↑

Properties

Label 2-570-95.8-c1-0-6
Degree $2$
Conductor $570$
Sign $0.862 - 0.505i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)2-s + (−0.258 − 0.965i)3-s + (0.866 − 0.499i)4-s + (0.0149 + 2.23i)5-s + (0.499 + 0.866i)6-s + (1.36 − 1.36i)7-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (−0.593 − 2.15i)10-s − 2.04·11-s + (−0.707 − 0.707i)12-s + (4.35 + 1.16i)13-s + (−0.962 + 1.66i)14-s + (2.15 − 0.593i)15-s + (0.500 − 0.866i)16-s + (0.419 + 1.56i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.149 − 0.557i)3-s + (0.433 − 0.249i)4-s + (0.00669 + 0.999i)5-s + (0.204 + 0.353i)6-s + (0.514 − 0.514i)7-s + (−0.249 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.187 − 0.681i)10-s − 0.617·11-s + (−0.204 − 0.204i)12-s + (1.20 + 0.323i)13-s + (−0.257 + 0.445i)14-s + (0.556 − 0.153i)15-s + (0.125 − 0.216i)16-s + (0.101 + 0.379i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.862 - 0.505i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.862 - 0.505i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02107 + 0.276909i\)
\(L(\frac12)\) \(\approx\) \(1.02107 + 0.276909i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0149 - 2.23i)T \)
19 \( 1 + (-1.91 - 3.91i)T \)
good7 \( 1 + (-1.36 + 1.36i)T - 7iT^{2} \)
11 \( 1 + 2.04T + 11T^{2} \)
13 \( 1 + (-4.35 - 1.16i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.419 - 1.56i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (-0.422 + 1.57i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.53 - 4.39i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.82iT - 31T^{2} \)
37 \( 1 + (-5.98 - 5.98i)T + 37iT^{2} \)
41 \( 1 + (1.54 + 0.894i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.44 + 1.46i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-11.1 - 2.98i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-13.4 - 3.60i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.15 - 5.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.71 + 2.97i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.29 + 8.57i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.15 + 1.82i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.18 - 0.853i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.26 - 2.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.18 + 8.18i)T + 83iT^{2} \)
89 \( 1 + (-3.55 - 6.14i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.767 + 0.205i)T + (84.0 - 48.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.73568680129652003189530887565, −10.18493430304166457797772526014, −8.913484027587007168123446768301, −7.922306487505678239454337550122, −7.43108530261197170660889643094, −6.39698254403187152290276081690, −5.70914091762967883683342521556, −4.03515955165793935154797234288, −2.67086830369017770381346952115, −1.29194022420247129684646408108, 0.921364794956918973197912437410, 2.55897418007084736347798265695, 4.00548281045842369883280434281, 5.17020388138407517369003218791, 5.85769184552164897558301157883, 7.35110865595423036393982401863, 8.395560131987705600490993243034, 8.839058352912072782292036686616, 9.689781755910501783535415839428, 10.64183880570345158093480252447

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