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Properties

Label 2-2717-2717.142-c0-0-2
Degree $2$
Conductor $2717$
Sign $-0.226 + 0.973i$
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  + (−0.333 − 1.89i)2-s + (0.856 − 0.718i)3-s + (−2.53 + 0.922i)4-s + (−1.64 − 1.38i)6-s + (−0.615 + 1.06i)7-s + (1.63 + 2.82i)8-s + (0.0435 − 0.246i)9-s + (0.5 + 0.866i)11-s + (−1.50 + 2.61i)12-s + (−0.766 − 0.642i)13-s + (2.22 + 0.809i)14-s + (2.73 − 2.29i)16-s − 0.482·18-s + (0.997 + 0.0697i)19-s + (0.239 + 1.35i)21-s + (1.47 − 1.23i)22-s + ⋯
L(s)  = 1  + (−0.333 − 1.89i)2-s + (0.856 − 0.718i)3-s + (−2.53 + 0.922i)4-s + (−1.64 − 1.38i)6-s + (−0.615 + 1.06i)7-s + (1.63 + 2.82i)8-s + (0.0435 − 0.246i)9-s + (0.5 + 0.866i)11-s + (−1.50 + 2.61i)12-s + (−0.766 − 0.642i)13-s + (2.22 + 0.809i)14-s + (2.73 − 2.29i)16-s − 0.482·18-s + (0.997 + 0.0697i)19-s + (0.239 + 1.35i)21-s + (1.47 − 1.23i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2717\)    =    \(11 \cdot 13 \cdot 19\)
Sign: $-0.226 + 0.973i$
Analytic conductor: \(1.35595\)
Root analytic conductor: \(1.16445\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2717} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2717,\ (\ :0),\ -0.226 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.088356420\)
\(L(\frac12)\) \(\approx\) \(1.088356420\)
\(L(1)\) 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 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (-0.997 - 0.0697i)T \)
good2 \( 1 + (0.333 + 1.89i)T + (-0.939 + 0.342i)T^{2} \)
3 \( 1 + (-0.856 + 0.718i)T + (0.173 - 0.984i)T^{2} \)
5 \( 1 + (-0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.615 - 1.06i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.939 - 0.342i)T^{2} \)
23 \( 1 + (-1.65 + 0.603i)T + (0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1.51 - 1.27i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (0.823 - 0.299i)T + (0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-1.23 + 1.04i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (-0.0348 + 0.0604i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.939 - 0.342i)T^{2} \)
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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

−9.103000909174831068314927571547, −8.389306482035165923986897605858, −7.62057297739178075549208002389, −6.78815564960111729248230275492, −5.24568690683112945151616503345, −4.72632568212766556404826068954, −3.17828566973889145059598531130, −3.02673562348262912331533093359, −2.14880461964949213278924289132, −1.25077857480098816280526102258, 0.857499514490901877232091431998, 3.21691738485329589679908761797, 3.82534563259582983648653605540, 4.70110892249301972091098089318, 5.39458759639959734526406620819, 6.58738286433477105449454100859, 6.90105058641183695789268319599, 7.63306796783351900681088275156, 8.544728840504647215965638317238, 9.019833810175458483517358559798

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