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Properties

Label 2-11e3-11.7-c0-0-2
Degree $2$
Conductor $1331$
Sign $1$
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  + (0.256 − 0.790i)3-s + (0.309 + 0.951i)4-s + (0.230 − 0.167i)5-s + (0.250 + 0.182i)9-s + 0.830·12-s + (−0.0730 − 0.224i)15-s + (−0.809 + 0.587i)16-s + (0.230 + 0.167i)20-s + 1.68·23-s + (−0.283 + 0.874i)25-s + (0.880 − 0.639i)27-s + (−1.36 − 0.988i)31-s + (−0.0957 + 0.294i)36-s + (−0.404 − 1.24i)37-s + 0.0881·45-s + ⋯
L(s)  = 1  + (0.256 − 0.790i)3-s + (0.309 + 0.951i)4-s + (0.230 − 0.167i)5-s + (0.250 + 0.182i)9-s + 0.830·12-s + (−0.0730 − 0.224i)15-s + (−0.809 + 0.587i)16-s + (0.230 + 0.167i)20-s + 1.68·23-s + (−0.283 + 0.874i)25-s + (0.880 − 0.639i)27-s + (−1.36 − 0.988i)31-s + (−0.0957 + 0.294i)36-s + (−0.404 − 1.24i)37-s + 0.0881·45-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1331\)    =    \(11^{3}\)
Sign: $1$
Analytic conductor: \(0.664255\)
Root analytic conductor: \(0.815018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1331} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1331,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.348444404\)
\(L(\frac12)\) \(\approx\) \(1.348444404\)
\(L(1)\) 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 \( 1 + (-0.256 + 0.790i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 + (-0.230 + 0.167i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 - 1.68T + T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (1.36 + 0.988i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.404 + 1.24i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.404 - 1.24i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (-1.55 - 1.12i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.592 + 1.82i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + 0.284T + T^{2} \)
71 \( 1 + (-1.05 + 0.769i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 + 1.91T + T^{2} \)
97 \( 1 + (1.36 + 0.988i)T + (0.309 + 0.951i)T^{2} \)
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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

−9.556018021178667258593969089825, −8.937578110224302910966690634135, −8.014380998298248514135363711225, −7.37003190035433347902823937327, −6.88862417761672350633697605215, −5.78225952331225686640665022401, −4.66944326717030601968764199555, −3.58405975131329532915825101491, −2.57872218986312447315675460325, −1.58511757954490734084886290531, 1.39127232227049611450809539498, 2.71515992428331541926843005493, 3.77735348047213346578201550901, 4.88652111160692234839603511548, 5.45092810105067994259763710340, 6.66999711282881372282782062010, 7.05056643738990634858083270620, 8.515572040745970818950576480451, 9.159264498606615825294635845272, 9.971684524123321734453882977136

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