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Properties

Label 2-3267-297.284-c0-0-0
Degree $2$
Conductor $3267$
Sign $0.970 + 0.240i$
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  + (−0.438 − 0.898i)3-s + (0.961 + 0.275i)4-s + (0.634 + 0.256i)5-s + (−0.615 + 0.788i)9-s + (−0.173 − 0.984i)12-s + (−0.0477 − 0.682i)15-s + (0.848 + 0.529i)16-s + (0.539 + 0.421i)20-s + (0.592 + 1.62i)23-s + (−0.382 − 0.369i)25-s + (0.978 + 0.207i)27-s + (−0.0121 − 0.347i)31-s + (−0.809 + 0.587i)36-s + (1.71 − 0.764i)37-s + (−0.592 + 0.342i)45-s + ⋯
L(s)  = 1  + (−0.438 − 0.898i)3-s + (0.961 + 0.275i)4-s + (0.634 + 0.256i)5-s + (−0.615 + 0.788i)9-s + (−0.173 − 0.984i)12-s + (−0.0477 − 0.682i)15-s + (0.848 + 0.529i)16-s + (0.539 + 0.421i)20-s + (0.592 + 1.62i)23-s + (−0.382 − 0.369i)25-s + (0.978 + 0.207i)27-s + (−0.0121 − 0.347i)31-s + (−0.809 + 0.587i)36-s + (1.71 − 0.764i)37-s + (−0.592 + 0.342i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.970 + 0.240i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (2066, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.970 + 0.240i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.552723507\)
\(L(\frac12)\) \(\approx\) \(1.552723507\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.438 + 0.898i)T \)
11 \( 1 \)
good2 \( 1 + (-0.961 - 0.275i)T^{2} \)
5 \( 1 + (-0.634 - 0.256i)T + (0.719 + 0.694i)T^{2} \)
7 \( 1 + (-0.374 + 0.927i)T^{2} \)
13 \( 1 + (0.559 + 0.829i)T^{2} \)
17 \( 1 + (-0.913 - 0.406i)T^{2} \)
19 \( 1 + (0.669 + 0.743i)T^{2} \)
23 \( 1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.615 + 0.788i)T^{2} \)
31 \( 1 + (0.0121 + 0.347i)T + (-0.997 + 0.0697i)T^{2} \)
37 \( 1 + (-1.71 + 0.764i)T + (0.669 - 0.743i)T^{2} \)
41 \( 1 + (0.615 - 0.788i)T^{2} \)
43 \( 1 + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (0.354 + 1.23i)T + (-0.848 + 0.529i)T^{2} \)
53 \( 1 + (-1.87 - 0.608i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (1.91 - 0.476i)T + (0.882 - 0.469i)T^{2} \)
61 \( 1 + (-0.997 - 0.0697i)T^{2} \)
67 \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.267 - 1.25i)T + (-0.913 - 0.406i)T^{2} \)
73 \( 1 + (-0.978 - 0.207i)T^{2} \)
79 \( 1 + (0.961 + 0.275i)T^{2} \)
83 \( 1 + (-0.559 + 0.829i)T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.573 + 1.42i)T + (-0.719 + 0.694i)T^{2} \)
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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

−8.622849035272399675504418174262, −7.76168207888003584422251889232, −7.26326897367712672928808470946, −6.62021138596393700660257764007, −5.81287686877350844558704114587, −5.46779285583475074288790776191, −4.02749455130820454830428925568, −2.87247539016318517213688786772, −2.20077224220236398056444465850, −1.29523977070590316367701773510, 1.14734180366905901326333585131, 2.43551813059524677306271610955, 3.18900305423068537436171040498, 4.39915802473833887315337234069, 5.04681824640938685541060595790, 6.01995172488290733601222761834, 6.25762507305480360363983571983, 7.22795897452659685801915315499, 8.181290839232321819884551521764, 9.068482012059878077227566650022

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