↑

Properties

Label 2-560-560.157-c1-0-47
Degree $2$
Conductor $560$
Sign $0.896 - 0.442i$
Analytic cond. $4.47162$
Root an. cond. $2.11462$
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  + (−1.29 − 0.569i)2-s + (2.95 + 1.70i)3-s + (1.35 + 1.47i)4-s + (2.23 − 0.0580i)5-s + (−2.85 − 3.89i)6-s + (−1.25 − 2.32i)7-s + (−0.909 − 2.67i)8-s + (4.31 + 7.47i)9-s + (−2.92 − 1.19i)10-s + (0.547 + 2.04i)11-s + (1.47 + 6.66i)12-s − 0.966i·13-s + (0.296 + 3.72i)14-s + (6.70 + 3.64i)15-s + (−0.347 + 3.98i)16-s + (−0.932 − 3.47i)17-s + ⋯
L(s)  = 1  + (−0.915 − 0.402i)2-s + (1.70 + 0.984i)3-s + (0.675 + 0.737i)4-s + (0.999 − 0.0259i)5-s + (−1.16 − 1.58i)6-s + (−0.473 − 0.880i)7-s + (−0.321 − 0.946i)8-s + (1.43 + 2.49i)9-s + (−0.925 − 0.378i)10-s + (0.164 + 0.615i)11-s + (0.426 + 1.92i)12-s − 0.268i·13-s + (0.0792 + 0.996i)14-s + (1.73 + 0.940i)15-s + (−0.0868 + 0.996i)16-s + (−0.226 − 0.843i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.896 - 0.442i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.896 - 0.442i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87528 + 0.437683i\)
\(L(\frac12)\) \(\approx\) \(1.87528 + 0.437683i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 + 0.569i)T \)
5 \( 1 + (-2.23 + 0.0580i)T \)
7 \( 1 + (1.25 + 2.32i)T \)
good3 \( 1 + (-2.95 - 1.70i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-0.547 - 2.04i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + 0.966iT - 13T^{2} \)
17 \( 1 + (0.932 + 3.47i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.0327 - 0.122i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.676 + 2.52i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.42 - 1.42i)T - 29iT^{2} \)
31 \( 1 + (7.08 + 4.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.64 - 4.58i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.15T + 41T^{2} \)
43 \( 1 - 0.958iT - 43T^{2} \)
47 \( 1 + (4.47 + 1.19i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.10 + 1.21i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.86 + 6.94i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-11.3 - 3.02i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-4.87 - 2.81i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.4iT - 71T^{2} \)
73 \( 1 + (1.00 + 3.76i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (11.6 - 6.70i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.64iT - 83T^{2} \)
89 \( 1 + (3.01 - 1.74i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.2 + 10.2i)T - 97iT^{2} \)
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

−10.29616145175424228086004223699, −9.806877407920333797637997856229, −9.328118717524149504161136012756, −8.503772764318448482386597766850, −7.54012239352871455860058595831, −6.76661209461178485809887865473, −4.86725818802840049483122057543, −3.72614778937616256455964676368, −2.82276155704098880677169649451, −1.80961311172126753241522915860, 1.53814547656455466281930803457, 2.33076239389046019724182541242, 3.38274118974699010157069397187, 5.67633711200985141228443150696, 6.47920245859038950681140338105, 7.19998541207898014981980817745, 8.337189276371466244951146728774, 8.856942924954796343200757399349, 9.394055278436681818009056836001, 10.18096747266062473112580404149

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