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Properties

Label 2-3267-297.257-c0-0-0
Degree $2$
Conductor $3267$
Sign $0.568 - 0.822i$
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  + (0.882 + 0.469i)3-s + (−0.615 − 0.788i)4-s + (−0.0477 + 0.682i)5-s + (0.559 + 0.829i)9-s + (−0.173 − 0.984i)12-s + (−0.362 + 0.580i)15-s + (−0.241 + 0.970i)16-s + (0.567 − 0.382i)20-s + (0.592 + 1.62i)23-s + (0.526 + 0.0740i)25-s + (0.104 + 0.994i)27-s + (−0.333 − 0.0957i)31-s + (0.309 − 0.951i)36-s + (−1.83 − 0.390i)37-s + (−0.592 + 0.342i)45-s + ⋯
L(s)  = 1  + (0.882 + 0.469i)3-s + (−0.615 − 0.788i)4-s + (−0.0477 + 0.682i)5-s + (0.559 + 0.829i)9-s + (−0.173 − 0.984i)12-s + (−0.362 + 0.580i)15-s + (−0.241 + 0.970i)16-s + (0.567 − 0.382i)20-s + (0.592 + 1.62i)23-s + (0.526 + 0.0740i)25-s + (0.104 + 0.994i)27-s + (−0.333 − 0.0957i)31-s + (0.309 − 0.951i)36-s + (−1.83 − 0.390i)37-s + (−0.592 + 0.342i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.568 - 0.822i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (3227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.568 - 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.452383398\)
\(L(\frac12)\) \(\approx\) \(1.452383398\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.882 - 0.469i)T \)
11 \( 1 \)
good2 \( 1 + (0.615 + 0.788i)T^{2} \)
5 \( 1 + (0.0477 - 0.682i)T + (-0.990 - 0.139i)T^{2} \)
7 \( 1 + (-0.997 - 0.0697i)T^{2} \)
13 \( 1 + (0.0348 - 0.999i)T^{2} \)
17 \( 1 + (0.978 - 0.207i)T^{2} \)
19 \( 1 + (0.913 - 0.406i)T^{2} \)
23 \( 1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.559 + 0.829i)T^{2} \)
31 \( 1 + (0.333 + 0.0957i)T + (0.848 + 0.529i)T^{2} \)
37 \( 1 + (1.83 + 0.390i)T + (0.913 + 0.406i)T^{2} \)
41 \( 1 + (-0.559 - 0.829i)T^{2} \)
43 \( 1 + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (-1.01 - 0.791i)T + (0.241 + 0.970i)T^{2} \)
53 \( 1 + (-1.15 + 1.59i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (-1.82 - 0.737i)T + (0.719 + 0.694i)T^{2} \)
61 \( 1 + (0.848 - 0.529i)T^{2} \)
67 \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (1.27 - 0.134i)T + (0.978 - 0.207i)T^{2} \)
73 \( 1 + (-0.104 - 0.994i)T^{2} \)
79 \( 1 + (-0.615 - 0.788i)T^{2} \)
83 \( 1 + (-0.0348 - 0.999i)T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.52 - 0.106i)T + (0.990 - 0.139i)T^{2} \)
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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

−8.897487083836565808935043728779, −8.515729705870092746560306245447, −7.32836512145501999319774248011, −6.96769013724975198623061734358, −5.65209044700718861875339787710, −5.20126518414026221920835292072, −4.12612645087311404973252851862, −3.51982005690316453976834509589, −2.51904761830802369041265295041, −1.44160108854718665358026356419, 0.875199310626575083151013802671, 2.27519358278358917081064040550, 3.11885814626869176513461649932, 4.00627405964541221474253100706, 4.64565023159409188826464093485, 5.56853571444854994439429136494, 6.87169147572046300240335308530, 7.22912562373162546980563933601, 8.237886133599202053448624881647, 8.786243673816766267915097628467

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