Properties

Label 2-42e2-28.27-c1-0-45
Degree $2$
Conductor $1764$
Sign $0.998 - 0.0536i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
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  + (−1.33 − 0.458i)2-s + (1.57 + 1.22i)4-s + 2.45i·5-s + (−1.55 − 2.36i)8-s + (1.12 − 3.28i)10-s + 1.26i·11-s − 2.99i·13-s + (0.992 + 3.87i)16-s − 1.83i·17-s + 4.15·19-s + (−3.00 + 3.87i)20-s + (0.579 − 1.69i)22-s − 6.73i·23-s − 1.01·25-s + (−1.37 + 4.01i)26-s + ⋯
L(s)  = 1  + (−0.946 − 0.324i)2-s + (0.789 + 0.613i)4-s + 1.09i·5-s + (−0.548 − 0.836i)8-s + (0.355 − 1.03i)10-s + 0.381i·11-s − 0.831i·13-s + (0.248 + 0.968i)16-s − 0.444i·17-s + 0.954·19-s + (−0.672 + 0.866i)20-s + (0.123 − 0.360i)22-s − 1.40i·23-s − 0.203·25-s + (−0.269 + 0.786i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.998 - 0.0536i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.998 - 0.0536i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.124471508\)
\(L(\frac12)\) \(\approx\) \(1.124471508\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 + 0.458i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 2.45iT - 5T^{2} \)
11 \( 1 - 1.26iT - 11T^{2} \)
13 \( 1 + 2.99iT - 13T^{2} \)
17 \( 1 + 1.83iT - 17T^{2} \)
19 \( 1 - 4.15T + 19T^{2} \)
23 \( 1 + 6.73iT - 23T^{2} \)
29 \( 1 - 9.42T + 29T^{2} \)
31 \( 1 + 9.43T + 31T^{2} \)
37 \( 1 - 7.51T + 37T^{2} \)
41 \( 1 + 1.08iT - 41T^{2} \)
43 \( 1 + 6.27iT - 43T^{2} \)
47 \( 1 - 7.35T + 47T^{2} \)
53 \( 1 - 0.0716T + 53T^{2} \)
59 \( 1 - 3.36T + 59T^{2} \)
61 \( 1 - 11.1iT - 61T^{2} \)
67 \( 1 + 2.80iT - 67T^{2} \)
71 \( 1 - 2.92iT - 71T^{2} \)
73 \( 1 - 8.10iT - 73T^{2} \)
79 \( 1 - 1.78iT - 79T^{2} \)
83 \( 1 - 5.33T + 83T^{2} \)
89 \( 1 - 8.57iT - 89T^{2} \)
97 \( 1 - 7.10iT - 97T^{2} \)
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

−9.374743821552024969254983117216, −8.572040551877208569346169571562, −7.69186149487275129156981582403, −7.09957449799630517661774420935, −6.44296236702154043389668248452, −5.39058114537172461911618741158, −4.02465092899890937192967249498, −2.94531777424141622825843341362, −2.43269225113038701698096456837, −0.825218718624443141646093779252, 0.889602165898357647300830584059, 1.79957215052806624055350374809, 3.21167633678758400375125305966, 4.51379119826783130106078751381, 5.40124504025774225263804136824, 6.11443145322907671833921719680, 7.10387691735099093188538727068, 7.86795450842091970502965398208, 8.563565129510119046662476437628, 9.305235919421759091765287265108

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