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Properties

Label 2-3267-297.146-c0-0-0
Degree $2$
Conductor $3267$
Sign $0.568 + 0.822i$
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  + (−0.848 − 0.529i)3-s + (0.990 + 0.139i)4-s + (1.63 − 1.10i)5-s + (0.438 + 0.898i)9-s + (−0.766 − 0.642i)12-s + (−1.96 + 0.0687i)15-s + (0.961 + 0.275i)16-s + (1.77 − 0.863i)20-s + (−1.70 + 0.300i)23-s + (1.07 − 2.66i)25-s + (0.104 − 0.994i)27-s + (1.10 + 1.06i)31-s + (0.309 + 0.951i)36-s + (0.339 − 0.0722i)37-s + (1.70 + 0.984i)45-s + ⋯
L(s)  = 1  + (−0.848 − 0.529i)3-s + (0.990 + 0.139i)4-s + (1.63 − 1.10i)5-s + (0.438 + 0.898i)9-s + (−0.766 − 0.642i)12-s + (−1.96 + 0.0687i)15-s + (0.961 + 0.275i)16-s + (1.77 − 0.863i)20-s + (−1.70 + 0.300i)23-s + (1.07 − 2.66i)25-s + (0.104 − 0.994i)27-s + (1.10 + 1.06i)31-s + (0.309 + 0.951i)36-s + (0.339 − 0.0722i)37-s + (1.70 + 0.984i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.568 + 0.822i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (1334, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.568 + 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.656660495\)
\(L(\frac12)\) \(\approx\) \(1.656660495\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.848 + 0.529i)T \)
11 \( 1 \)
good2 \( 1 + (-0.990 - 0.139i)T^{2} \)
5 \( 1 + (-1.63 + 1.10i)T + (0.374 - 0.927i)T^{2} \)
7 \( 1 + (0.559 + 0.829i)T^{2} \)
13 \( 1 + (-0.882 - 0.469i)T^{2} \)
17 \( 1 + (0.978 + 0.207i)T^{2} \)
19 \( 1 + (0.913 + 0.406i)T^{2} \)
23 \( 1 + (1.70 - 0.300i)T + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.438 + 0.898i)T^{2} \)
31 \( 1 + (-1.10 - 1.06i)T + (0.0348 + 0.999i)T^{2} \)
37 \( 1 + (-0.339 + 0.0722i)T + (0.913 - 0.406i)T^{2} \)
41 \( 1 + (-0.438 - 0.898i)T^{2} \)
43 \( 1 + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (-0.0952 - 0.677i)T + (-0.961 + 0.275i)T^{2} \)
53 \( 1 + (0.755 + 1.04i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-1.01 - 0.791i)T + (0.241 + 0.970i)T^{2} \)
61 \( 1 + (0.0348 - 0.999i)T^{2} \)
67 \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.680 + 0.0715i)T + (0.978 + 0.207i)T^{2} \)
73 \( 1 + (-0.104 + 0.994i)T^{2} \)
79 \( 1 + (0.990 + 0.139i)T^{2} \)
83 \( 1 + (0.882 - 0.469i)T^{2} \)
89 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.05 - 1.55i)T + (-0.374 - 0.927i)T^{2} \)
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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

−8.545905002478211210567583561881, −8.003731634421338418731652107880, −6.96064248613560689157784830430, −6.31201910901425374717134521265, −5.81262365722620490783292739717, −5.21257525315089186326933393298, −4.29645924801665380740766312984, −2.69507873007885707706161528818, −1.86606113241617415364915335791, −1.23612967333083022947580551824, 1.52607123399347225913304649564, 2.40262458341313793041152527560, 3.20307889884968796085697716315, 4.38366274961087479552819070170, 5.56580220840992990309777959413, 5.98496456961289850757427377190, 6.45756700865933892647559977038, 7.09166011717856809910002653088, 8.050913954674918186660609041387, 9.410053394750038362495143994908

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