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Properties

Label 2-3267-297.139-c0-0-0
Degree $2$
Conductor $3267$
Sign $-0.975 + 0.219i$
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  + (0.0348 + 0.999i)3-s + (−0.374 + 0.927i)4-s + (0.671 + 1.37i)5-s + (−0.997 + 0.0697i)9-s + (−0.939 − 0.342i)12-s + (−1.35 + 0.719i)15-s + (−0.719 − 0.694i)16-s + (−1.52 + 0.106i)20-s + (−0.766 − 0.642i)23-s + (−0.829 + 1.06i)25-s + (−0.104 − 0.994i)27-s + (0.454 + 1.82i)31-s + (0.309 − 0.951i)36-s + (−1.49 − 0.318i)37-s + (−0.766 − 1.32i)45-s + ⋯
L(s)  = 1  + (0.0348 + 0.999i)3-s + (−0.374 + 0.927i)4-s + (0.671 + 1.37i)5-s + (−0.997 + 0.0697i)9-s + (−0.939 − 0.342i)12-s + (−1.35 + 0.719i)15-s + (−0.719 − 0.694i)16-s + (−1.52 + 0.106i)20-s + (−0.766 − 0.642i)23-s + (−0.829 + 1.06i)25-s + (−0.104 − 0.994i)27-s + (0.454 + 1.82i)31-s + (0.309 − 0.951i)36-s + (−1.49 − 0.318i)37-s + (−0.766 − 1.32i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $-0.975 + 0.219i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (2218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ -0.975 + 0.219i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.033436507\)
\(L(\frac12)\) \(\approx\) \(1.033436507\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0348 - 0.999i)T \)
11 \( 1 \)
good2 \( 1 + (0.374 - 0.927i)T^{2} \)
5 \( 1 + (-0.671 - 1.37i)T + (-0.615 + 0.788i)T^{2} \)
7 \( 1 + (-0.438 - 0.898i)T^{2} \)
13 \( 1 + (-0.848 - 0.529i)T^{2} \)
17 \( 1 + (0.978 - 0.207i)T^{2} \)
19 \( 1 + (-0.913 + 0.406i)T^{2} \)
23 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (0.997 + 0.0697i)T^{2} \)
31 \( 1 + (-0.454 - 1.82i)T + (-0.882 + 0.469i)T^{2} \)
37 \( 1 + (1.49 + 0.318i)T + (0.913 + 0.406i)T^{2} \)
41 \( 1 + (0.997 - 0.0697i)T^{2} \)
43 \( 1 + (0.939 - 0.342i)T^{2} \)
47 \( 1 + (0.130 + 0.322i)T + (-0.719 + 0.694i)T^{2} \)
53 \( 1 + (-1.52 - 1.10i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (1.86 - 0.261i)T + (0.961 - 0.275i)T^{2} \)
61 \( 1 + (0.882 + 0.469i)T^{2} \)
67 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.0363 + 0.345i)T + (-0.978 + 0.207i)T^{2} \)
73 \( 1 + (0.104 + 0.994i)T^{2} \)
79 \( 1 + (0.374 - 0.927i)T^{2} \)
83 \( 1 + (-0.848 + 0.529i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.152 + 0.312i)T + (-0.615 - 0.788i)T^{2} \)
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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

−9.163691875044275139721241129125, −8.656960534827194305659851194525, −7.78587809576762266489894740611, −6.94955176095427793570899953997, −6.27690657958335047845512974639, −5.35244606358748322326864709834, −4.49773144400840864106648606993, −3.59982103746515758633815157196, −3.02581071231501398260372964040, −2.24115796663308787247774337896, 0.60785079986049719637201268572, 1.60282010808325718558881819902, 2.23458574459395686682136314870, 3.82611480385635774073704762827, 4.86769217470825353711110381614, 5.50173566643029886922456687498, 5.99786858720630045207262472812, 6.79186307505036459283362161135, 7.83460119904702094624985863111, 8.514397184208831706938746101527

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