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Properties

Label 2-3267-297.14-c0-0-0
Degree $2$
Conductor $3267$
Sign $-0.988 - 0.150i$
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  + (−0.990 + 0.139i)3-s + (0.0348 + 0.999i)4-s + (−0.663 + 0.165i)5-s + (0.961 − 0.275i)9-s + (−0.173 − 0.984i)12-s + (0.634 − 0.256i)15-s + (−0.997 + 0.0697i)16-s + (−0.188 − 0.657i)20-s + (0.592 + 1.62i)23-s + (−0.469 + 0.249i)25-s + (−0.913 + 0.406i)27-s + (−0.194 + 0.287i)31-s + (0.309 + 0.951i)36-s + (1.25 + 1.39i)37-s + (−0.592 + 0.342i)45-s + ⋯
L(s)  = 1  + (−0.990 + 0.139i)3-s + (0.0348 + 0.999i)4-s + (−0.663 + 0.165i)5-s + (0.961 − 0.275i)9-s + (−0.173 − 0.984i)12-s + (0.634 − 0.256i)15-s + (−0.997 + 0.0697i)16-s + (−0.188 − 0.657i)20-s + (0.592 + 1.62i)23-s + (−0.469 + 0.249i)25-s + (−0.913 + 0.406i)27-s + (−0.194 + 0.287i)31-s + (0.309 + 0.951i)36-s + (1.25 + 1.39i)37-s + (−0.592 + 0.342i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $-0.988 - 0.150i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (608, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ -0.988 - 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4309923862\)
\(L(\frac12)\) \(\approx\) \(0.4309923862\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.990 - 0.139i)T \)
11 \( 1 \)
good2 \( 1 + (-0.0348 - 0.999i)T^{2} \)
5 \( 1 + (0.663 - 0.165i)T + (0.882 - 0.469i)T^{2} \)
7 \( 1 + (-0.241 - 0.970i)T^{2} \)
13 \( 1 + (-0.615 + 0.788i)T^{2} \)
17 \( 1 + (-0.669 + 0.743i)T^{2} \)
19 \( 1 + (-0.104 - 0.994i)T^{2} \)
23 \( 1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.961 - 0.275i)T^{2} \)
31 \( 1 + (0.194 - 0.287i)T + (-0.374 - 0.927i)T^{2} \)
37 \( 1 + (-1.25 - 1.39i)T + (-0.104 + 0.994i)T^{2} \)
41 \( 1 + (-0.961 + 0.275i)T^{2} \)
43 \( 1 + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (1.28 + 0.0448i)T + (0.997 + 0.0697i)T^{2} \)
53 \( 1 + (1.15 + 1.59i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (1.04 + 1.67i)T + (-0.438 + 0.898i)T^{2} \)
61 \( 1 + (-0.374 + 0.927i)T^{2} \)
67 \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.522 + 1.17i)T + (-0.669 + 0.743i)T^{2} \)
73 \( 1 + (0.913 - 0.406i)T^{2} \)
79 \( 1 + (0.0348 + 0.999i)T^{2} \)
83 \( 1 + (0.615 + 0.788i)T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.370 - 1.48i)T + (-0.882 - 0.469i)T^{2} \)
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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

−9.358804989389630257693957117559, −8.096494982661967987127977886620, −7.76604763841432705601275098330, −6.92987579211623075792428843376, −6.33869705112723009756047232303, −5.26592798792356416673333184547, −4.56547058028809388259613148488, −3.70517295805917779162594089635, −3.08567889031389853964624247026, −1.55878364820597287055738298212, 0.31107208434711384387016312711, 1.45050175155305111811924097416, 2.64709461768174517553248449546, 4.23799905652438875176905332397, 4.54850656894304302038756499882, 5.57251119587918491108831307308, 6.09857393584215489460560895307, 6.86817424538423577013830040744, 7.55177775625253933474770519692, 8.465115423942366338454973351464

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