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Properties

Label 2-570-95.64-c1-0-16
Degree $2$
Conductor $570$
Sign $-0.150 + 0.988i$
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  + (0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (0.919 − 2.03i)5-s + (−0.499 + 0.866i)6-s − 1.07i·7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.223 − 2.22i)10-s + 0.410·11-s + 0.999i·12-s + (−3.30 − 1.91i)13-s + (−0.537 − 0.931i)14-s + (0.223 + 2.22i)15-s + (−0.5 − 0.866i)16-s + (−1.08 + 0.627i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (0.410 − 0.911i)5-s + (−0.204 + 0.353i)6-s − 0.406i·7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.0706 − 0.703i)10-s + 0.123·11-s + 0.288i·12-s + (−0.917 − 0.529i)13-s + (−0.143 − 0.248i)14-s + (0.0576 + 0.574i)15-s + (−0.125 − 0.216i)16-s + (−0.263 + 0.152i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.150 + 0.988i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.150 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11101 - 1.29247i\)
\(L(\frac12)\) \(\approx\) \(1.11101 - 1.29247i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-0.919 + 2.03i)T \)
19 \( 1 + (-3.85 + 2.03i)T \)
good7 \( 1 + 1.07iT - 7T^{2} \)
11 \( 1 - 0.410T + 11T^{2} \)
13 \( 1 + (3.30 + 1.91i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.08 - 0.627i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.23 + 1.86i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.18 + 2.05i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.75T + 31T^{2} \)
37 \( 1 - 2.08iT - 37T^{2} \)
41 \( 1 + (2.80 + 4.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.92 + 1.10i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.58 - 2.07i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.32 + 2.49i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.25 - 2.17i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.37 - 5.84i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.07 - 4.66i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.79 - 8.30i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.02 - 3.47i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.47 - 7.74i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.12iT - 83T^{2} \)
89 \( 1 + (2.72 - 4.72i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.5 + 6.08i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.36102234741441114166508684945, −9.948245646147675402161369998133, −8.949026944040508768661922849836, −7.77695842234557450597013028648, −6.62215934290873181921811442505, −5.60566351262887146389200715622, −4.86014430811842035135277006318, −4.03806072998021917654700816917, −2.51198839456964371157507443787, −0.863000752122615408681113861171, 2.04577928983534869703768239162, 3.18455745269861130847299066400, 4.58952769051961388690452670356, 5.60493783291235409567012491437, 6.38970141571583448623279681957, 7.15005512534631017827354151327, 7.981885316650843249961087547734, 9.379765162231682209747529605605, 10.15258707227683201831710513690, 11.19845704555758893240876489500

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