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Properties

Label 2-3267-297.13-c0-0-0
Degree $2$
Conductor $3267$
Sign $0.990 - 0.138i$
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  + (0.438 + 0.898i)3-s + (0.961 + 0.275i)4-s + (0.704 − 1.74i)5-s + (−0.615 + 0.788i)9-s + (0.173 + 0.984i)12-s + (1.87 − 0.131i)15-s + (0.848 + 0.529i)16-s + (1.15 − 1.48i)20-s + (0.939 − 0.342i)23-s + (−1.82 − 1.75i)25-s + (−0.978 − 0.207i)27-s + (0.0121 + 0.347i)31-s + (−0.809 + 0.587i)36-s + (−1.71 + 0.764i)37-s + (0.939 + 1.62i)45-s + ⋯
L(s)  = 1  + (0.438 + 0.898i)3-s + (0.961 + 0.275i)4-s + (0.704 − 1.74i)5-s + (−0.615 + 0.788i)9-s + (0.173 + 0.984i)12-s + (1.87 − 0.131i)15-s + (0.848 + 0.529i)16-s + (1.15 − 1.48i)20-s + (0.939 − 0.342i)23-s + (−1.82 − 1.75i)25-s + (−0.978 − 0.207i)27-s + (0.0121 + 0.347i)31-s + (−0.809 + 0.587i)36-s + (−1.71 + 0.764i)37-s + (0.939 + 1.62i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.990 - 0.138i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.990 - 0.138i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.061491441\)
\(L(\frac12)\) \(\approx\) \(2.061491441\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.438 - 0.898i)T \)
11 \( 1 \)
good2 \( 1 + (-0.961 - 0.275i)T^{2} \)
5 \( 1 + (-0.704 + 1.74i)T + (-0.719 - 0.694i)T^{2} \)
7 \( 1 + (0.374 - 0.927i)T^{2} \)
13 \( 1 + (-0.559 - 0.829i)T^{2} \)
17 \( 1 + (-0.913 - 0.406i)T^{2} \)
19 \( 1 + (-0.669 - 0.743i)T^{2} \)
23 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.615 + 0.788i)T^{2} \)
31 \( 1 + (-0.0121 - 0.347i)T + (-0.997 + 0.0697i)T^{2} \)
37 \( 1 + (1.71 - 0.764i)T + (0.669 - 0.743i)T^{2} \)
41 \( 1 + (0.615 - 0.788i)T^{2} \)
43 \( 1 + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (-1.47 + 0.422i)T + (0.848 - 0.529i)T^{2} \)
53 \( 1 + (-0.107 + 0.330i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.0840 + 0.336i)T + (-0.882 + 0.469i)T^{2} \)
61 \( 1 + (0.997 + 0.0697i)T^{2} \)
67 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (1.49 + 0.318i)T + (0.913 + 0.406i)T^{2} \)
73 \( 1 + (0.978 + 0.207i)T^{2} \)
79 \( 1 + (-0.961 - 0.275i)T^{2} \)
83 \( 1 + (-0.559 + 0.829i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.573 + 1.42i)T + (-0.719 + 0.694i)T^{2} \)
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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

−8.830229065042818446435898011001, −8.362022848573165915572328709557, −7.53996800540065989219592396443, −6.49959737211994574743338404599, −5.58702077493648664175377619135, −5.04996032944503141321571799162, −4.26921181497280876039410201966, −3.27920695354294147415523577543, −2.29804706101768272394118859647, −1.37042470119783972349165009494, 1.52313331061645885916056321115, 2.33128946896980655562396356309, 2.92823833144669977402999159120, 3.64799283549902490086836126624, 5.48811383377481780904857637605, 6.00360172394265549348963061573, 6.78521339239600798282900029607, 7.14682856603976762981306261175, 7.66421841468708950137810713214, 8.794670528503894229608786733961

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