Properties

Label 2-42e2-7.2-c1-0-7
Degree $2$
Conductor $1764$
Sign $0.605 - 0.795i$
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 + 1.73i)5-s + (1 + 1.73i)11-s + 4·13-s + (−3 − 5.19i)17-s + (4 − 6.92i)19-s + (−3 + 5.19i)23-s + (0.500 + 0.866i)25-s + 10·29-s + (2 + 3.46i)31-s + (−3 + 5.19i)37-s − 6·41-s + 4·43-s + (−4 + 6.92i)47-s + (1 + 1.73i)53-s − 3.99·55-s + ⋯
L(s)  = 1  + (−0.447 + 0.774i)5-s + (0.301 + 0.522i)11-s + 1.10·13-s + (−0.727 − 1.26i)17-s + (0.917 − 1.58i)19-s + (−0.625 + 1.08i)23-s + (0.100 + 0.173i)25-s + 1.85·29-s + (0.359 + 0.622i)31-s + (−0.493 + 0.854i)37-s − 0.937·41-s + 0.609·43-s + (−0.583 + 1.01i)47-s + (0.137 + 0.237i)53-s − 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.623397038\)
\(L(\frac12)\) \(\approx\) \(1.623397038\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4 + 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-7 + 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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

−9.360744082005070130876122216836, −8.701715136667653329038702155890, −7.71877403305596125776327766156, −6.90523684432299866215992131695, −6.55042227123339253103176996654, −5.23651417419348683409569018204, −4.47716268805571381575434138277, −3.35051073617788004427920111886, −2.67596701654423032832564116924, −1.11424805813388068093017046870, 0.75522123158939433778133858115, 1.92330405248765734572686974635, 3.49646891531374989788758422672, 4.06264070557333237350571060364, 5.03869372266193042569691126211, 6.12833013177810949718136754426, 6.52352173358036477463193674405, 8.055250166016411363199403881654, 8.279065044700320875082612490360, 8.929437560417049530649254143739

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