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Properties

Label 2-3267-99.13-c0-0-2
Degree $2$
Conductor $3267$
Sign $0.416 + 0.909i$
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  + (0.913 − 0.406i)4-s + (−0.978 − 0.207i)5-s + (0.669 − 0.743i)16-s + (−0.978 + 0.207i)20-s + (1 − 1.73i)23-s + (−0.669 − 0.743i)31-s + (0.809 + 0.587i)37-s + (0.913 + 0.406i)47-s + (−0.978 − 0.207i)49-s + (0.309 − 0.951i)53-s + (0.913 − 0.406i)59-s + (0.309 − 0.951i)64-s + (0.5 − 0.866i)67-s + (0.309 + 0.951i)71-s + (−0.809 + 0.587i)80-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)4-s + (−0.978 − 0.207i)5-s + (0.669 − 0.743i)16-s + (−0.978 + 0.207i)20-s + (1 − 1.73i)23-s + (−0.669 − 0.743i)31-s + (0.809 + 0.587i)37-s + (0.913 + 0.406i)47-s + (−0.978 − 0.207i)49-s + (0.309 − 0.951i)53-s + (0.913 − 0.406i)59-s + (0.309 − 0.951i)64-s + (0.5 − 0.866i)67-s + (0.309 + 0.951i)71-s + (−0.809 + 0.587i)80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.416 + 0.909i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (1927, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.416 + 0.909i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.265640143\)
\(L(\frac12)\) \(\approx\) \(1.265640143\)
\(L(1)\) 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 \( 1 + (-0.913 + 0.406i)T^{2} \)
5 \( 1 + (0.978 + 0.207i)T + (0.913 + 0.406i)T^{2} \)
7 \( 1 + (0.978 + 0.207i)T^{2} \)
13 \( 1 + (0.104 - 0.994i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.978 + 0.207i)T^{2} \)
31 \( 1 + (0.669 + 0.743i)T + (-0.104 + 0.994i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.978 - 0.207i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.913 - 0.406i)T + (0.669 + 0.743i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.913 + 0.406i)T + (0.669 - 0.743i)T^{2} \)
61 \( 1 + (0.104 + 0.994i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.913 + 0.406i)T^{2} \)
83 \( 1 + (0.104 + 0.994i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + (-0.978 + 0.207i)T + (0.913 - 0.406i)T^{2} \)
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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

−8.474978679652681421567315868138, −7.971155542157201027194167287847, −7.12191940935626008613679799663, −6.59643259153843119387385979509, −5.72373134070882238622013364457, −4.82908448380022384129633386190, −4.01939688444183918602044045407, −3.02362605317537372332403189386, −2.15532165277758020528918703446, −0.794313357271090733583329391944, 1.41434960442159240740548448588, 2.62581748094913698932389524942, 3.45635155482939134579688994336, 4.03061327381775408531982035429, 5.23178067842072607698088210814, 6.01006307483197628224189943995, 7.06708377553536741996008481831, 7.34775328829252194603115191576, 8.022188099316178882698328030906, 8.836364804355641303714488906406

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