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Properties

Label 2-3267-297.218-c0-0-0
Degree $2$
Conductor $3267$
Sign $-0.491 - 0.871i$
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  + (0.719 + 0.694i)3-s + (0.559 + 0.829i)4-s + (−0.362 + 0.580i)5-s + (0.0348 + 0.999i)9-s + (−0.173 + 0.984i)12-s + (−0.663 + 0.165i)15-s + (−0.374 + 0.927i)16-s + (−0.683 + 0.0238i)20-s + (0.592 − 1.62i)23-s + (0.233 + 0.478i)25-s + (−0.669 + 0.743i)27-s + (0.213 − 0.273i)31-s + (−0.809 + 0.587i)36-s + (−0.196 + 1.86i)37-s + (−0.592 − 0.342i)45-s + ⋯
L(s)  = 1  + (0.719 + 0.694i)3-s + (0.559 + 0.829i)4-s + (−0.362 + 0.580i)5-s + (0.0348 + 0.999i)9-s + (−0.173 + 0.984i)12-s + (−0.663 + 0.165i)15-s + (−0.374 + 0.927i)16-s + (−0.683 + 0.0238i)20-s + (0.592 − 1.62i)23-s + (0.233 + 0.478i)25-s + (−0.669 + 0.743i)27-s + (0.213 − 0.273i)31-s + (−0.809 + 0.587i)36-s + (−0.196 + 1.86i)37-s + (−0.592 − 0.342i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.491 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.491 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $-0.491 - 0.871i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (1703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ -0.491 - 0.871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.681286197\)
\(L(\frac12)\) \(\approx\) \(1.681286197\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.719 - 0.694i)T \)
11 \( 1 \)
good2 \( 1 + (-0.559 - 0.829i)T^{2} \)
5 \( 1 + (0.362 - 0.580i)T + (-0.438 - 0.898i)T^{2} \)
7 \( 1 + (0.848 + 0.529i)T^{2} \)
13 \( 1 + (0.961 + 0.275i)T^{2} \)
17 \( 1 + (0.104 + 0.994i)T^{2} \)
19 \( 1 + (-0.978 + 0.207i)T^{2} \)
23 \( 1 + (-0.592 + 1.62i)T + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (-0.0348 + 0.999i)T^{2} \)
31 \( 1 + (-0.213 + 0.273i)T + (-0.241 - 0.970i)T^{2} \)
37 \( 1 + (0.196 - 1.86i)T + (-0.978 - 0.207i)T^{2} \)
41 \( 1 + (-0.0348 - 0.999i)T^{2} \)
43 \( 1 + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (-1.06 - 0.718i)T + (0.374 + 0.927i)T^{2} \)
53 \( 1 + (1.87 + 0.608i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.137 + 1.96i)T + (-0.990 + 0.139i)T^{2} \)
61 \( 1 + (-0.241 + 0.970i)T^{2} \)
67 \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.955 - 0.860i)T + (0.104 + 0.994i)T^{2} \)
73 \( 1 + (0.669 - 0.743i)T^{2} \)
79 \( 1 + (0.559 + 0.829i)T^{2} \)
83 \( 1 + (-0.961 + 0.275i)T^{2} \)
89 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.29 + 0.811i)T + (0.438 - 0.898i)T^{2} \)
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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

−8.904477768936840765613089423218, −8.263348385790334478858858618742, −7.76782254025616508561616628740, −6.90152633880237492791320625127, −6.35368721439858958269970256100, −4.97229133646969003812438804999, −4.31465404658728883471450745900, −3.28659064336659132485726710519, −2.97385751962109424156135784143, −1.94326039576427091551968707886, 0.935502749689012095142879481255, 1.82315825275304369047462986932, 2.79102544962195219944644480162, 3.74790233172537215960234868281, 4.79536662341060716833943364993, 5.68465968049867360811502720290, 6.35787252008748781665464366154, 7.32146091143061809941592972470, 7.60338969911480095824305150541, 8.662890704260960457085892173666

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