↑

Properties

Label 2-820-205.137-c1-0-7
Degree $2$
Conductor $820$
Sign $0.999 - 0.0319i$
Analytic cond. $6.54773$
Root an. cond. $2.55885$
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  + (−1.41 − 0.585i)3-s + (1.93 + 1.11i)5-s + (−0.549 + 1.32i)7-s + (−0.464 − 0.464i)9-s + (5.38 − 2.23i)11-s + (−2.48 − 1.03i)13-s + (−2.08 − 2.71i)15-s + (−3.07 + 1.27i)17-s + (5.44 + 2.25i)19-s + (1.55 − 1.55i)21-s + (0.656 + 0.656i)23-s + (2.49 + 4.33i)25-s + (2.14 + 5.17i)27-s + (1.75 − 0.725i)29-s − 4.97i·31-s + ⋯
L(s)  = 1  + (−0.816 − 0.338i)3-s + (0.865 + 0.500i)5-s + (−0.207 + 0.501i)7-s + (−0.154 − 0.154i)9-s + (1.62 − 0.672i)11-s + (−0.690 − 0.285i)13-s + (−0.537 − 0.701i)15-s + (−0.745 + 0.308i)17-s + (1.24 + 0.517i)19-s + (0.339 − 0.339i)21-s + (0.136 + 0.136i)23-s + (0.499 + 0.866i)25-s + (0.412 + 0.995i)27-s + (0.325 − 0.134i)29-s − 0.893i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.999 - 0.0319i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.999 - 0.0319i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38574 + 0.0221225i\)
\(L(\frac12)\) \(\approx\) \(1.38574 + 0.0221225i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.93 - 1.11i)T \)
41 \( 1 + (6.38 + 0.465i)T \)
good3 \( 1 + (1.41 + 0.585i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (0.549 - 1.32i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-5.38 + 2.23i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (2.48 + 1.03i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + (3.07 - 1.27i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (-5.44 - 2.25i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-0.656 - 0.656i)T + 23iT^{2} \)
29 \( 1 + (-1.75 + 0.725i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 4.97iT - 31T^{2} \)
37 \( 1 + (-5.97 - 5.97i)T + 37iT^{2} \)
43 \( 1 - 8.73T + 43T^{2} \)
47 \( 1 + (-12.5 + 5.18i)T + (33.2 - 33.2i)T^{2} \)
53 \( 1 + (-1.29 + 3.13i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + 10.4iT - 59T^{2} \)
61 \( 1 + (1.06 + 1.06i)T + 61iT^{2} \)
67 \( 1 + (0.682 - 0.282i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-5.84 - 14.1i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + 7.69T + 73T^{2} \)
79 \( 1 + (1.10 + 2.67i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-0.371 + 0.371i)T - 83iT^{2} \)
89 \( 1 + (-15.0 + 6.22i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (-1.42 - 3.44i)T + (-68.5 + 68.5i)T^{2} \)
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

−10.22128882282962100029368071443, −9.392225103858099908004370698139, −8.805466842832392062373070216464, −7.40674109154963072538301194332, −6.45697173683838247542722627766, −6.04185289972864094703493093662, −5.24160646604954044077294505411, −3.73287845403226282673033372753, −2.56225365066644818071985132494, −1.09536528232929407654497208115, 1.02155171885317467982146566174, 2.48819246956365450966523257962, 4.19311048722215928348462661384, 4.84942803738513100455718924836, 5.78303794149404918002090455822, 6.67830198984534284629559852507, 7.39371690101718657958585547438, 9.009536258613385831816141707731, 9.313180753743006112852785056184, 10.23653831181182473941839610115

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