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Properties

Label 2-11e3-11.7-c0-0-3
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  + (−0.404 + 1.24i)3-s + (0.309 + 0.951i)4-s + (1.55 − 1.12i)5-s + (−0.578 − 0.420i)9-s − 1.30·12-s + (0.776 + 2.39i)15-s + (−0.809 + 0.587i)16-s + (1.55 + 1.12i)20-s + 0.830·23-s + (0.828 − 2.55i)25-s + (−0.301 + 0.219i)27-s + (−0.672 − 0.488i)31-s + (0.221 − 0.680i)36-s + (−0.0879 − 0.270i)37-s − 1.37·45-s + ⋯
L(s)  = 1  + (−0.404 + 1.24i)3-s + (0.309 + 0.951i)4-s + (1.55 − 1.12i)5-s + (−0.578 − 0.420i)9-s − 1.30·12-s + (0.776 + 2.39i)15-s + (−0.809 + 0.587i)16-s + (1.55 + 1.12i)20-s + 0.830·23-s + (0.828 − 2.55i)25-s + (−0.301 + 0.219i)27-s + (−0.672 − 0.488i)31-s + (0.221 − 0.680i)36-s + (−0.0879 − 0.270i)37-s − 1.37·45-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: $\chi_{1331} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1331,\ (\ :0),\ 0.309 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.306647134\)
\(L(\frac12)\) \(\approx\) \(1.306647134\)
\(L(1)\) 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 \( 1 + (0.404 - 1.24i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 + (-1.55 + 1.12i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 - 0.830T + T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.672 + 0.488i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.0879 + 0.270i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.0879 - 0.270i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (1.36 + 0.988i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.519 - 1.60i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + 1.91T + T^{2} \)
71 \( 1 + (-0.230 + 0.167i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 - 1.68T + T^{2} \)
97 \( 1 + (0.672 + 0.488i)T + (0.309 + 0.951i)T^{2} \)
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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

−9.871152857461554940304897699978, −9.193942851516843697673070914860, −8.754321341184062969470039802591, −7.65844394901027765001086454358, −6.46679939830289156352716573231, −5.63626600446155534750820470182, −4.88256199007158035484353472364, −4.19995752609921633399147646597, −2.98795578337115650828954193881, −1.75390355149563486657709909846, 1.40013686751848431108800247719, 2.07114346706589476641891385131, 3.06475965101882535152434078494, 5.07984221044910510872484868268, 5.77211288020463843571276440607, 6.53542639879672823830409593265, 6.77950548323629368803544670704, 7.66397565380443105315526872796, 9.120361364563608130185012786172, 9.740900821296738975950631601473

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