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Properties

Label 2-2717-2717.1429-c0-0-2
Degree $2$
Conductor $2717$
Sign $-0.312 + 0.949i$
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  + (−1.87 − 0.682i)2-s + (−0.0840 − 0.476i)3-s + (2.28 + 1.91i)4-s + (−0.167 + 0.950i)6-s + (0.848 − 1.46i)7-s + (−1.97 − 3.42i)8-s + (0.719 − 0.261i)9-s + (0.5 + 0.866i)11-s + (0.721 − 1.24i)12-s + (−0.173 + 0.984i)13-s + (−2.59 + 2.17i)14-s + (0.851 + 4.82i)16-s − 1.52·18-s + (0.719 − 0.694i)19-s + (−0.771 − 0.280i)21-s + (−0.346 − 1.96i)22-s + ⋯
L(s)  = 1  + (−1.87 − 0.682i)2-s + (−0.0840 − 0.476i)3-s + (2.28 + 1.91i)4-s + (−0.167 + 0.950i)6-s + (0.848 − 1.46i)7-s + (−1.97 − 3.42i)8-s + (0.719 − 0.261i)9-s + (0.5 + 0.866i)11-s + (0.721 − 1.24i)12-s + (−0.173 + 0.984i)13-s + (−2.59 + 2.17i)14-s + (0.851 + 4.82i)16-s − 1.52·18-s + (0.719 − 0.694i)19-s + (−0.771 − 0.280i)21-s + (−0.346 − 1.96i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2717\)    =    \(11 \cdot 13 \cdot 19\)
Sign: $-0.312 + 0.949i$
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: \((2,\ 2717,\ (\ :0),\ -0.312 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6173370983\)
\(L(\frac12)\) \(\approx\) \(0.6173370983\)
\(L(1)\) 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 \( 1 + (-0.719 + 0.694i)T \)
good2 \( 1 + (1.87 + 0.682i)T + (0.766 + 0.642i)T^{2} \)
3 \( 1 + (0.0840 + 0.476i)T + (-0.939 + 0.342i)T^{2} \)
5 \( 1 + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (-0.848 + 1.46i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (0.943 + 0.791i)T + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.0121 + 0.0687i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (-1.47 - 1.23i)T + (0.173 + 0.984i)T^{2} \)
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 \( 1 + (-0.280 - 1.59i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.374 - 0.648i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.816482317921814072441864105489, −8.074302514414984218025237682282, −7.26208451296680981269798434185, −7.09183456863362862272954223706, −6.39883572281414926613117854844, −4.35727221415778055157997046292, −3.97662670000536791222486889627, −2.44287902446317438221797576635, −1.61675823798770280219010353059, −0.866246506359621768826081250832, 1.30760507613651658153104125518, 2.13408356122024047575905306743, 3.33718247664778354944056942002, 5.18868553752006660434555978031, 5.53099012690230163518611277634, 6.23122845552399193509569567074, 7.36996570962543065875826224063, 7.901179316542296920787914519406, 8.516483066296914308343205323684, 9.158535649190805633750899850842

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