Properties

Label 2-3267-11.6-c0-0-0
Degree $2$
Conductor $3267$
Sign $-0.105 - 0.994i$
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 + (−1.13 − 1.56i)7-s + (−0.492 + 0.159i)13-s + (0.309 − 0.951i)16-s + (−0.831 + 1.14i)19-s + (0.809 + 0.587i)25-s + (1.83 + 0.596i)28-s + (0.535 + 1.64i)31-s + 1.41i·43-s + (−0.844 + 2.59i)49-s + (0.304 − 0.418i)52-s + (1.34 + 0.437i)61-s + (0.309 + 0.951i)64-s − 1.73·67-s + (−0.304 − 0.418i)73-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)4-s + (−1.13 − 1.56i)7-s + (−0.492 + 0.159i)13-s + (0.309 − 0.951i)16-s + (−0.831 + 1.14i)19-s + (0.809 + 0.587i)25-s + (1.83 + 0.596i)28-s + (0.535 + 1.64i)31-s + 1.41i·43-s + (−0.844 + 2.59i)49-s + (0.304 − 0.418i)52-s + (1.34 + 0.437i)61-s + (0.309 + 0.951i)64-s − 1.73·67-s + (−0.304 − 0.418i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $-0.105 - 0.994i$
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: \((2,\ 3267,\ (\ :0),\ -0.105 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5084928248\)
\(L(\frac12)\) \(\approx\) \(0.5084928248\)
\(L(1)\) 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 \( 1 + (1.13 + 1.56i)T + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.492 - 0.159i)T + (0.809 - 0.587i)T^{2} \)
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 \( 1 + (0.304 + 0.418i)T + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.492 + 0.159i)T + (0.809 - 0.587i)T^{2} \)
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

−9.051231326211761942788019023430, −8.257182242482379676395163267013, −7.50389787019201477383881240553, −6.90681732541223864727579472436, −6.18244061513063455055842729299, −4.97212523801421561005704019331, −4.27904039296474071752662941925, −3.58348915984394351229030868609, −2.91714526215407673227707678162, −1.17067295121211329376219747123, 0.34669964667033080075886463480, 2.19168363399318001586107407332, 2.85759782762158150729223672888, 4.02893505616840435110266225467, 4.90896047274438158454242020589, 5.61301910866459448418082226058, 6.25339330649354929453520665190, 6.91281092020145091713910841683, 8.172122468793051809769186250228, 8.848465226937882216976576090273

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