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Properties

Label 2-3267-297.211-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  + (−0.997 − 0.0697i)3-s + (−0.719 + 0.694i)4-s + (−0.213 − 0.273i)5-s + (0.990 + 0.139i)9-s + (0.766 − 0.642i)12-s + (0.194 + 0.287i)15-s + (0.0348 − 0.999i)16-s + (0.343 + 0.0483i)20-s + (−0.173 + 0.984i)23-s + (0.212 − 0.853i)25-s + (−0.978 − 0.207i)27-s + (−1.35 − 0.719i)31-s + (−0.809 + 0.587i)36-s + (0.317 − 0.141i)37-s + (−0.173 − 0.300i)45-s + ⋯
L(s)  = 1  + (−0.997 − 0.0697i)3-s + (−0.719 + 0.694i)4-s + (−0.213 − 0.273i)5-s + (0.990 + 0.139i)9-s + (0.766 − 0.642i)12-s + (0.194 + 0.287i)15-s + (0.0348 − 0.999i)16-s + (0.343 + 0.0483i)20-s + (−0.173 + 0.984i)23-s + (0.212 − 0.853i)25-s + (−0.978 − 0.207i)27-s + (−1.35 − 0.719i)31-s + (−0.809 + 0.587i)36-s + (0.317 − 0.141i)37-s + (−0.173 − 0.300i)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: $\chi_{3267} (2290, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.990 - 0.138i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6508908626\)
\(L(\frac12)\) \(\approx\) \(0.6508908626\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.997 + 0.0697i)T \)
11 \( 1 \)
good2 \( 1 + (0.719 - 0.694i)T^{2} \)
5 \( 1 + (0.213 + 0.273i)T + (-0.241 + 0.970i)T^{2} \)
7 \( 1 + (0.615 + 0.788i)T^{2} \)
13 \( 1 + (-0.438 + 0.898i)T^{2} \)
17 \( 1 + (-0.913 - 0.406i)T^{2} \)
19 \( 1 + (-0.669 - 0.743i)T^{2} \)
23 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.990 + 0.139i)T^{2} \)
31 \( 1 + (1.35 + 0.719i)T + (0.559 + 0.829i)T^{2} \)
37 \( 1 + (-0.317 + 0.141i)T + (0.669 - 0.743i)T^{2} \)
41 \( 1 + (-0.990 - 0.139i)T^{2} \)
43 \( 1 + (-0.766 - 0.642i)T^{2} \)
47 \( 1 + (-1.35 - 1.30i)T + (0.0348 + 0.999i)T^{2} \)
53 \( 1 + (-0.473 + 1.45i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-1.47 - 0.422i)T + (0.848 + 0.529i)T^{2} \)
61 \( 1 + (-0.559 + 0.829i)T^{2} \)
67 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (-1.83 - 0.390i)T + (0.913 + 0.406i)T^{2} \)
73 \( 1 + (0.978 + 0.207i)T^{2} \)
79 \( 1 + (0.719 - 0.694i)T^{2} \)
83 \( 1 + (-0.438 - 0.898i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.15 + 1.48i)T + (-0.241 - 0.970i)T^{2} \)
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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

−8.827648087031363460252139350174, −8.009650262252897905807972915197, −7.41227310693537339411205257317, −6.64250230183267538040847130354, −5.61432827046001019360106581746, −5.10016547516724391473062642894, −4.15777220445873866841395436090, −3.66550110468472873017526381073, −2.23504489843166330584808424711, −0.73578757731614749582260975555, 0.77825359633759190514835373397, 1.99505196803293913037905705067, 3.54448972894009661293961525664, 4.29937541914633843991584279529, 5.13233998589830067944629771211, 5.63562914111147393959519608380, 6.50502151084413194993483755141, 7.10869441303675363276423080177, 8.061074237641672842766911901798, 9.018945306597615350117114849852

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