↑

Properties

Label 2-756-63.59-c1-0-5
Degree $2$
Conductor $756$
Sign $0.999 + 0.00713i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
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  + 2.96·5-s + (2.38 + 1.14i)7-s − 4.72i·11-s + (3.54 + 2.04i)13-s + (−0.835 + 1.44i)17-s + (−4.25 + 2.45i)19-s − 4.91i·23-s + 3.82·25-s + (−0.238 + 0.137i)29-s + (−1.38 + 0.801i)31-s + (7.08 + 3.40i)35-s + (−1.69 − 2.93i)37-s + (−3.55 + 6.15i)41-s + (5.22 + 9.05i)43-s + (5.49 − 9.52i)47-s + ⋯
L(s)  = 1  + 1.32·5-s + (0.901 + 0.433i)7-s − 1.42i·11-s + (0.981 + 0.566i)13-s + (−0.202 + 0.350i)17-s + (−0.975 + 0.563i)19-s − 1.02i·23-s + 0.764·25-s + (−0.0442 + 0.0255i)29-s + (−0.249 + 0.143i)31-s + (1.19 + 0.575i)35-s + (−0.278 − 0.483i)37-s + (−0.555 + 0.961i)41-s + (0.797 + 1.38i)43-s + (0.802 − 1.38i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.999 + 0.00713i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.999 + 0.00713i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12901 - 0.00759598i\)
\(L(\frac12)\) \(\approx\) \(2.12901 - 0.00759598i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-2.38 - 1.14i)T \)
good5 \( 1 - 2.96T + 5T^{2} \)
11 \( 1 + 4.72iT - 11T^{2} \)
13 \( 1 + (-3.54 - 2.04i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.835 - 1.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.25 - 2.45i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.91iT - 23T^{2} \)
29 \( 1 + (0.238 - 0.137i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.38 - 0.801i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.69 + 2.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.55 - 6.15i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.22 - 9.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.49 + 9.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.707 - 0.408i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.37 + 2.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.23 + 3.60i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.80 + 10.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 + (-13.6 - 7.88i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.15 + 10.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.03 + 6.99i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.60 - 7.98i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.00 - 4.04i)T + (48.5 - 84.0i)T^{2} \)
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

−10.60578964904028812578159512697, −9.322145397416565437893682674854, −8.667723572122167290013480569140, −8.075890626268204720589550120301, −6.40700289288500094626198510569, −6.06983790598099931689245522596, −5.11458422151969803270369918298, −3.88513156994801540832752766976, −2.44238050045193640384992303625, −1.43540632838984603906886753733, 1.47498386771565482263240631387, 2.36239789697066434802809700441, 4.02031564420913789595218640013, 5.02296991283414914451966978965, 5.82608759538117582634245768780, 6.87624850723479414156029067245, 7.67888288598046275855310801984, 8.802319199601620016096559498611, 9.493641623378948405205475088444, 10.44733670634349470120998909702

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