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Properties

Label 2-3311-3311.2617-c0-0-0
Degree $2$
Conductor $3311$
Sign $0.0861 - 0.996i$
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  − 2-s + (−0.5 − 0.866i)7-s + 8-s + (0.5 + 0.866i)9-s − 11-s + (0.5 + 0.866i)14-s − 16-s + (−0.5 − 0.866i)18-s + 22-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s + (−0.5 − 0.866i)29-s + (0.5 + 0.866i)43-s + (0.5 − 0.866i)46-s + (−0.499 + 0.866i)49-s + (−0.5 − 0.866i)50-s + ⋯
L(s)  = 1  − 2-s + (−0.5 − 0.866i)7-s + 8-s + (0.5 + 0.866i)9-s − 11-s + (0.5 + 0.866i)14-s − 16-s + (−0.5 − 0.866i)18-s + 22-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s + (−0.5 − 0.866i)29-s + (0.5 + 0.866i)43-s + (0.5 − 0.866i)46-s + (−0.499 + 0.866i)49-s + (−0.5 − 0.866i)50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0861 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0861 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $0.0861 - 0.996i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (2617, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ 0.0861 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4149270310\)
\(L(\frac12)\) \(\approx\) \(0.4149270310\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + T \)
43 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + T + T^{2} \)
3 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.112860950565938731514113436508, −8.064566212915065591467689186197, −7.63735122057590743814225764132, −7.22072018731187464289128010980, −6.09271201701078416946244237758, −5.07308330752722874204787660299, −4.42699818808930828064291025242, −3.47580879859884773262890132369, −2.24229554579504582589387798399, −1.15748325579433729887650177531, 0.39743745266832884022661834852, 1.85714824858982537581885111984, 2.85176793702396942544236257487, 3.91818398430117083753993103722, 4.85215557259427249711659480731, 5.67052592256757500466721999823, 6.62199255909859965034407923991, 7.20667921059858492085810800260, 8.252376740547545125563627767433, 8.578974908846151423175098503947

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