Properties

Label 2-570-15.2-c1-0-14
Degree $2$
Conductor $570$
Sign $-0.113 - 0.993i$
Analytic cond. $4.55147$
Root an. cond. $2.13341$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.577 + 1.63i)3-s − 1.00i·4-s + (2.23 − 0.0135i)5-s + (−1.56 − 0.746i)6-s + (3.38 + 3.38i)7-s + (0.707 + 0.707i)8-s + (−2.33 + 1.88i)9-s + (−1.57 + 1.59i)10-s − 1.22i·11-s + (1.63 − 0.577i)12-s + (3.51 − 3.51i)13-s − 4.78·14-s + (1.31 + 3.64i)15-s − 1.00·16-s + (0.826 − 0.826i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.333 + 0.942i)3-s − 0.500i·4-s + (0.999 − 0.00606i)5-s + (−0.638 − 0.304i)6-s + (1.27 + 1.27i)7-s + (0.250 + 0.250i)8-s + (−0.777 + 0.628i)9-s + (−0.496 + 0.503i)10-s − 0.369i·11-s + (0.471 − 0.166i)12-s + (0.975 − 0.975i)13-s − 1.27·14-s + (0.339 + 0.940i)15-s − 0.250·16-s + (0.200 − 0.200i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.113 - 0.993i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.113 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12839 + 1.26526i\)
\(L(\frac12)\) \(\approx\) \(1.12839 + 1.26526i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.577 - 1.63i)T \)
5 \( 1 + (-2.23 + 0.0135i)T \)
19 \( 1 + iT \)
good7 \( 1 + (-3.38 - 3.38i)T + 7iT^{2} \)
11 \( 1 + 1.22iT - 11T^{2} \)
13 \( 1 + (-3.51 + 3.51i)T - 13iT^{2} \)
17 \( 1 + (-0.826 + 0.826i)T - 17iT^{2} \)
23 \( 1 + (1.04 + 1.04i)T + 23iT^{2} \)
29 \( 1 + 8.14T + 29T^{2} \)
31 \( 1 - 7.41T + 31T^{2} \)
37 \( 1 + (5.83 + 5.83i)T + 37iT^{2} \)
41 \( 1 + 3.53iT - 41T^{2} \)
43 \( 1 + (2.68 - 2.68i)T - 43iT^{2} \)
47 \( 1 + (6.84 - 6.84i)T - 47iT^{2} \)
53 \( 1 + (3.40 + 3.40i)T + 53iT^{2} \)
59 \( 1 + 9.91T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 + (-6.00 - 6.00i)T + 67iT^{2} \)
71 \( 1 + 8.35iT - 71T^{2} \)
73 \( 1 + (3.99 - 3.99i)T - 73iT^{2} \)
79 \( 1 - 6.75iT - 79T^{2} \)
83 \( 1 + (-0.717 - 0.717i)T + 83iT^{2} \)
89 \( 1 + 0.384T + 89T^{2} \)
97 \( 1 + (12.5 + 12.5i)T + 97iT^{2} \)
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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

−10.86226333910404088901496194809, −9.928116156162677416044370839593, −9.051633540673283752172351643785, −8.543530388902812518774266625052, −7.81187381356449460853688039455, −6.05817665444561266599212575712, −5.55628434775201960306391044696, −4.75829729418449954361275559765, −3.02720202039811421703698220135, −1.79638523556042124547928492351, 1.38237209253487188079518830034, 1.84045039813779506590546768091, 3.51403742489743781115176641624, 4.76448565861648576708628867687, 6.21467182508335585111279028828, 7.06961721137720124867455913456, 7.938461159903597647245528614770, 8.667763685134642970680802564538, 9.638947303384769098579610192052, 10.55062135694944555391581438817

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