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Properties

Label 2-11e3-11.7-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  + (−0.592 + 1.82i)3-s + (0.309 + 0.951i)4-s + (−0.672 + 0.488i)5-s + (−2.17 − 1.57i)9-s − 1.91·12-s + (−0.492 − 1.51i)15-s + (−0.809 + 0.587i)16-s + (−0.672 − 0.488i)20-s − 0.284·23-s + (−0.0957 + 0.294i)25-s + (2.61 − 1.89i)27-s + (0.230 + 0.167i)31-s + (0.828 − 2.55i)36-s + (0.519 + 1.60i)37-s + 2.22·45-s + ⋯
L(s)  = 1  + (−0.592 + 1.82i)3-s + (0.309 + 0.951i)4-s + (−0.672 + 0.488i)5-s + (−2.17 − 1.57i)9-s − 1.91·12-s + (−0.492 − 1.51i)15-s + (−0.809 + 0.587i)16-s + (−0.672 − 0.488i)20-s − 0.284·23-s + (−0.0957 + 0.294i)25-s + (2.61 − 1.89i)27-s + (0.230 + 0.167i)31-s + (0.828 − 2.55i)36-s + (0.519 + 1.60i)37-s + 2.22·45-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: $\chi_{1331} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1331,\ (\ :0),\ -0.809 + 0.587i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6100210796\)
\(L(\frac12)\) \(\approx\) \(0.6100210796\)
\(L(1)\) 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 \( 1 + (0.592 - 1.82i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 + (0.672 - 0.488i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 + 0.284T + T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.230 - 0.167i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.519 - 1.60i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.519 + 1.60i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (-1.05 - 0.769i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.404 + 1.24i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 - 0.830T + T^{2} \)
71 \( 1 + (1.36 - 0.988i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 + 1.30T + T^{2} \)
97 \( 1 + (-0.230 - 0.167i)T + (0.309 + 0.951i)T^{2} \)
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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

−10.37813010444126940178092895116, −9.629763100495720066982513319651, −8.740377199804456917656831482153, −8.057533393109080520362415233870, −6.99381478701265740862416936692, −6.11451309088917505219384283135, −5.04350759962848583160884913060, −4.18530239687396054063397526333, −3.56183544593405503928765547949, −2.80048102981688872793813200093, 0.54819995392615220150138307518, 1.64217527698553263815979520721, 2.63321535825133369513306475650, 4.41759537398838226663349337892, 5.50745830476552497254320726995, 6.04731300919445060850483583110, 6.88501932575080990368804429338, 7.56037310558843433890732030583, 8.261660616225223796001430142611, 9.178596873011065582554129755236

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