Properties

Label 2-570-95.49-c1-0-8
Degree $2$
Conductor $570$
Sign $0.246 - 0.969i$
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 + (−1.60 − 1.55i)5-s + (0.499 + 0.866i)6-s + 3.53i·7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.618 − 2.14i)10-s + 3.34·11-s + 0.999i·12-s + (1.48 − 0.854i)13-s + (−1.76 + 3.06i)14-s + (−0.618 − 2.14i)15-s + (−0.5 + 0.866i)16-s + (3.29 + 1.90i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.719 − 0.694i)5-s + (0.204 + 0.353i)6-s + 1.33i·7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.195 − 0.679i)10-s + 1.00·11-s + 0.288i·12-s + (0.410 − 0.237i)13-s + (−0.472 + 0.819i)14-s + (−0.159 − 0.554i)15-s + (−0.125 + 0.216i)16-s + (0.798 + 0.460i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.246 - 0.969i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.246 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76514 + 1.37203i\)
\(L(\frac12)\) \(\approx\) \(1.76514 + 1.37203i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.60 + 1.55i)T \)
19 \( 1 + (-1.30 - 4.15i)T \)
good7 \( 1 - 3.53iT - 7T^{2} \)
11 \( 1 - 3.34T + 11T^{2} \)
13 \( 1 + (-1.48 + 0.854i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.29 - 1.90i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (7.33 - 4.23i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.97 + 6.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.07T + 31T^{2} \)
37 \( 1 + 4.70iT - 37T^{2} \)
41 \( 1 + (-5.96 + 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.39 - 2.53i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.73 - 2.73i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.8 + 6.25i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.93 + 5.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.92 + 11.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.82 - 4.51i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.04 + 7.00i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.11 + 2.95i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.53 - 6.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.68iT - 83T^{2} \)
89 \( 1 + (-4.66 - 8.07i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.06 + 2.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.23169784403058468626418453181, −9.772459540818616684672620251470, −9.038401723529222164786402871202, −8.177358354425803440647229868728, −7.62042133106987167078898349128, −5.99569424298633120452913948898, −5.52556246021839400468680015991, −4.09248992308455009985318158115, −3.55287046100137211936146618332, −1.92629532654439267617418932374, 1.12906673731350389457720736591, 2.85934769513652608281318332075, 3.82733431178586181918262707747, 4.43101357309760504236248953878, 6.16584292392040430363368641742, 7.04136318619879758847038281574, 7.57951902351156428921055847672, 8.780879648087726333103242145534, 9.930219447542802771321743483292, 10.64973950017767220725553525305

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