↑

Properties

Label 2-11e3-11.6-c0-0-4
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.230 + 0.167i)3-s + (−0.809 + 0.587i)4-s + (0.519 − 1.60i)5-s + (−0.283 − 0.874i)9-s − 0.284·12-s + (0.387 − 0.281i)15-s + (0.309 − 0.951i)16-s + (0.519 + 1.60i)20-s − 1.30·23-s + (−1.48 − 1.07i)25-s + (0.168 − 0.519i)27-s + (−0.404 − 1.24i)31-s + (0.743 + 0.540i)36-s + (1.55 − 1.12i)37-s − 1.54·45-s + ⋯
L(s)  = 1  + (0.230 + 0.167i)3-s + (−0.809 + 0.587i)4-s + (0.519 − 1.60i)5-s + (−0.283 − 0.874i)9-s − 0.284·12-s + (0.387 − 0.281i)15-s + (0.309 − 0.951i)16-s + (0.519 + 1.60i)20-s − 1.30·23-s + (−1.48 − 1.07i)25-s + (0.168 − 0.519i)27-s + (−0.404 − 1.24i)31-s + (0.743 + 0.540i)36-s + (1.55 − 1.12i)37-s − 1.54·45-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: $\chi_{1331} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1331,\ (\ :0),\ 0.309 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9416692304\)
\(L(\frac12)\) \(\approx\) \(0.9416692304\)
\(L(1)\) 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 \( 1 + (0.809 - 0.587i)T^{2} \)
3 \( 1 + (-0.230 - 0.167i)T + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.519 + 1.60i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + 1.30T + T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.404 + 1.24i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-1.55 + 1.12i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-1.55 - 1.12i)T + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.256 - 0.790i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.672 - 0.488i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 - 1.68T + T^{2} \)
71 \( 1 + (0.592 - 1.82i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 - 0.830T + T^{2} \)
97 \( 1 + (0.404 + 1.24i)T + (-0.809 + 0.587i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.412397443440432357841062313997, −8.969098448696220097160468355188, −8.267984125555782838457960156937, −7.54761106146349996957922205854, −6.02429968551097391027111287938, −5.48025291801082192548156885669, −4.28691464047454845530236392244, −3.96717275352085414746820209330, −2.44857591627705761634283245239, −0.812435283290659250667624295587, 1.84546903389970258791212010733, 2.79370040319787209309081043676, 3.87973814056114463076564936873, 5.05329710071332257414607160238, 5.90767006580290715741230101976, 6.59528893621422752254406777065, 7.58331224739838081687275792073, 8.325704186557341239516357118220, 9.330698195144209155717716230788, 10.08899718265880167279658574412

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