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Properties

Label 2-3267-11.6-c0-0-3
Degree $2$
Conductor $3267$
Sign $0.292 + 0.956i$
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  + (−0.809 + 0.587i)4-s + (−0.304 − 0.418i)7-s + (−1.83 + 0.596i)13-s + (0.309 − 0.951i)16-s + (0.831 − 1.14i)19-s + (0.809 + 0.587i)25-s + (0.492 + 0.159i)28-s + (−0.535 − 1.64i)31-s − 1.41i·43-s + (0.226 − 0.696i)49-s + (1.13 − 1.56i)52-s + (−1.34 − 0.437i)61-s + (0.309 + 0.951i)64-s + 1.73·67-s + (−1.13 − 1.56i)73-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)4-s + (−0.304 − 0.418i)7-s + (−1.83 + 0.596i)13-s + (0.309 − 0.951i)16-s + (0.831 − 1.14i)19-s + (0.809 + 0.587i)25-s + (0.492 + 0.159i)28-s + (−0.535 − 1.64i)31-s − 1.41i·43-s + (0.226 − 0.696i)49-s + (1.13 − 1.56i)52-s + (−1.34 − 0.437i)61-s + (0.309 + 0.951i)64-s + 1.73·67-s + (−1.13 − 1.56i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.292 + 0.956i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (2998, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.292 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6441723538\)
\(L(\frac12)\) \(\approx\) \(0.6441723538\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.809 - 0.587i)T^{2} \)
5 \( 1 + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (0.304 + 0.418i)T + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (1.83 - 0.596i)T + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.535 + 1.64i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2} \)
67 \( 1 - 1.73T + T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (1.13 + 1.56i)T + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (-1.83 + 0.596i)T + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
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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

−8.869676825760762830205283667505, −7.72256321691022467921170622070, −7.35683961957015230975891221853, −6.66676201213035514869613039975, −5.33131276202724699023675114036, −4.84887751292469121028238542200, −4.03273756133781538159435564945, −3.15197777062040201778723862841, −2.23750292159158818923295133690, −0.43068129681576193715264096459, 1.22554977726817142538065734547, 2.55117617851007652666873958086, 3.42251186699316920350359868858, 4.57340288780439488799194435188, 5.15921035436385732826333329337, 5.77369127872901694884807182199, 6.69492827332307969169185530593, 7.58087061093774244561794553613, 8.274175481943559679583579421570, 9.108693262732714004245426640441

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