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Properties

Label 2-11e3-11.2-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  + (−0.672 + 0.488i)3-s + (−0.809 − 0.587i)4-s + (−0.0879 − 0.270i)5-s + (−0.0957 + 0.294i)9-s + 0.830·12-s + (0.191 + 0.138i)15-s + (0.309 + 0.951i)16-s + (−0.0879 + 0.270i)20-s + 1.68·23-s + (0.743 − 0.540i)25-s + (−0.336 − 1.03i)27-s + (0.519 − 1.60i)31-s + (0.250 − 0.182i)36-s + (1.05 + 0.769i)37-s + 0.0881·45-s + ⋯
L(s)  = 1  + (−0.672 + 0.488i)3-s + (−0.809 − 0.587i)4-s + (−0.0879 − 0.270i)5-s + (−0.0957 + 0.294i)9-s + 0.830·12-s + (0.191 + 0.138i)15-s + (0.309 + 0.951i)16-s + (−0.0879 + 0.270i)20-s + 1.68·23-s + (0.743 − 0.540i)25-s + (−0.336 − 1.03i)27-s + (0.519 − 1.60i)31-s + (0.250 − 0.182i)36-s + (1.05 + 0.769i)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: $\chi_{1331} (596, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1331,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6895029471\)
\(L(\frac12)\) \(\approx\) \(0.6895029471\)
\(L(1)\) 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 \( 1 + (0.672 - 0.488i)T + (0.309 - 0.951i)T^{2} \)
5 \( 1 + (0.0879 + 0.270i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 - 1.68T + T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.519 + 1.60i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-1.05 - 0.769i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-1.05 + 0.769i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.592 - 1.82i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-1.55 - 1.12i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 + 0.284T + T^{2} \)
71 \( 1 + (0.404 + 1.24i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + 1.91T + T^{2} \)
97 \( 1 + (-0.519 + 1.60i)T + (-0.809 - 0.587i)T^{2} \)
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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

−9.864401839269452120711782419085, −9.108265213525785403236630440127, −8.407096148428256569029194402935, −7.42009196960413041229756274965, −6.19728854887104068043129638996, −5.56264459540220542314452016298, −4.68459158220548203298898390345, −4.25194053726174400891170954967, −2.69692539728970287020408130001, −0.951139192621951683066811718955, 0.981635519143627430834342897639, 2.86394402475023642000813169368, 3.71108347751307166887678354504, 4.88308193773829424051297576160, 5.54185604596991230558490333891, 6.78374058032854058936318349685, 7.14346915770421613088607715917, 8.282023384014114367240851405874, 8.956612362801077762126231950197, 9.682764248829263649462019621170

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