Properties

Label 2-570-15.8-c1-0-16
Degree $2$
Conductor $570$
Sign $-0.624 - 0.781i$
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 + (0.830 + 1.52i)3-s + 1.00i·4-s + (2.23 + 0.129i)5-s + (−0.487 + 1.66i)6-s + (−2.47 + 2.47i)7-s + (−0.707 + 0.707i)8-s + (−1.62 + 2.52i)9-s + (1.48 + 1.66i)10-s − 0.319i·11-s + (−1.52 + 0.830i)12-s + (−3.95 − 3.95i)13-s − 3.49·14-s + (1.65 + 3.50i)15-s − 1.00·16-s + (4.30 + 4.30i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.479 + 0.877i)3-s + 0.500i·4-s + (0.998 + 0.0577i)5-s + (−0.199 + 0.678i)6-s + (−0.935 + 0.935i)7-s + (−0.250 + 0.250i)8-s + (−0.540 + 0.841i)9-s + (0.470 + 0.528i)10-s − 0.0963i·11-s + (−0.438 + 0.239i)12-s + (−1.09 − 1.09i)13-s − 0.935·14-s + (0.427 + 0.903i)15-s − 0.250·16-s + (1.04 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.624 - 0.781i$
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: \((2,\ 570,\ (\ :1/2),\ -0.624 - 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.960440 + 1.99663i\)
\(L(\frac12)\) \(\approx\) \(0.960440 + 1.99663i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.830 - 1.52i)T \)
5 \( 1 + (-2.23 - 0.129i)T \)
19 \( 1 + iT \)
good7 \( 1 + (2.47 - 2.47i)T - 7iT^{2} \)
11 \( 1 + 0.319iT - 11T^{2} \)
13 \( 1 + (3.95 + 3.95i)T + 13iT^{2} \)
17 \( 1 + (-4.30 - 4.30i)T + 17iT^{2} \)
23 \( 1 + (-2.89 + 2.89i)T - 23iT^{2} \)
29 \( 1 - 8.60T + 29T^{2} \)
31 \( 1 + 2.73T + 31T^{2} \)
37 \( 1 + (-1.42 + 1.42i)T - 37iT^{2} \)
41 \( 1 + 7.07iT - 41T^{2} \)
43 \( 1 + (-6.45 - 6.45i)T + 43iT^{2} \)
47 \( 1 + (-4.12 - 4.12i)T + 47iT^{2} \)
53 \( 1 + (1.15 - 1.15i)T - 53iT^{2} \)
59 \( 1 + 7.51T + 59T^{2} \)
61 \( 1 - 8.09T + 61T^{2} \)
67 \( 1 + (11.1 - 11.1i)T - 67iT^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 + (9.22 + 9.22i)T + 73iT^{2} \)
79 \( 1 - 3.00iT - 79T^{2} \)
83 \( 1 + (2.31 - 2.31i)T - 83iT^{2} \)
89 \( 1 - 4.95T + 89T^{2} \)
97 \( 1 + (2.21 - 2.21i)T - 97iT^{2} \)
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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

−10.69242521181350086886426990932, −10.09372362600715795696218013473, −9.289352958125138087124082836138, −8.570146863827419312359383976574, −7.49071084699070191393376992185, −6.12031366733232210011464944448, −5.61212790734844586706733378105, −4.66581585587042042342008068404, −3.14217673276019446225749687565, −2.59650850232582216832570530297, 1.07232395508182878054290311245, 2.42785326076087705220191903074, 3.32451606678098557137736093251, 4.72588621986845550003016279070, 5.91624086159108162318278624100, 6.87965760716323799402025087340, 7.36769957285379066080094411700, 8.954447695192883974224953443400, 9.711536189448212389273075262183, 10.18077060647308623476132834042

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