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Properties

Label 2-570-95.12-c1-0-8
Degree $2$
Conductor $570$
Sign $0.499 + 0.866i$
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  + (−0.965 − 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 + 0.499i)4-s + (−2.23 + 0.0982i)5-s + (0.499 − 0.866i)6-s + (−0.368 − 0.368i)7-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (2.18 + 0.483i)10-s + 0.0597·11-s + (−0.707 + 0.707i)12-s + (−0.741 + 0.198i)13-s + (0.260 + 0.451i)14-s + (0.483 − 2.18i)15-s + (0.500 + 0.866i)16-s + (1.89 − 7.07i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.149 + 0.557i)3-s + (0.433 + 0.249i)4-s + (−0.999 + 0.0439i)5-s + (0.204 − 0.353i)6-s + (−0.139 − 0.139i)7-s + (−0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.690 + 0.152i)10-s + 0.0180·11-s + (−0.204 + 0.204i)12-s + (−0.205 + 0.0550i)13-s + (0.0696 + 0.120i)14-s + (0.124 − 0.563i)15-s + (0.125 + 0.216i)16-s + (0.459 − 1.71i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.499 + 0.866i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.499 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.549245 - 0.317246i\)
\(L(\frac12)\) \(\approx\) \(0.549245 - 0.317246i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (2.23 - 0.0982i)T \)
19 \( 1 + (-2.26 - 3.72i)T \)
good7 \( 1 + (0.368 + 0.368i)T + 7iT^{2} \)
11 \( 1 - 0.0597T + 11T^{2} \)
13 \( 1 + (0.741 - 0.198i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-1.89 + 7.07i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (1.71 + 6.38i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.71 - 4.70i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.60iT - 31T^{2} \)
37 \( 1 + (-6.95 + 6.95i)T - 37iT^{2} \)
41 \( 1 + (-10.3 + 5.95i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.94 - 2.66i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (3.81 - 1.02i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (12.2 - 3.27i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.464 + 0.804i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.593 + 1.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.06 + 11.4i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-7.61 + 4.39i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (13.6 + 3.66i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.71 - 2.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.15 + 8.15i)T - 83iT^{2} \)
89 \( 1 + (1.67 - 2.90i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.51 + 1.74i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.68380595524956225711877485592, −9.606732155215669978555531240628, −9.070484180679869484988817986908, −7.82172773192918633164807198387, −7.38719035634072627774472333988, −6.10574478454738954779176720699, −4.81505659292062406322670728004, −3.78953667830529774315718562194, −2.70340489021676288229300684593, −0.52655981884922020773571985131, 1.23934697957400368748409045363, 2.89851522534208567136303665671, 4.20607746959583567167090953018, 5.63192191643248788430263814687, 6.51329115781710801465874870581, 7.67156177311003517971407748281, 7.922146412013460831765201422554, 9.026370052160797846921109565148, 9.921483663576767628581102170117, 11.06613970861889931685931799474

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