Properties

Label 2-570-15.8-c1-0-30
Degree $2$
Conductor $570$
Sign $0.0910 + 0.995i$
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 + (1.72 − 0.162i)3-s + 1.00i·4-s + (0.520 − 2.17i)5-s + (−1.33 − 1.10i)6-s + (0.496 − 0.496i)7-s + (0.707 − 0.707i)8-s + (2.94 − 0.561i)9-s + (−1.90 + 1.17i)10-s − 1.60i·11-s + (0.162 + 1.72i)12-s + (0.465 + 0.465i)13-s − 0.702·14-s + (0.542 − 3.83i)15-s − 1.00·16-s + (−0.134 − 0.134i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.995 − 0.0939i)3-s + 0.500i·4-s + (0.232 − 0.972i)5-s + (−0.544 − 0.450i)6-s + (0.187 − 0.187i)7-s + (0.250 − 0.250i)8-s + (0.982 − 0.187i)9-s + (−0.602 + 0.370i)10-s − 0.482i·11-s + (0.0469 + 0.497i)12-s + (0.128 + 0.128i)13-s − 0.187·14-s + (0.140 − 0.990i)15-s − 0.250·16-s + (−0.0326 − 0.0326i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0910 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0910 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.0910 + 0.995i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.0910 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24314 - 1.13467i\)
\(L(\frac12)\) \(\approx\) \(1.24314 - 1.13467i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.162i)T \)
5 \( 1 + (-0.520 + 2.17i)T \)
19 \( 1 - iT \)
good7 \( 1 + (-0.496 + 0.496i)T - 7iT^{2} \)
11 \( 1 + 1.60iT - 11T^{2} \)
13 \( 1 + (-0.465 - 0.465i)T + 13iT^{2} \)
17 \( 1 + (0.134 + 0.134i)T + 17iT^{2} \)
23 \( 1 + (-0.307 + 0.307i)T - 23iT^{2} \)
29 \( 1 + 2.93T + 29T^{2} \)
31 \( 1 + 1.63T + 31T^{2} \)
37 \( 1 + (-4.03 + 4.03i)T - 37iT^{2} \)
41 \( 1 + 3.54iT - 41T^{2} \)
43 \( 1 + (-3.15 - 3.15i)T + 43iT^{2} \)
47 \( 1 + (-4.45 - 4.45i)T + 47iT^{2} \)
53 \( 1 + (-2.36 + 2.36i)T - 53iT^{2} \)
59 \( 1 + 8.58T + 59T^{2} \)
61 \( 1 - 6.10T + 61T^{2} \)
67 \( 1 + (6.28 - 6.28i)T - 67iT^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 + (-2.24 - 2.24i)T + 73iT^{2} \)
79 \( 1 - 3.48iT - 79T^{2} \)
83 \( 1 + (1.76 - 1.76i)T - 83iT^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 + (0.0173 - 0.0173i)T - 97iT^{2} \)
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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

−10.36923274276292355287057466894, −9.385841381931350968808390446592, −8.946088341675459683483061882510, −8.080979851279347386084236987051, −7.38372385006418706371207056339, −5.96649533576352473620106192047, −4.57992765309885508006910592024, −3.65486336687663407653357114482, −2.32188880269892482446507525105, −1.12757434178198856835997053594, 1.86383626902215839690378320155, 2.96649366352391126785672845270, 4.24179092102609071606128568867, 5.59290032378768426332438755594, 6.76577497750212203983223287766, 7.41248829412705139420955982666, 8.258806420266666517947601359127, 9.184808921149449129153902824658, 9.908992337634166132894351457177, 10.62023141762984220244535536003

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