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Properties

Label 2-2717-2717.2144-c0-0-9
Degree $2$
Conductor $2717$
Sign $-0.956 + 0.290i$
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  + (−0.671 − 0.563i)2-s + (1.35 − 0.492i)3-s + (−0.0401 − 0.227i)4-s + (−1.18 − 0.431i)6-s + (−0.882 − 1.52i)7-s + (−0.539 + 0.934i)8-s + (0.819 − 0.687i)9-s + (0.5 − 0.866i)11-s + (−0.166 − 0.288i)12-s + (0.939 + 0.342i)13-s + (−0.268 + 1.52i)14-s + (0.672 − 0.244i)16-s − 0.937·18-s + (−0.961 + 0.275i)19-s + (−1.94 − 1.63i)21-s + (−0.823 + 0.299i)22-s + ⋯
L(s)  = 1  + (−0.671 − 0.563i)2-s + (1.35 − 0.492i)3-s + (−0.0401 − 0.227i)4-s + (−1.18 − 0.431i)6-s + (−0.882 − 1.52i)7-s + (−0.539 + 0.934i)8-s + (0.819 − 0.687i)9-s + (0.5 − 0.866i)11-s + (−0.166 − 0.288i)12-s + (0.939 + 0.342i)13-s + (−0.268 + 1.52i)14-s + (0.672 − 0.244i)16-s − 0.937·18-s + (−0.961 + 0.275i)19-s + (−1.94 − 1.63i)21-s + (−0.823 + 0.299i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2717\)    =    \(11 \cdot 13 \cdot 19\)
Sign: $-0.956 + 0.290i$
Analytic conductor: \(1.35595\)
Root analytic conductor: \(1.16445\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2717} (2144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2717,\ (\ :0),\ -0.956 + 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.110234741\)
\(L(\frac12)\) \(\approx\) \(1.110234741\)
\(L(1)\) 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 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (0.961 - 0.275i)T \)
good2 \( 1 + (0.671 + 0.563i)T + (0.173 + 0.984i)T^{2} \)
3 \( 1 + (-1.35 + 0.492i)T + (0.766 - 0.642i)T^{2} \)
5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.882 + 1.52i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.130 + 0.737i)T + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-1.59 + 0.580i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (0.0840 + 0.476i)T + (-0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (1.52 - 0.553i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.990 - 1.71i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (-0.173 - 0.984i)T^{2} \)
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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

−8.664215845772001025823998459520, −8.236295278899498819560047095186, −7.37356019334305156744663445372, −6.48751738282722609301382619002, −5.93549555164558724895355871865, −4.20580164097377512128623753905, −3.68505750646153790884245722587, −2.76900797386778000093359706609, −1.76128826984647580236650158411, −0.75432382611091824128435719312, 1.99877812464406169443764345390, 2.92605089884156832470624563887, 3.58953667111549886640115367809, 4.38331035336484629899048747364, 5.86466681089499344986292795530, 6.33773602797744266433405123223, 7.40209501771769190724497879695, 8.047231338221069387065985281700, 8.787516547311054927973536844165, 9.152833503010630852544678718086

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