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Properties

Label 2-570-95.12-c1-0-7
Degree $2$
Conductor $570$
Sign $0.993 - 0.114i$
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 + (1.56 + 1.59i)5-s + (−0.499 + 0.866i)6-s + (0.819 + 0.819i)7-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−1.09 − 1.94i)10-s + 2.57·11-s + (0.707 − 0.707i)12-s + (−0.103 + 0.0277i)13-s + (−0.579 − 1.00i)14-s + (1.94 − 1.09i)15-s + (0.500 + 0.866i)16-s + (−1.57 + 5.87i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.149 − 0.557i)3-s + (0.433 + 0.249i)4-s + (0.700 + 0.713i)5-s + (−0.204 + 0.353i)6-s + (0.309 + 0.309i)7-s + (−0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.347 − 0.615i)10-s + 0.775·11-s + (0.204 − 0.204i)12-s + (−0.0287 + 0.00770i)13-s + (−0.154 − 0.268i)14-s + (0.502 − 0.283i)15-s + (0.125 + 0.216i)16-s + (−0.381 + 1.42i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.993 - 0.114i$
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: \((2,\ 570,\ (\ :1/2),\ 0.993 - 0.114i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33287 + 0.0764856i\)
\(L(\frac12)\) \(\approx\) \(1.33287 + 0.0764856i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.56 - 1.59i)T \)
19 \( 1 + (-2.43 + 3.61i)T \)
good7 \( 1 + (-0.819 - 0.819i)T + 7iT^{2} \)
11 \( 1 - 2.57T + 11T^{2} \)
13 \( 1 + (0.103 - 0.0277i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.57 - 5.87i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-1.25 - 4.66i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (0.920 - 1.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.46iT - 31T^{2} \)
37 \( 1 + (-2.36 + 2.36i)T - 37iT^{2} \)
41 \( 1 + (-3.53 + 2.04i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.26 - 1.67i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.52 - 0.677i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.972 + 0.260i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.34 - 4.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.67 + 9.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.91 + 7.15i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-5.46 + 3.15i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.16 + 1.38i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.79 - 11.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.39 - 1.39i)T - 83iT^{2} \)
89 \( 1 + (5.04 - 8.74i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.8 + 3.97i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.90691906029917773339662887149, −9.687306969152407422013166626024, −9.128581505411868502426903721915, −8.143302347881269647584944088910, −7.17881677050473641896162282655, −6.43414561584219000515791956399, −5.52448177644621127861465219990, −3.75001636578493507316415563822, −2.48593088212973673196484876682, −1.47887081100449041645452396240, 1.09385641448154127553412904032, 2.59489160318730838612013733460, 4.21741080707504244821701689505, 5.16194546549534421886212699868, 6.18099194720480427601040389173, 7.25485723057779799753374092988, 8.310209331114367544032332109439, 9.116734818384458142449407780610, 9.655151859191097012261080152033, 10.46353117570893419222878160055

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