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Properties

Label 2-2717-2717.1429-c0-0-4
Degree $2$
Conductor $2717$
Sign $0.811 - 0.584i$
Analytic cond. $1.35595$
Root an. cond. $1.16445$
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  + (1.59 + 0.580i)2-s + (−0.130 − 0.737i)3-s + (1.43 + 1.20i)4-s + (0.220 − 1.25i)6-s + (−0.241 + 0.419i)7-s + (0.743 + 1.28i)8-s + (0.412 − 0.150i)9-s + (0.5 + 0.866i)11-s + (0.703 − 1.21i)12-s + (−0.173 + 0.984i)13-s + (−0.628 + 0.527i)14-s + (0.112 + 0.635i)16-s + 0.744·18-s + (−0.990 + 0.139i)19-s + (0.340 + 0.123i)21-s + (0.294 + 1.67i)22-s + ⋯
L(s)  = 1  + (1.59 + 0.580i)2-s + (−0.130 − 0.737i)3-s + (1.43 + 1.20i)4-s + (0.220 − 1.25i)6-s + (−0.241 + 0.419i)7-s + (0.743 + 1.28i)8-s + (0.412 − 0.150i)9-s + (0.5 + 0.866i)11-s + (0.703 − 1.21i)12-s + (−0.173 + 0.984i)13-s + (−0.628 + 0.527i)14-s + (0.112 + 0.635i)16-s + 0.744·18-s + (−0.990 + 0.139i)19-s + (0.340 + 0.123i)21-s + (0.294 + 1.67i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2717\)    =    \(11 \cdot 13 \cdot 19\)
Sign: $0.811 - 0.584i$
Analytic conductor: \(1.35595\)
Root analytic conductor: \(1.16445\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2717} (1429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2717,\ (\ :0),\ 0.811 - 0.584i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.949531115\)
\(L(\frac12)\) \(\approx\) \(2.949531115\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (0.990 - 0.139i)T \)
good2 \( 1 + (-1.59 - 0.580i)T + (0.766 + 0.642i)T^{2} \)
3 \( 1 + (0.130 + 0.737i)T + (-0.939 + 0.342i)T^{2} \)
5 \( 1 + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.241 - 0.419i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.856 - 0.718i)T + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.333 + 1.89i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (0.943 + 0.791i)T + (0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (0.107 + 0.608i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.997 - 1.72i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.939 + 0.342i)T^{2} \)
97 \( 1 + (-0.766 - 0.642i)T^{2} \)
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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

−9.015264999938226617937443078894, −7.934468544922990492674892672972, −7.03090112816554610037496187387, −6.77945517820325524003980761874, −6.12779321339511502195038691145, −5.21497382401704531634131670540, −4.35729584954150082741723123013, −3.84407886372942230040369583415, −2.56790361375951040212264020786, −1.74463996976702558433706954706, 1.37367954432414737424597964922, 2.80188806670403154914115405048, 3.41460510533341057161225696052, 4.21120377712706302969977888549, 4.82142779517673540643808227862, 5.55326740058384434515273035070, 6.35487408531034979334260300346, 7.06475335200958366692826510940, 8.192042621748345247764649632940, 9.128389549944718722830401925677

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