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Properties

Label 2-3267-297.277-c0-0-0
Degree $2$
Conductor $3267$
Sign $0.151 - 0.988i$
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  + (0.961 − 0.275i)3-s + (−0.997 + 0.0697i)4-s + (−1.35 + 0.719i)5-s + (0.848 − 0.529i)9-s + (−0.939 + 0.342i)12-s + (−1.10 + 1.06i)15-s + (0.990 − 0.139i)16-s + (1.29 − 0.811i)20-s + (−0.766 + 0.642i)23-s + (0.753 − 1.11i)25-s + (0.669 − 0.743i)27-s + (0.704 + 1.74i)31-s + (−0.809 + 0.587i)36-s + (−0.160 + 1.52i)37-s + (−0.766 + 1.32i)45-s + ⋯
L(s)  = 1  + (0.961 − 0.275i)3-s + (−0.997 + 0.0697i)4-s + (−1.35 + 0.719i)5-s + (0.848 − 0.529i)9-s + (−0.939 + 0.342i)12-s + (−1.10 + 1.06i)15-s + (0.990 − 0.139i)16-s + (1.29 − 0.811i)20-s + (−0.766 + 0.642i)23-s + (0.753 − 1.11i)25-s + (0.669 − 0.743i)27-s + (0.704 + 1.74i)31-s + (−0.809 + 0.587i)36-s + (−0.160 + 1.52i)37-s + (−0.766 + 1.32i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.151 - 0.988i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (2653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.151 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8680507198\)
\(L(\frac12)\) \(\approx\) \(0.8680507198\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.961 + 0.275i)T \)
11 \( 1 \)
good2 \( 1 + (0.997 - 0.0697i)T^{2} \)
5 \( 1 + (1.35 - 0.719i)T + (0.559 - 0.829i)T^{2} \)
7 \( 1 + (0.882 - 0.469i)T^{2} \)
13 \( 1 + (0.241 + 0.970i)T^{2} \)
17 \( 1 + (0.104 + 0.994i)T^{2} \)
19 \( 1 + (0.978 - 0.207i)T^{2} \)
23 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.848 - 0.529i)T^{2} \)
31 \( 1 + (-0.704 - 1.74i)T + (-0.719 + 0.694i)T^{2} \)
37 \( 1 + (0.160 - 1.52i)T + (-0.978 - 0.207i)T^{2} \)
41 \( 1 + (-0.848 + 0.529i)T^{2} \)
43 \( 1 + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (0.346 + 0.0242i)T + (0.990 + 0.139i)T^{2} \)
53 \( 1 + (0.580 - 1.78i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.823 - 1.68i)T + (-0.615 - 0.788i)T^{2} \)
61 \( 1 + (0.719 + 0.694i)T^{2} \)
67 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.232 + 0.258i)T + (-0.104 - 0.994i)T^{2} \)
73 \( 1 + (-0.669 + 0.743i)T^{2} \)
79 \( 1 + (0.997 - 0.0697i)T^{2} \)
83 \( 1 + (0.241 - 0.970i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.306 + 0.163i)T + (0.559 + 0.829i)T^{2} \)
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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

−8.722026641510043402151018357890, −8.269129185057035762949779875579, −7.66698494098351558048589886242, −7.05669089736005151067797188738, −6.15395519389401923147960740631, −4.81766417223422414370009374245, −4.21198067986061189233507645973, −3.38726010387875776439692946785, −2.94060810399454822788398986474, −1.32626303071026643828651602744, 0.52770281301175657073652918369, 2.08722488310536080284327126781, 3.47212564402094297603754001784, 3.93537601387006867706501193033, 4.59623290321174017881501594513, 5.23333251534221142632396835139, 6.50468082805607201984388958334, 7.66328055715460596115705015293, 8.130920676955524340394795725756, 8.430005064634152539291184514706

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