Properties

Label 2-1776-111.101-c0-0-0
Degree $2$
Conductor $1776$
Sign $-0.227 - 0.973i$
Analytic cond. $0.886339$
Root an. cond. $0.941456$
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)3-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−1.5 + 0.866i)13-s + (−0.499 + 0.866i)21-s + (0.5 − 0.866i)25-s − 0.999·27-s + 1.73i·31-s + 37-s + (−1.5 − 0.866i)39-s − 1.73i·43-s − 0.999·63-s + (0.5 + 0.866i)67-s + 73-s + 0.999·75-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−1.5 + 0.866i)13-s + (−0.499 + 0.866i)21-s + (0.5 − 0.866i)25-s − 0.999·27-s + 1.73i·31-s + 37-s + (−1.5 − 0.866i)39-s − 1.73i·43-s − 0.999·63-s + (0.5 + 0.866i)67-s + 73-s + 0.999·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $-0.227 - 0.973i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1776} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1776,\ (\ :0),\ -0.227 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.250265656\)
\(L(\frac12)\) \(\approx\) \(1.250265656\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.73iT - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + 1.73iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - 1.73iT - T^{2} \)
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

−9.613009786047755710339410458238, −8.933022611431709029939379201112, −8.381170347648151226574497511031, −7.46381363676700004773407029293, −6.55378443777929638949203500807, −5.29213107875301605217683294288, −4.89250490777436438282645973149, −3.95239635753751879776209999070, −2.73881059035946788918972354560, −2.07070886115653362971446260255, 0.896464788541896387308152225520, 2.21759540581430457238434984357, 3.10741552901677517301310799421, 4.24042279354031696152018996599, 5.18283535932327000913724474644, 6.19271244943421759707244801744, 7.12753286825809124388889063578, 7.75157693361575687240855413046, 8.084263798150192620438116216965, 9.361676175737549223754440137754

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