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Properties

Label 2-2717-2717.142-c0-0-4
Degree $2$
Conductor $2717$
Sign $-0.389 - 0.921i$
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  + (0.213 + 1.21i)2-s + (0.0534 − 0.0448i)3-s + (−0.485 + 0.176i)4-s + (0.0658 + 0.0552i)6-s + (0.559 − 0.968i)7-s + (0.297 + 0.515i)8-s + (−0.172 + 0.980i)9-s + (0.5 + 0.866i)11-s + (−0.0180 + 0.0312i)12-s + (−0.766 − 0.642i)13-s + (1.29 + 0.470i)14-s + (−0.957 + 0.803i)16-s − 1.22·18-s + (−0.848 + 0.529i)19-s + (−0.0135 − 0.0768i)21-s + (−0.943 + 0.791i)22-s + ⋯
L(s)  = 1  + (0.213 + 1.21i)2-s + (0.0534 − 0.0448i)3-s + (−0.485 + 0.176i)4-s + (0.0658 + 0.0552i)6-s + (0.559 − 0.968i)7-s + (0.297 + 0.515i)8-s + (−0.172 + 0.980i)9-s + (0.5 + 0.866i)11-s + (−0.0180 + 0.0312i)12-s + (−0.766 − 0.642i)13-s + (1.29 + 0.470i)14-s + (−0.957 + 0.803i)16-s − 1.22·18-s + (−0.848 + 0.529i)19-s + (−0.0135 − 0.0768i)21-s + (−0.943 + 0.791i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2717\)    =    \(11 \cdot 13 \cdot 19\)
Sign: $-0.389 - 0.921i$
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: \((2,\ 2717,\ (\ :0),\ -0.389 - 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.561731725\)
\(L(\frac12)\) \(\approx\) \(1.561731725\)
\(L(1)\) 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 \( 1 + (0.848 - 0.529i)T \)
good2 \( 1 + (-0.213 - 1.21i)T + (-0.939 + 0.342i)T^{2} \)
3 \( 1 + (-0.0534 + 0.0448i)T + (0.173 - 0.984i)T^{2} \)
5 \( 1 + (-0.766 - 0.642i)T^{2} \)
7 \( 1 + (-0.559 + 0.968i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.939 - 0.342i)T^{2} \)
23 \( 1 + (-1.35 + 0.492i)T + (0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.671 - 0.563i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (-1.65 + 0.603i)T + (0.766 - 0.642i)T^{2} \)
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 \( 1 + (0.473 - 0.397i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (-0.961 + 1.66i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.885907664518672956238297183123, −8.241368057227889750251342131001, −7.45327097776409116715529768581, −7.17283332403121547327739772742, −6.38445404692705359579947049923, −5.21524884942887401935913320744, −4.88342373475077160395213906659, −4.08210939676175843358929525833, −2.62682554889289761309815283060, −1.56017488620558073024264239358, 1.01580357996056208996165149512, 2.21986343955362245483333383364, 2.90099583856628127287812527528, 3.78377052400710031976637217875, 4.63524382535391740655275324223, 5.49686266604826012819885977822, 6.56634101570391218429270715532, 7.06425018060434893735882033820, 8.406531323897500376251737023967, 9.027533095572320165728150028160

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