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Properties

Label 2-3311-3311.1140-c0-0-1
Degree $2$
Conductor $3311$
Sign $-0.915 + 0.401i$
Analytic cond. $1.65240$
Root an. cond. $1.28545$
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  + (−0.813 − 1.51i)2-s + (−1.07 + 1.62i)4-s + (−0.309 − 0.951i)7-s + (1.62 + 0.145i)8-s + (0.963 − 0.266i)9-s + (−0.473 + 0.880i)11-s + (−1.18 + 1.24i)14-s + (−0.332 − 0.777i)16-s + (−1.18 − 1.24i)18-s + 1.71·22-s + (1.24 − 1.55i)23-s + (−0.134 − 0.990i)25-s + (1.87 + 0.518i)28-s + (−0.258 + 1.91i)29-s + (0.108 − 0.136i)32-s + ⋯
L(s)  = 1  + (−0.813 − 1.51i)2-s + (−1.07 + 1.62i)4-s + (−0.309 − 0.951i)7-s + (1.62 + 0.145i)8-s + (0.963 − 0.266i)9-s + (−0.473 + 0.880i)11-s + (−1.18 + 1.24i)14-s + (−0.332 − 0.777i)16-s + (−1.18 − 1.24i)18-s + 1.71·22-s + (1.24 − 1.55i)23-s + (−0.134 − 0.990i)25-s + (1.87 + 0.518i)28-s + (−0.258 + 1.91i)29-s + (0.108 − 0.136i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $-0.915 + 0.401i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (1140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ -0.915 + 0.401i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7195815585\)
\(L(\frac12)\) \(\approx\) \(0.7195815585\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.473 - 0.880i)T \)
43 \( 1 + (0.0448 + 0.998i)T \)
good2 \( 1 + (0.813 + 1.51i)T + (-0.550 + 0.834i)T^{2} \)
3 \( 1 + (-0.963 + 0.266i)T^{2} \)
5 \( 1 + (0.134 + 0.990i)T^{2} \)
13 \( 1 + (-0.691 - 0.722i)T^{2} \)
17 \( 1 + (-0.691 + 0.722i)T^{2} \)
19 \( 1 + (-0.753 + 0.657i)T^{2} \)
23 \( 1 + (-1.24 + 1.55i)T + (-0.222 - 0.974i)T^{2} \)
29 \( 1 + (0.258 - 1.91i)T + (-0.963 - 0.266i)T^{2} \)
31 \( 1 + (0.550 - 0.834i)T^{2} \)
37 \( 1 + (-0.170 + 0.0554i)T + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.0448 - 0.998i)T^{2} \)
47 \( 1 + (-0.753 + 0.657i)T^{2} \)
53 \( 1 + (-1.13 + 0.990i)T + (0.134 - 0.990i)T^{2} \)
59 \( 1 + (-0.983 + 0.178i)T^{2} \)
61 \( 1 + (0.550 + 0.834i)T^{2} \)
67 \( 1 + (-0.590 - 0.740i)T + (-0.222 + 0.974i)T^{2} \)
71 \( 1 + (-0.312 + 1.13i)T + (-0.858 - 0.512i)T^{2} \)
73 \( 1 + (-0.473 + 0.880i)T^{2} \)
79 \( 1 + (-1.17 + 1.61i)T + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.550 - 0.834i)T^{2} \)
89 \( 1 + (0.623 - 0.781i)T^{2} \)
97 \( 1 + (-0.858 + 0.512i)T^{2} \)
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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

−8.781263559833687727749697617193, −7.949470178099230308208635925257, −7.08841011785141626533430816390, −6.66254500452550399177030231117, −5.01518772150325742305640249726, −4.30835209479065881571743769869, −3.59253528486211104058637590374, −2.66292337562520133583496028647, −1.70918256152360472771578679828, −0.67571343682153771685779833218, 1.18648313601589832595367484220, 2.62461851405153370803145732050, 3.81741357734663313090767754776, 5.09056476041983879686623769036, 5.52240315569140797884291860441, 6.24543162651498641382970565807, 6.99386799059869727324168639488, 7.75592643682187841012227254520, 8.144985437691981214299048552144, 9.177083222419687913957426594968

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