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Properties

Label 2-3267-99.52-c0-0-0
Degree $2$
Conductor $3267$
Sign $-0.509 - 0.860i$
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  + (−0.978 + 0.207i)4-s + (−0.104 + 0.994i)5-s + (0.913 − 0.406i)16-s + (−0.104 − 0.994i)20-s + (1 + 1.73i)23-s + (−0.913 − 0.406i)31-s + (−0.309 + 0.951i)37-s + (−0.978 − 0.207i)47-s + (−0.104 + 0.994i)49-s + (−0.809 + 0.587i)53-s + (−0.978 + 0.207i)59-s + (−0.809 + 0.587i)64-s + (0.5 + 0.866i)67-s + (−0.809 − 0.587i)71-s + (0.309 + 0.951i)80-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)4-s + (−0.104 + 0.994i)5-s + (0.913 − 0.406i)16-s + (−0.104 − 0.994i)20-s + (1 + 1.73i)23-s + (−0.913 − 0.406i)31-s + (−0.309 + 0.951i)37-s + (−0.978 − 0.207i)47-s + (−0.104 + 0.994i)49-s + (−0.809 + 0.587i)53-s + (−0.978 + 0.207i)59-s + (−0.809 + 0.587i)64-s + (0.5 + 0.866i)67-s + (−0.809 − 0.587i)71-s + (0.309 + 0.951i)80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $-0.509 - 0.860i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (1207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ -0.509 - 0.860i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7481495083\)
\(L(\frac12)\) \(\approx\) \(0.7481495083\)
\(L(1)\) 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 \( 1 + (0.978 - 0.207i)T^{2} \)
5 \( 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2} \)
7 \( 1 + (0.104 - 0.994i)T^{2} \)
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 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.104 - 0.994i)T^{2} \)
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 \( 1 + (-0.669 + 0.743i)T^{2} \)
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 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.978 - 0.207i)T^{2} \)
83 \( 1 + (-0.669 + 0.743i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
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

−9.202606488734546123972439270977, −8.308142820759727689462562170620, −7.54966288156008324667718041845, −7.01254372397019116801852401184, −6.01474678339651415569891320995, −5.24132896100674764644662322412, −4.42468690809598752094662416111, −3.43077939147794961493488788904, −2.97825849042611852837872480355, −1.45490200566881147822610228346, 0.49757716454336614095278597182, 1.67144306035248999820036698691, 3.09851753439624664637011447344, 4.09163191980560521742651966030, 4.82109951353587343163821252300, 5.24496523306584725108749674603, 6.22283269815819783694168032239, 7.12552822275765368173513627696, 8.136548456855850905701658031250, 8.676325850053130464016220119829

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