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Properties

Label 2-3267-297.185-c0-0-0
Degree $2$
Conductor $3267$
Sign $-0.693 + 0.720i$
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  + (−0.559 + 0.829i)3-s + (−0.241 − 0.970i)4-s + (−0.178 − 1.27i)5-s + (−0.374 − 0.927i)9-s + (0.939 + 0.342i)12-s + (1.15 + 0.563i)15-s + (−0.882 + 0.469i)16-s + (−1.19 + 0.481i)20-s + (1.11 − 1.32i)23-s + (−0.627 + 0.179i)25-s + (0.978 + 0.207i)27-s + (1.59 − 0.995i)31-s + (−0.809 + 0.587i)36-s + (−1.39 + 0.623i)37-s + (−1.11 + 0.642i)45-s + ⋯
L(s)  = 1  + (−0.559 + 0.829i)3-s + (−0.241 − 0.970i)4-s + (−0.178 − 1.27i)5-s + (−0.374 − 0.927i)9-s + (0.939 + 0.342i)12-s + (1.15 + 0.563i)15-s + (−0.882 + 0.469i)16-s + (−1.19 + 0.481i)20-s + (1.11 − 1.32i)23-s + (−0.627 + 0.179i)25-s + (0.978 + 0.207i)27-s + (1.59 − 0.995i)31-s + (−0.809 + 0.587i)36-s + (−1.39 + 0.623i)37-s + (−1.11 + 0.642i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $-0.693 + 0.720i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (3155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ -0.693 + 0.720i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6501924790\)
\(L(\frac12)\) \(\approx\) \(0.6501924790\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.559 - 0.829i)T \)
11 \( 1 \)
good2 \( 1 + (0.241 + 0.970i)T^{2} \)
5 \( 1 + (0.178 + 1.27i)T + (-0.961 + 0.275i)T^{2} \)
7 \( 1 + (0.990 - 0.139i)T^{2} \)
13 \( 1 + (-0.997 + 0.0697i)T^{2} \)
17 \( 1 + (-0.913 - 0.406i)T^{2} \)
19 \( 1 + (0.669 + 0.743i)T^{2} \)
23 \( 1 + (-1.11 + 1.32i)T + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (0.374 - 0.927i)T^{2} \)
31 \( 1 + (-1.59 + 0.995i)T + (0.438 - 0.898i)T^{2} \)
37 \( 1 + (1.39 - 0.623i)T + (0.669 - 0.743i)T^{2} \)
41 \( 1 + (0.374 + 0.927i)T^{2} \)
43 \( 1 + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (1.91 + 0.476i)T + (0.882 + 0.469i)T^{2} \)
53 \( 1 + (0.650 + 0.211i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.475 + 0.492i)T + (-0.0348 + 0.999i)T^{2} \)
61 \( 1 + (0.438 + 0.898i)T^{2} \)
67 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.409 + 1.92i)T + (-0.913 - 0.406i)T^{2} \)
73 \( 1 + (-0.978 - 0.207i)T^{2} \)
79 \( 1 + (-0.241 - 0.970i)T^{2} \)
83 \( 1 + (0.997 + 0.0697i)T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.343 - 0.0483i)T + (0.961 + 0.275i)T^{2} \)
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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

−8.699270719202463907257935943439, −8.164292483877205235540415414668, −6.68772133871202578362728135512, −6.21054684664485165061008286720, −5.18654805117585717387904045219, −4.81637060162895372814910173874, −4.30603242937156611023074841064, −3.05914474112378017396563099545, −1.50313559005068318091479531662, −0.44071445480457966093853497144, 1.61515187155055594884412285846, 2.95895230182246679567539679593, 3.20287331435200343042479137010, 4.51791994366053555474358870438, 5.35139395071938377512284261684, 6.43787476747485227933277965020, 6.92967957563006920954426504418, 7.46632166639522894215269108869, 8.139816230131963718551403911697, 8.886530643884279111147083488388

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