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Properties

Label 2-3267-27.14-c0-0-0
Degree $2$
Conductor $3267$
Sign $-0.396 - 0.918i$
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  + (−0.173 + 0.984i)3-s + (−0.939 + 0.342i)4-s + (0.439 − 0.524i)5-s + (−0.939 − 0.342i)9-s + (−0.173 − 0.984i)12-s + (0.439 + 0.524i)15-s + (0.766 − 0.642i)16-s + (−0.233 + 0.642i)20-s + (0.592 + 1.62i)23-s + (0.0923 + 0.524i)25-s + (0.5 − 0.866i)27-s + (0.326 − 0.118i)31-s + 0.999·36-s + (−0.939 + 1.62i)37-s + (−0.592 + 0.342i)45-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)3-s + (−0.939 + 0.342i)4-s + (0.439 − 0.524i)5-s + (−0.939 − 0.342i)9-s + (−0.173 − 0.984i)12-s + (0.439 + 0.524i)15-s + (0.766 − 0.642i)16-s + (−0.233 + 0.642i)20-s + (0.592 + 1.62i)23-s + (0.0923 + 0.524i)25-s + (0.5 − 0.866i)27-s + (0.326 − 0.118i)31-s + 0.999·36-s + (−0.939 + 1.62i)37-s + (−0.592 + 0.342i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $-0.396 - 0.918i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ -0.396 - 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8285332546\)
\(L(\frac12)\) \(\approx\) \(0.8285332546\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 \)
good2 \( 1 + (0.939 - 0.342i)T^{2} \)
5 \( 1 + (-0.439 + 0.524i)T + (-0.173 - 0.984i)T^{2} \)
7 \( 1 + (0.766 + 0.642i)T^{2} \)
13 \( 1 + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \)
37 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (0.439 - 1.20i)T + (-0.766 - 0.642i)T^{2} \)
53 \( 1 - 1.96iT - T^{2} \)
59 \( 1 + (-1.26 + 1.50i)T + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.766 + 0.642i)T^{2} \)
67 \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (-1.11 - 0.642i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.939 - 0.342i)T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
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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

−9.212309334705313436622465708168, −8.500649698285444552211723301284, −7.86025401553258196937696328288, −6.75289819809402636083907571815, −5.64785702612269900454509086233, −5.18324758556152947685935760475, −4.54345887630728799729863111730, −3.65019555839717160049966558737, −2.97473811417611200024454703847, −1.29875893146418546164645615957, 0.57480218064960379523749741278, 1.88656625822407242026446373898, 2.78007155965612752466602809950, 3.90255912381001307393038261701, 4.97214837081066650101141859059, 5.57235417199948006801225353451, 6.48940886095712393827728523943, 6.86700539859842131713615084433, 7.950286028258937349553921911758, 8.566205431770709856485264672456

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