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Properties

Label 2-11e3-11.8-c0-0-3
Degree $2$
Conductor $1331$
Sign $-0.809 + 0.587i$
Analytic cond. $0.664255$
Root an. cond. $0.815018$
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  + (−0.0879 − 0.270i)3-s + (0.309 − 0.951i)4-s + (−1.36 − 0.988i)5-s + (0.743 − 0.540i)9-s − 0.284·12-s + (−0.147 + 0.455i)15-s + (−0.809 − 0.587i)16-s + (−1.36 + 0.988i)20-s − 1.30·23-s + (0.565 + 1.74i)25-s + (−0.441 − 0.321i)27-s + (1.05 − 0.769i)31-s + (−0.283 − 0.874i)36-s + (−0.592 + 1.82i)37-s − 1.54·45-s + ⋯
L(s)  = 1  + (−0.0879 − 0.270i)3-s + (0.309 − 0.951i)4-s + (−1.36 − 0.988i)5-s + (0.743 − 0.540i)9-s − 0.284·12-s + (−0.147 + 0.455i)15-s + (−0.809 − 0.587i)16-s + (−1.36 + 0.988i)20-s − 1.30·23-s + (0.565 + 1.74i)25-s + (−0.441 − 0.321i)27-s + (1.05 − 0.769i)31-s + (−0.283 − 0.874i)36-s + (−0.592 + 1.82i)37-s − 1.54·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1331\)    =    \(11^{3}\)
Sign: $-0.809 + 0.587i$
Analytic conductor: \(0.664255\)
Root analytic conductor: \(0.815018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1331} (1207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1331,\ (\ :0),\ -0.809 + 0.587i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7815010030\)
\(L(\frac12)\) \(\approx\) \(0.7815010030\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T^{2} \)
3 \( 1 + (0.0879 + 0.270i)T + (-0.809 + 0.587i)T^{2} \)
5 \( 1 + (1.36 + 0.988i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + 1.30T + T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-1.05 + 0.769i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.592 - 1.82i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.592 + 1.82i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.672 - 0.488i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.256 + 0.790i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 - 1.68T + T^{2} \)
71 \( 1 + (-1.55 - 1.12i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 - 0.830T + T^{2} \)
97 \( 1 + (-1.05 + 0.769i)T + (0.309 - 0.951i)T^{2} \)
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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

−9.703834858561387117271590797412, −8.511264106370933074516497923889, −7.982885895064531357878224016477, −6.97228114182517077033325408706, −6.31515295700400639372811031007, −5.16170542152996583448036561193, −4.44937017289304686762714613905, −3.55960426397694812054542243359, −1.83874668153421823065654875332, −0.67298068428479325757427945693, 2.23650065638581581160866804457, 3.34375834192526889457091687192, 3.96788906112860932433350918658, 4.76334434035789920705405263292, 6.33728241948049274654932749383, 7.03829983235158069307098795840, 7.86508523830649239362479282764, 8.051725168956934141775096188731, 9.328472381698280177852390172438, 10.41026397651418457774535273652

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