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Properties

Label 2-11e3-11.8-c0-0-0
Degree $2$
Conductor $1331$
Sign $0.309 - 0.951i$
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  + (0.519 + 1.60i)3-s + (0.309 − 0.951i)4-s + (1.05 + 0.769i)5-s + (−1.48 + 1.07i)9-s + 1.68·12-s + (−0.680 + 2.09i)15-s + (−0.809 − 0.587i)16-s + (1.05 − 0.769i)20-s − 1.91·23-s + (0.221 + 0.680i)25-s + (−1.13 − 0.821i)27-s + (1.55 − 1.12i)31-s + (0.565 + 1.74i)36-s + (0.256 − 0.790i)37-s − 2.39·45-s + ⋯
L(s)  = 1  + (0.519 + 1.60i)3-s + (0.309 − 0.951i)4-s + (1.05 + 0.769i)5-s + (−1.48 + 1.07i)9-s + 1.68·12-s + (−0.680 + 2.09i)15-s + (−0.809 − 0.587i)16-s + (1.05 − 0.769i)20-s − 1.91·23-s + (0.221 + 0.680i)25-s + (−1.13 − 0.821i)27-s + (1.55 − 1.12i)31-s + (0.565 + 1.74i)36-s + (0.256 − 0.790i)37-s − 2.39·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1331\)    =    \(11^{3}\)
Sign: $0.309 - 0.951i$
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: \((2,\ 1331,\ (\ :0),\ 0.309 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.535989965\)
\(L(\frac12)\) \(\approx\) \(1.535989965\)
\(L(1)\) 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 \( 1 + (-0.519 - 1.60i)T + (-0.809 + 0.587i)T^{2} \)
5 \( 1 + (-1.05 - 0.769i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + 1.91T + T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-1.55 + 1.12i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.256 + 0.790i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.256 - 0.790i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.230 + 0.167i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.0879 - 0.270i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + 1.30T + T^{2} \)
71 \( 1 + (0.672 + 0.488i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + 0.284T + T^{2} \)
97 \( 1 + (-1.55 + 1.12i)T + (0.309 - 0.951i)T^{2} \)
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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

−9.965618505806687349636692354041, −9.607664686126343277911923132058, −8.710171748698903722591594936807, −7.61778812025058756689646618403, −6.22849376082595726787915407763, −5.94369303278102486620494349431, −4.88360479320888780899155856469, −4.03846525659286494854444504653, −2.81617293040663643274586352083, −2.06059751667065391828083876081, 1.44555140690865457603447390606, 2.23662004123553542176174588292, 3.14633781865876696699388297217, 4.52171489747595964815957243454, 5.86415004892171914931522701250, 6.46422490233456165069011711778, 7.27412056369696162497196935739, 8.187776847901538094899820123022, 8.457400656318352547034082273431, 9.404610174796893004876199143311

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