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Properties

Label 2-3267-297.205-c0-0-0
Degree $2$
Conductor $3267$
Sign $-0.0692 + 0.997i$
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  + (−0.374 − 0.927i)3-s + (−0.882 + 0.469i)4-s + (0.333 − 0.0957i)5-s + (−0.719 + 0.694i)9-s + (0.766 + 0.642i)12-s + (−0.213 − 0.273i)15-s + (0.559 − 0.829i)16-s + (−0.249 + 0.241i)20-s + (−0.173 − 0.984i)23-s + (−0.745 + 0.466i)25-s + (0.913 + 0.406i)27-s + (0.671 − 1.37i)31-s + (0.309 − 0.951i)36-s + (0.232 − 0.258i)37-s + (−0.173 + 0.300i)45-s + ⋯
L(s)  = 1  + (−0.374 − 0.927i)3-s + (−0.882 + 0.469i)4-s + (0.333 − 0.0957i)5-s + (−0.719 + 0.694i)9-s + (0.766 + 0.642i)12-s + (−0.213 − 0.273i)15-s + (0.559 − 0.829i)16-s + (−0.249 + 0.241i)20-s + (−0.173 − 0.984i)23-s + (−0.745 + 0.466i)25-s + (0.913 + 0.406i)27-s + (0.671 − 1.37i)31-s + (0.309 − 0.951i)36-s + (0.232 − 0.258i)37-s + (−0.173 + 0.300i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0692 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0692 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $-0.0692 + 0.997i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (2581, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ -0.0692 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7560046310\)
\(L(\frac12)\) \(\approx\) \(0.7560046310\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.374 + 0.927i)T \)
11 \( 1 \)
good2 \( 1 + (0.882 - 0.469i)T^{2} \)
5 \( 1 + (-0.333 + 0.0957i)T + (0.848 - 0.529i)T^{2} \)
7 \( 1 + (-0.961 + 0.275i)T^{2} \)
13 \( 1 + (-0.990 + 0.139i)T^{2} \)
17 \( 1 + (-0.669 - 0.743i)T^{2} \)
19 \( 1 + (0.104 - 0.994i)T^{2} \)
23 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (0.719 + 0.694i)T^{2} \)
31 \( 1 + (-0.671 + 1.37i)T + (-0.615 - 0.788i)T^{2} \)
37 \( 1 + (-0.232 + 0.258i)T + (-0.104 - 0.994i)T^{2} \)
41 \( 1 + (0.719 - 0.694i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (-1.65 - 0.882i)T + (0.559 + 0.829i)T^{2} \)
53 \( 1 + (1.23 + 0.900i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.0534 + 1.53i)T + (-0.997 - 0.0697i)T^{2} \)
61 \( 1 + (0.615 - 0.788i)T^{2} \)
67 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (1.71 + 0.764i)T + (0.669 + 0.743i)T^{2} \)
73 \( 1 + (-0.913 - 0.406i)T^{2} \)
79 \( 1 + (0.882 - 0.469i)T^{2} \)
83 \( 1 + (-0.990 - 0.139i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.80 + 0.518i)T + (0.848 + 0.529i)T^{2} \)
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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

−8.480100029152596744588295178462, −7.903223454764158464403172269909, −7.28026871162704384566345343401, −6.29023150315534666351779658869, −5.70401542022193622890351206143, −4.83838935009889222380741696731, −4.04601829995605203434967643702, −2.90644044279521253690512904335, −1.94613481840525231356875573787, −0.55940335244794128740168162423, 1.18804230737929630620371305895, 2.70902136989793618229298369817, 3.81607552773239057140214304109, 4.37246417972263713973267633988, 5.29506522032021903731567946071, 5.73680649814235703277274320778, 6.51665097578850996416212719698, 7.62151081508282466530704420395, 8.637158030728076315017741389300, 9.050960136289366419593745938357

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