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Properties

Label 2-3267-99.40-c0-0-2
Degree $2$
Conductor $3267$
Sign $-0.165 + 0.986i$
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  + (1.40 + 0.147i)2-s + (0.978 + 0.207i)4-s + (−0.104 − 0.994i)5-s + (−1.05 − 0.946i)7-s − 1.41i·10-s + (−1.33 − 1.48i)14-s + (−0.913 − 0.406i)16-s + (0.104 − 0.994i)20-s + (−0.831 − 1.14i)28-s + (−1.05 − 0.946i)29-s + (0.913 − 0.406i)31-s + (−1.22 − 0.707i)32-s + (−0.831 + 1.14i)35-s + (−0.309 − 0.951i)37-s + (−0.978 + 0.207i)47-s + ⋯
L(s)  = 1  + (1.40 + 0.147i)2-s + (0.978 + 0.207i)4-s + (−0.104 − 0.994i)5-s + (−1.05 − 0.946i)7-s − 1.41i·10-s + (−1.33 − 1.48i)14-s + (−0.913 − 0.406i)16-s + (0.104 − 0.994i)20-s + (−0.831 − 1.14i)28-s + (−1.05 − 0.946i)29-s + (0.913 − 0.406i)31-s + (−1.22 − 0.707i)32-s + (−0.831 + 1.14i)35-s + (−0.309 − 0.951i)37-s + (−0.978 + 0.207i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $-0.165 + 0.986i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (766, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ -0.165 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.863043163\)
\(L(\frac12)\) \(\approx\) \(1.863043163\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-1.40 - 0.147i)T + (0.978 + 0.207i)T^{2} \)
5 \( 1 + (0.104 + 0.994i)T + (-0.978 + 0.207i)T^{2} \)
7 \( 1 + (1.05 + 0.946i)T + (0.104 + 0.994i)T^{2} \)
13 \( 1 + (-0.669 + 0.743i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (1.05 + 0.946i)T + (0.104 + 0.994i)T^{2} \)
31 \( 1 + (-0.913 + 0.406i)T + (0.669 - 0.743i)T^{2} \)
37 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.104 - 0.994i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.978 - 0.207i)T + (0.913 - 0.406i)T^{2} \)
53 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.978 - 0.207i)T + (0.913 + 0.406i)T^{2} \)
61 \( 1 + (-0.575 + 1.29i)T + (-0.669 - 0.743i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.978 + 0.207i)T^{2} \)
83 \( 1 + (0.575 - 1.29i)T + (-0.669 - 0.743i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2} \)
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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

−8.593854736033989733762383155611, −7.66449787655207837804233516704, −6.86364901675603511333815398255, −6.22680745287204869794609895919, −5.45368702477807064557538525819, −4.69712282445656984679934571940, −3.97860295629957547862195025534, −3.46719624172469146832711879784, −2.32212971261426102653017932273, −0.66363393707226686707493003493, 2.09398522342391422671063326131, 3.06566255256318579287432348131, 3.26218971121575888348562575499, 4.31980219003254571432734812613, 5.33269479648490522544252580214, 5.84770657118862845938142528685, 6.75147577689689344433273353473, 6.94986669738967590067744759258, 8.341076730281621821946585216526, 9.026775839333196717395495560932

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