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Properties

Label 2-3267-11.2-c0-0-1
Degree $2$
Conductor $3267$
Sign $0.681 + 0.731i$
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  + (−0.809 − 0.587i)4-s + (0.304 − 0.418i)7-s + (1.83 + 0.596i)13-s + (0.309 + 0.951i)16-s + (−0.831 − 1.14i)19-s + (0.809 − 0.587i)25-s + (−0.492 + 0.159i)28-s + (−0.535 + 1.64i)31-s − 1.41i·43-s + (0.226 + 0.696i)49-s + (−1.13 − 1.56i)52-s + (1.34 − 0.437i)61-s + (0.309 − 0.951i)64-s + 1.73·67-s + (1.13 − 1.56i)73-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)4-s + (0.304 − 0.418i)7-s + (1.83 + 0.596i)13-s + (0.309 + 0.951i)16-s + (−0.831 − 1.14i)19-s + (0.809 − 0.587i)25-s + (−0.492 + 0.159i)28-s + (−0.535 + 1.64i)31-s − 1.41i·43-s + (0.226 + 0.696i)49-s + (−1.13 − 1.56i)52-s + (1.34 − 0.437i)61-s + (0.309 − 0.951i)64-s + 1.73·67-s + (1.13 − 1.56i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.681 + 0.731i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (838, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.681 + 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.133699143\)
\(L(\frac12)\) \(\approx\) \(1.133699143\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.304 + 0.418i)T + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-1.83 - 0.596i)T + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.831 + 1.14i)T + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.535 - 1.64i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2} \)
67 \( 1 - 1.73T + T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-1.13 + 1.56i)T + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (1.83 + 0.596i)T + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
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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

−8.600318455223118400348272185968, −8.437462614021968044226496234106, −7.04515517372000576502886756315, −6.51690755325409044029723532081, −5.65650055307991812059617099624, −4.80831694470678973350776581739, −4.16858829870369060677789181889, −3.37066188118227708094761357023, −1.89995998755732687185511280733, −0.886188653004514832453301012965, 1.16619913820001205251355152875, 2.51323951846356027015164846669, 3.68462016643140450619198518862, 3.99878968579795002560419198531, 5.18830002093268639229146036758, 5.79209469452651247114826755125, 6.61353958824165358962593857295, 7.74723202868378783724316144784, 8.300577029821686795597087425657, 8.696172754021391535641798082821

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