Properties

Label 2-570-15.2-c1-0-6
Degree $2$
Conductor $570$
Sign $0.464 - 0.885i$
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 + (−0.274 − 1.71i)3-s − 1.00i·4-s + (2.05 + 0.893i)5-s + (1.40 + 1.01i)6-s + (2.76 + 2.76i)7-s + (0.707 + 0.707i)8-s + (−2.84 + 0.940i)9-s + (−2.08 + 0.818i)10-s + 4.37i·11-s + (−1.71 + 0.274i)12-s + (−0.0858 + 0.0858i)13-s − 3.91·14-s + (0.963 − 3.75i)15-s − 1.00·16-s + (−4.90 + 4.90i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.158 − 0.987i)3-s − 0.500i·4-s + (0.916 + 0.399i)5-s + (0.573 + 0.414i)6-s + (1.04 + 1.04i)7-s + (0.250 + 0.250i)8-s + (−0.949 + 0.313i)9-s + (−0.658 + 0.258i)10-s + 1.32i·11-s + (−0.493 + 0.0793i)12-s + (−0.0238 + 0.0238i)13-s − 1.04·14-s + (0.248 − 0.968i)15-s − 0.250·16-s + (−1.18 + 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.464 - 0.885i$
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: \((2,\ 570,\ (\ :1/2),\ 0.464 - 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05793 + 0.639620i\)
\(L(\frac12)\) \(\approx\) \(1.05793 + 0.639620i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.274 + 1.71i)T \)
5 \( 1 + (-2.05 - 0.893i)T \)
19 \( 1 - iT \)
good7 \( 1 + (-2.76 - 2.76i)T + 7iT^{2} \)
11 \( 1 - 4.37iT - 11T^{2} \)
13 \( 1 + (0.0858 - 0.0858i)T - 13iT^{2} \)
17 \( 1 + (4.90 - 4.90i)T - 17iT^{2} \)
23 \( 1 + (4.04 + 4.04i)T + 23iT^{2} \)
29 \( 1 - 7.73T + 29T^{2} \)
31 \( 1 + 4.80T + 31T^{2} \)
37 \( 1 + (1.15 + 1.15i)T + 37iT^{2} \)
41 \( 1 - 3.99iT - 41T^{2} \)
43 \( 1 + (2.22 - 2.22i)T - 43iT^{2} \)
47 \( 1 + (-9.37 + 9.37i)T - 47iT^{2} \)
53 \( 1 + (1.61 + 1.61i)T + 53iT^{2} \)
59 \( 1 - 13.6T + 59T^{2} \)
61 \( 1 - 13.0T + 61T^{2} \)
67 \( 1 + (9.57 + 9.57i)T + 67iT^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 + (3.04 - 3.04i)T - 73iT^{2} \)
79 \( 1 - 1.24iT - 79T^{2} \)
83 \( 1 + (9.27 + 9.27i)T + 83iT^{2} \)
89 \( 1 - 4.47T + 89T^{2} \)
97 \( 1 + (1.05 + 1.05i)T + 97iT^{2} \)
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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

−10.78416293714511673978814206396, −10.01261617796786835347168581020, −8.774023456496664502237781111063, −8.355508121520360050851526681023, −7.21505915607804996609071383482, −6.44417854704474972133356411812, −5.69175375102091638929931207150, −4.70143007212103969479268513016, −2.23224570528786533834322633129, −1.85326460056093780107634239531, 0.869626704867355035412930934756, 2.54179826417870870111736455119, 3.93161056460595743986462824579, 4.83529971211167423517186584802, 5.76896300426050096987428418756, 7.08854407531942653614837004118, 8.391719697135739618589740440707, 8.902597930104034263480760443752, 9.844661280453393128020319870178, 10.57207085385125412537880924050

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