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Properties

Label 2-3311-3311.3310-c0-0-19
Degree $2$
Conductor $3311$
Sign $-0.5 + 0.866i$
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 − 9-s − 11-s + (0.5 − 0.866i)14-s − 16-s + 18-s + 1.73i·19-s + 22-s + 23-s − 25-s + 29-s − 1.73i·31-s − 1.73i·38-s + ⋯
L(s)  = 1  − 2-s + (−0.5 + 0.866i)7-s + 8-s − 9-s − 11-s + (0.5 − 0.866i)14-s − 16-s + 18-s + 1.73i·19-s + 22-s + 23-s − 25-s + 29-s − 1.73i·31-s − 1.73i·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $-0.5 + 0.866i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (3310, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ -0.5 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1193146913\)
\(L(\frac12)\) \(\approx\) \(0.1193146913\)
\(L(1)\) 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 \( 1 + T \)
good2 \( 1 + T + T^{2} \)
3 \( 1 + T^{2} \)
5 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 1.73iT - T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
47 \( 1 + 1.73iT - T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 - 1.73iT - T^{2} \)
61 \( 1 + 1.73iT - T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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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

−8.450361966855849921635811453407, −8.153226481502350323242030210615, −7.40114576199530353302863881532, −6.25166546449569114774525191881, −5.64391875104492880305826513093, −4.90452578923521867940498540895, −3.70543618305378118697274646827, −2.74862020248336242659602560803, −1.82637020642320401159290164248, −0.11450355215158241144927256593, 1.09774670390797591481437883832, 2.61376858094486118421661035279, 3.33328702464501934454540160028, 4.66959389478211386517829951693, 5.06326508530147604712346056837, 6.30979673092127123479491680445, 7.04995725516377102337934571241, 7.70828962419818778877108707674, 8.421049987936147605340650570703, 9.025039414325233654726678122075

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