-0
Degree $2$
Conductor $3267$
Sign $0.428 + 0.903i$
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  + (−0.0348 − 0.999i)3-s + (−0.374 + 0.927i)4-s + (1.15 − 0.563i)5-s + (−0.997 + 0.0697i)9-s + (0.939 + 0.342i)12-s + (−0.603 − 1.13i)15-s + (−0.719 − 0.694i)16-s + (0.0896 + 1.28i)20-s + (1.11 − 1.32i)23-s + (0.401 − 0.514i)25-s + (0.104 + 0.994i)27-s + (−0.454 − 1.82i)31-s + (0.309 − 0.951i)36-s + (1.49 + 0.318i)37-s + (−1.11 + 0.642i)45-s + ⋯
L(s)  = 1  + (−0.0348 − 0.999i)3-s + (−0.374 + 0.927i)4-s + (1.15 − 0.563i)5-s + (−0.997 + 0.0697i)9-s + (0.939 + 0.342i)12-s + (−0.603 − 1.13i)15-s + (−0.719 − 0.694i)16-s + (0.0896 + 1.28i)20-s + (1.11 − 1.32i)23-s + (0.401 − 0.514i)25-s + (0.104 + 0.994i)27-s + (−0.454 − 1.82i)31-s + (0.309 − 0.951i)36-s + (1.49 + 0.318i)37-s + (−1.11 + 0.642i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.428 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.428 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.428 + 0.903i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.428 + 0.903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.329852080\)
\(L(\frac12)\) \(\approx\) \(1.329852080\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0348 + 0.999i)T \)
11 \( 1 \)
good2 \( 1 + (0.374 - 0.927i)T^{2} \)
5 \( 1 + (-1.15 + 0.563i)T + (0.615 - 0.788i)T^{2} \)
7 \( 1 + (0.438 + 0.898i)T^{2} \)
13 \( 1 + (0.848 + 0.529i)T^{2} \)
17 \( 1 + (0.978 - 0.207i)T^{2} \)
19 \( 1 + (0.913 - 0.406i)T^{2} \)
23 \( 1 + (-1.11 + 1.32i)T + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (0.997 + 0.0697i)T^{2} \)
31 \( 1 + (0.454 + 1.82i)T + (-0.882 + 0.469i)T^{2} \)
37 \( 1 + (-1.49 - 0.318i)T + (0.913 + 0.406i)T^{2} \)
41 \( 1 + (0.997 - 0.0697i)T^{2} \)
43 \( 1 + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (-1.82 + 0.737i)T + (0.719 - 0.694i)T^{2} \)
53 \( 1 + (0.402 - 0.553i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.0952 - 0.677i)T + (-0.961 + 0.275i)T^{2} \)
61 \( 1 + (-0.882 - 0.469i)T^{2} \)
67 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (-1.95 + 0.205i)T + (0.978 - 0.207i)T^{2} \)
73 \( 1 + (-0.104 - 0.994i)T^{2} \)
79 \( 1 + (-0.374 + 0.927i)T^{2} \)
83 \( 1 + (-0.848 + 0.529i)T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.152 + 0.312i)T + (-0.615 - 0.788i)T^{2} \)
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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

−8.752544608188804136786971587728, −7.930059867750753447183870094585, −7.29179508910941487555217269238, −6.45284561768029791490473736738, −5.76402184455431215544939662292, −4.96161544440508355587307390003, −4.03670406759278505817351691631, −2.75968366102954301768252776899, −2.21380853041113456481375720206, −0.886415029839822622995746434946, 1.36922855986393513196576637478, 2.53034558394621677465092626156, 3.42403743863222781782715264398, 4.52692121041129336031409514288, 5.23041926588618319198932055540, 5.81254977417600958777009461679, 6.40593644250300371951645024933, 7.36432308179305909587269554650, 8.572321343073514423920514442130, 9.377438221580441885957466907657

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