↑

Properties

Label 2-770-385.164-c1-0-33
Degree $2$
Conductor $770$
Sign $0.991 - 0.128i$
Analytic cond. $6.14848$
Root an. cond. $2.47961$
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  + (0.5 + 0.866i)2-s + (−0.987 + 1.71i)3-s + (−0.499 + 0.866i)4-s + (1.00 − 1.99i)5-s − 1.97·6-s + (−1.58 − 2.12i)7-s − 0.999·8-s + (−0.451 − 0.782i)9-s + (2.23 − 0.127i)10-s + (2.12 − 2.54i)11-s + (−0.987 − 1.71i)12-s − 4.36i·13-s + (1.04 − 2.42i)14-s + (2.42 + 3.69i)15-s + (−0.5 − 0.866i)16-s + (0.154 + 0.0893i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.570 + 0.987i)3-s + (−0.249 + 0.433i)4-s + (0.449 − 0.893i)5-s − 0.806·6-s + (−0.597 − 0.802i)7-s − 0.353·8-s + (−0.150 − 0.260i)9-s + (0.705 − 0.0404i)10-s + (0.640 − 0.768i)11-s + (−0.285 − 0.493i)12-s − 1.21i·13-s + (0.280 − 0.649i)14-s + (0.625 + 0.953i)15-s + (−0.125 − 0.216i)16-s + (0.0375 + 0.0216i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.991 - 0.128i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.991 - 0.128i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43472 + 0.0926475i\)
\(L(\frac12)\) \(\approx\) \(1.43472 + 0.0926475i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-1.00 + 1.99i)T \)
7 \( 1 + (1.58 + 2.12i)T \)
11 \( 1 + (-2.12 + 2.54i)T \)
good3 \( 1 + (0.987 - 1.71i)T + (-1.5 - 2.59i)T^{2} \)
13 \( 1 + 4.36iT - 13T^{2} \)
17 \( 1 + (-0.154 - 0.0893i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.57 - 2.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.01 + 3.47i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.86iT - 29T^{2} \)
31 \( 1 + (-6.35 - 3.67i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.87 - 1.08i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.35T + 41T^{2} \)
43 \( 1 - 1.81T + 43T^{2} \)
47 \( 1 + (-1.99 - 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.2 - 5.91i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.30 + 3.63i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.80 + 13.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.07 - 2.92i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.02T + 71T^{2} \)
73 \( 1 + (-4.65 - 2.68i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.13 - 2.96i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.33iT - 83T^{2} \)
89 \( 1 + (5.10 - 2.94i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.52T + 97T^{2} \)
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

−10.20043698658806604043245651448, −9.623376813003566318280706754184, −8.634423580781102180024432161799, −7.80808041208005327338128702151, −6.52803833230181821796061746091, −5.78224905689830153481505467979, −4.98192244588125490621570946144, −4.18402565769824435463566292202, −3.20223694920954397290129545515, −0.76285801272651325174670072857, 1.47058367506093818211200271155, 2.45690355850669229858487210227, 3.58960213509863836007629583512, 5.02954178230378969361774022016, 6.02344597513304432160137172836, 6.84739048088001153971432878715, 7.12278153578840730113262754487, 8.974731623719396384646817678930, 9.473960274538547769682455715662, 10.41143068018300367787827166492

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