↑

Properties

Label 2-42e2-28.11-c0-0-0
Degree $2$
Conductor $1764$
Sign $-0.900 - 0.435i$
Analytic cond. $0.880350$
Root an. cond. $0.938270$
Motivic weight $0$
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 + (−0.499 + 0.866i)4-s + (0.707 + 1.22i)5-s − 0.999·8-s + (−0.707 + 1.22i)10-s − 1.41·13-s + (−0.5 − 0.866i)16-s + (−0.707 + 1.22i)17-s − 1.41·20-s + (−0.499 + 0.866i)25-s + (−0.707 − 1.22i)26-s + 2·29-s + (0.499 − 0.866i)32-s − 1.41·34-s + (−0.707 − 1.22i)40-s + 1.41·41-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.707 + 1.22i)5-s − 0.999·8-s + (−0.707 + 1.22i)10-s − 1.41·13-s + (−0.5 − 0.866i)16-s + (−0.707 + 1.22i)17-s − 1.41·20-s + (−0.499 + 0.866i)25-s + (−0.707 − 1.22i)26-s + 2·29-s + (0.499 − 0.866i)32-s − 1.41·34-s + (−0.707 − 1.22i)40-s + 1.41·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.900 - 0.435i$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ -0.900 - 0.435i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.345293552\)
\(L(\frac12)\) \(\approx\) \(1.345293552\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + 1.41T + T^{2} \)
17 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - 2T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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

−9.838927621599197261452861429917, −8.938276247429830080626657954625, −8.030046484485869010993647865580, −7.25369437398030691426589929893, −6.52476244805830565071570121698, −6.07792784409190766043957656997, −5.01952014799094913429764524117, −4.18356412122964584526306206137, −2.99830200218569670949194476655, −2.29484146976005498225225813765, 0.861575488933812684743512913508, 2.15695355467461341467190766697, 2.91437810629171799123164382310, 4.54157595715655184726207980485, 4.73963629398135902466231837803, 5.60104984311348044057391435008, 6.52727337123881425159423266145, 7.62570827676043482650655861657, 8.842282077696305665502102269566, 9.202358177545345413255521613292

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