↑

Properties

Label 2-3267-297.151-c0-0-0
Degree $2$
Conductor $3267$
Sign $0.678 - 0.734i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.848 + 0.529i)3-s + (0.990 + 0.139i)4-s + (0.194 + 0.287i)5-s + (0.438 + 0.898i)9-s + (0.766 + 0.642i)12-s + (0.0121 + 0.347i)15-s + (0.961 + 0.275i)16-s + (0.152 + 0.312i)20-s + (−0.173 − 0.984i)23-s + (0.329 − 0.815i)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 + (−0.173 + 0.300i)45-s + ⋯
L(s)  = 1  + (0.848 + 0.529i)3-s + (0.990 + 0.139i)4-s + (0.194 + 0.287i)5-s + (0.438 + 0.898i)9-s + (0.766 + 0.642i)12-s + (0.0121 + 0.347i)15-s + (0.961 + 0.275i)16-s + (0.152 + 0.312i)20-s + (−0.173 − 0.984i)23-s + (0.329 − 0.815i)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 + (−0.173 + 0.300i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.678 - 0.734i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (1933, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.678 - 0.734i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.230004592\)
\(L(\frac12)\) \(\approx\) \(2.230004592\)
\(L(1)\) 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 \( 1 + (-0.194 - 0.287i)T + (-0.374 + 0.927i)T^{2} \)
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 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
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 \( 1 + (1.86 - 0.261i)T + (0.961 - 0.275i)T^{2} \)
53 \( 1 + (1.23 - 0.900i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.943 - 1.20i)T + (-0.241 - 0.970i)T^{2} \)
61 \( 1 + (-0.0348 + 0.999i)T^{2} \)
67 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (-0.196 + 1.86i)T + (-0.978 - 0.207i)T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.05 - 1.55i)T + (-0.374 - 0.927i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.976936863968865578028225167269, −7.985226984998374166432933372178, −7.67713307576127856242046543416, −6.66388007420520096137495277248, −6.11063615845759152528347669963, −5.01086584447541748678942268240, −4.12169208942505485625628809896, −3.20545598068377814103627879102, −2.53414665220086209248899717660, −1.71375073558755386762051288790, 1.40470458283148781529755115434, 1.98975326336163472577572265463, 3.12605791648857449031212808300, 3.65268341973964640695031185196, 5.06020421503548414906469472664, 5.78418806136414986409747419730, 6.82762687731641417225478150979, 7.07805665650930320622031487392, 8.012520709576343049629035908563, 8.553904191098403958093766194518

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