↑

Properties

Label 2-2200-440.123-c0-0-6
Degree $2$
Conductor $2200$
Sign $0.496 + 0.867i$
Analytic cond. $1.09794$
Root an. cond. $1.04782$
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  + (0.891 − 0.453i)2-s + (0.410 + 0.0650i)3-s + (0.587 − 0.809i)4-s + (0.395 − 0.128i)6-s + (0.156 − 0.987i)8-s + (−0.786 − 0.255i)9-s + (0.913 + 0.406i)11-s + (0.294 − 0.294i)12-s + (−0.309 − 0.951i)16-s + (0.0949 − 0.186i)17-s + (−0.816 + 0.129i)18-s + (0.658 − 0.478i)19-s + (0.998 − 0.0523i)22-s + (0.128 − 0.395i)24-s + (−0.676 − 0.344i)27-s + ⋯
L(s)  = 1  + (0.891 − 0.453i)2-s + (0.410 + 0.0650i)3-s + (0.587 − 0.809i)4-s + (0.395 − 0.128i)6-s + (0.156 − 0.987i)8-s + (−0.786 − 0.255i)9-s + (0.913 + 0.406i)11-s + (0.294 − 0.294i)12-s + (−0.309 − 0.951i)16-s + (0.0949 − 0.186i)17-s + (−0.816 + 0.129i)18-s + (0.658 − 0.478i)19-s + (0.998 − 0.0523i)22-s + (0.128 − 0.395i)24-s + (−0.676 − 0.344i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $0.496 + 0.867i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (1443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ 0.496 + 0.867i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.260685161\)
\(L(\frac12)\) \(\approx\) \(2.260685161\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.891 + 0.453i)T \)
5 \( 1 \)
11 \( 1 + (-0.913 - 0.406i)T \)
good3 \( 1 + (-0.410 - 0.0650i)T + (0.951 + 0.309i)T^{2} \)
7 \( 1 + (0.951 - 0.309i)T^{2} \)
13 \( 1 + (0.587 - 0.809i)T^{2} \)
17 \( 1 + (-0.0949 + 0.186i)T + (-0.587 - 0.809i)T^{2} \)
19 \( 1 + (-0.658 + 0.478i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (-0.873 - 1.20i)T + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + (0.437 - 0.437i)T - iT^{2} \)
47 \( 1 + (0.951 + 0.309i)T^{2} \)
53 \( 1 + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (1.40 - 1.40i)T - iT^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (1.32 - 0.209i)T + (0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (1.19 + 0.607i)T + (0.587 + 0.809i)T^{2} \)
89 \( 1 - 1.95iT - T^{2} \)
97 \( 1 + (-0.863 - 1.69i)T + (-0.587 + 0.809i)T^{2} \)
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

−9.398763820069315715633507479819, −8.415666747795254156134893851467, −7.42262811313577026928703108580, −6.60077566933996754896704663848, −5.89933124083126123656416561878, −4.99303853895425507031499726566, −4.15516053944104724903921404440, −3.27844141720740833447340677772, −2.55449627235467199907657908561, −1.30522106148309244481648996893, 1.78580285551843107885212948157, 2.94869952797943716812602530821, 3.60928863188773185963514572081, 4.51509961291300287524186686700, 5.60134619194551156582920222613, 6.01974859617678873995799290172, 7.04796040946777913133857917870, 7.69986897557016762189598923607, 8.559545493094159321745422948051, 9.011505054228162784923994553063

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