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Properties

Label 2-1776-444.167-c0-0-0
Degree $2$
Conductor $1776$
Sign $0.999 + 0.0143i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.766 + 0.642i)3-s + (−0.118 + 0.326i)7-s + (0.173 − 0.984i)9-s + (0.939 − 1.34i)13-s + (0.168 − 1.92i)19-s + (−0.118 − 0.326i)21-s + (−0.642 + 0.766i)25-s + (0.500 + 0.866i)27-s + (1.28 + 1.28i)31-s + (0.5 − 0.866i)37-s + (0.142 + 1.63i)39-s + (0.123 − 0.123i)43-s + (0.673 + 0.565i)49-s + (1.10 + 1.58i)57-s + (1.15 + 0.811i)61-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)3-s + (−0.118 + 0.326i)7-s + (0.173 − 0.984i)9-s + (0.939 − 1.34i)13-s + (0.168 − 1.92i)19-s + (−0.118 − 0.326i)21-s + (−0.642 + 0.766i)25-s + (0.500 + 0.866i)27-s + (1.28 + 1.28i)31-s + (0.5 − 0.866i)37-s + (0.142 + 1.63i)39-s + (0.123 − 0.123i)43-s + (0.673 + 0.565i)49-s + (1.10 + 1.58i)57-s + (1.15 + 0.811i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $0.999 + 0.0143i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1776} (1055, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1776,\ (\ :0),\ 0.999 + 0.0143i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8894347892\)
\(L(\frac12)\) \(\approx\) \(0.8894347892\)
\(L(1)\) 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 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.642 - 0.766i)T^{2} \)
7 \( 1 + (0.118 - 0.326i)T + (-0.766 - 0.642i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.939 + 1.34i)T + (-0.342 - 0.939i)T^{2} \)
17 \( 1 + (-0.342 + 0.939i)T^{2} \)
19 \( 1 + (-0.168 + 1.92i)T + (-0.984 - 0.173i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.866 - 0.5i)T^{2} \)
31 \( 1 + (-1.28 - 1.28i)T + iT^{2} \)
41 \( 1 + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.123 + 0.123i)T - iT^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.642 + 0.766i)T^{2} \)
61 \( 1 + (-1.15 - 0.811i)T + (0.342 + 0.939i)T^{2} \)
67 \( 1 + (1.20 + 0.439i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (0.173 - 0.984i)T^{2} \)
73 \( 1 + 1.53iT - T^{2} \)
79 \( 1 + (-1.80 + 0.842i)T + (0.642 - 0.766i)T^{2} \)
83 \( 1 + (-0.939 - 0.342i)T^{2} \)
89 \( 1 + (0.642 + 0.766i)T^{2} \)
97 \( 1 + (0.515 - 1.92i)T + (-0.866 - 0.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.383663339568549473110883159580, −8.949736118802917239130181755298, −7.943445942084408338184024103774, −6.94307774616865699630603853682, −6.12835395028296835633538173712, −5.41453803179289137180417669866, −4.70385411443643816746808564157, −3.61275199917875218361136976216, −2.76489513223198140896301659448, −0.907532060440997599218338915139, 1.22565143922426595703629594892, 2.24278708139112142336183727353, 3.81651521436736200966262565503, 4.47516110852867504016992045669, 5.74553164716207783864945946710, 6.24698065855360867426321324980, 6.95132876878384191300255766566, 7.950722608347887224447525744598, 8.416096210426291470291109658011, 9.718593773124922141714239023127

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