Properties

Label 2-570-57.8-c1-0-15
Degree $2$
Conductor $570$
Sign $0.752 - 0.658i$
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  + (−0.5 + 0.866i)2-s + (1.72 + 0.162i)3-s + (−0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + (−1.00 + 1.41i)6-s + 3.36·7-s + 0.999·8-s + (2.94 + 0.561i)9-s + (−0.866 + 0.499i)10-s − 0.795i·11-s + (−0.721 − 1.57i)12-s + (1.59 − 0.922i)13-s + (−1.68 + 2.91i)14-s + (1.41 + 1.00i)15-s + (−0.5 + 0.866i)16-s + (−6.17 − 3.56i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.995 + 0.0939i)3-s + (−0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + (−0.409 + 0.576i)6-s + 1.27·7-s + 0.353·8-s + (0.982 + 0.187i)9-s + (−0.273 + 0.158i)10-s − 0.239i·11-s + (−0.208 − 0.454i)12-s + (0.443 − 0.255i)13-s + (−0.449 + 0.779i)14-s + (0.364 + 0.259i)15-s + (−0.125 + 0.216i)16-s + (−1.49 − 0.865i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.752 - 0.658i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.752 - 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87161 + 0.703617i\)
\(L(\frac12)\) \(\approx\) \(1.87161 + 0.703617i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-1.72 - 0.162i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.831 + 4.27i)T \)
good7 \( 1 - 3.36T + 7T^{2} \)
11 \( 1 + 0.795iT - 11T^{2} \)
13 \( 1 + (-1.59 + 0.922i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (6.17 + 3.56i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.72 - 0.997i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.95 - 3.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.28iT - 31T^{2} \)
37 \( 1 + 3.63iT - 37T^{2} \)
41 \( 1 + (2.08 - 3.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.75 - 3.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.13 - 1.80i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.21 - 5.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.40 + 7.62i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.34 + 5.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.16 - 1.82i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.800 + 1.38i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.05 - 12.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.3 + 5.97i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.9iT - 83T^{2} \)
89 \( 1 + (1.83 + 3.18i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (15.9 + 9.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.77640727770169291530274428981, −9.717927924702171499360374475326, −8.768583039123595938515145970375, −8.423698613179889148682671613953, −7.32755938385154938034405541086, −6.62349969169094531142969029117, −5.16666871274970020871814811313, −4.41119908739430887010625138167, −2.83074386570320039646389391408, −1.57243614971032717757109471937, 1.62251441672412676404855265629, 2.26606247165588683611625820309, 3.92054806319785561275818650516, 4.58613316196062775156976789821, 6.14325067033159692636221543608, 7.41288488589663101463087949991, 8.430781705911927868198038382838, 8.615455913254154124141377246366, 9.788023820595025926768552997333, 10.49868714340263291900600089510

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