Properties

Label 2-11e3-11.2-c0-0-0
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  + (−1.36 + 0.988i)3-s + (−0.809 − 0.587i)4-s + (−0.404 − 1.24i)5-s + (0.565 − 1.74i)9-s + 1.68·12-s + (1.78 + 1.29i)15-s + (0.309 + 0.951i)16-s + (−0.404 + 1.24i)20-s − 1.91·23-s + (−0.578 + 0.420i)25-s + (0.431 + 1.32i)27-s + (−0.592 + 1.82i)31-s + (−1.48 + 1.07i)36-s + (−0.672 − 0.488i)37-s − 2.39·45-s + ⋯
L(s)  = 1  + (−1.36 + 0.988i)3-s + (−0.809 − 0.587i)4-s + (−0.404 − 1.24i)5-s + (0.565 − 1.74i)9-s + 1.68·12-s + (1.78 + 1.29i)15-s + (0.309 + 0.951i)16-s + (−0.404 + 1.24i)20-s − 1.91·23-s + (−0.578 + 0.420i)25-s + (0.431 + 1.32i)27-s + (−0.592 + 1.82i)31-s + (−1.48 + 1.07i)36-s + (−0.672 − 0.488i)37-s − 2.39·45-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: $\chi_{1331} (596, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1331,\ (\ :0),\ -0.809 - 0.587i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08468834112\)
\(L(\frac12)\) \(\approx\) \(0.08468834112\)
\(L(1)\) 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 \( 1 + (1.36 - 0.988i)T + (0.309 - 0.951i)T^{2} \)
5 \( 1 + (0.404 + 1.24i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + 1.91T + T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.592 - 1.82i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.672 + 0.488i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.672 - 0.488i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.0879 - 0.270i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.230 - 0.167i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 + 1.30T + T^{2} \)
71 \( 1 + (-0.256 - 0.790i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + 0.284T + T^{2} \)
97 \( 1 + (0.592 - 1.82i)T + (-0.809 - 0.587i)T^{2} \)
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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

−10.25132548616641103829530560870, −9.374942441158280503374500295154, −8.840913048242415624967441679595, −7.923995237524650304546313950565, −6.47897599686832003470152907076, −5.62703530897772423036995458997, −5.10665903087951618533043727291, −4.40050957636114654562732970058, −3.79965436967675944039204753310, −1.31359029467698936436623054699, 0.094203593162691583514429885827, 2.06548028031878964011245206911, 3.43278416229258840300236509988, 4.37920296286079656381582381364, 5.52620930853638338400892866584, 6.23767982440381064270892361462, 7.08499761351543973471650429882, 7.65283907611406537729419571192, 8.361596304660442579447016430722, 9.723360119786147504650262256043

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