↑

Properties

Label 2-21e2-63.20-c1-0-20
Degree $2$
Conductor $441$
Sign $-0.636 + 0.770i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
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  + (−2.34 − 1.35i)2-s + (1.70 − 0.322i)3-s + (2.66 + 4.62i)4-s + (−0.601 − 1.04i)5-s + (−4.42 − 1.54i)6-s − 9.04i·8-s + (2.79 − 1.09i)9-s + 3.25i·10-s + (−2.15 − 1.24i)11-s + (6.03 + 7.00i)12-s + (−1.63 + 0.942i)13-s + (−1.35 − 1.57i)15-s + (−6.90 + 11.9i)16-s + 1.20·17-s + (−8.03 − 1.21i)18-s − 7.47i·19-s + ⋯
L(s)  = 1  + (−1.65 − 0.957i)2-s + (0.982 − 0.185i)3-s + (1.33 + 2.31i)4-s + (−0.268 − 0.465i)5-s + (−1.80 − 0.632i)6-s − 3.19i·8-s + (0.930 − 0.365i)9-s + 1.03i·10-s + (−0.649 − 0.374i)11-s + (1.74 + 2.02i)12-s + (−0.452 + 0.261i)13-s + (−0.350 − 0.407i)15-s + (−1.72 + 2.99i)16-s + 0.291·17-s + (−1.89 − 0.285i)18-s − 1.71i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 + 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.636 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.636 + 0.770i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.636 + 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.326740 - 0.693768i\)
\(L(\frac12)\) \(\approx\) \(0.326740 - 0.693768i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 + 0.322i)T \)
7 \( 1 \)
good2 \( 1 + (2.34 + 1.35i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.601 + 1.04i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.15 + 1.24i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.63 - 0.942i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.20T + 17T^{2} \)
19 \( 1 + 7.47iT - 19T^{2} \)
23 \( 1 + (-2.63 + 1.52i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.173 + 0.100i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.03 + 1.75i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.73T + 37T^{2} \)
41 \( 1 + (3.36 + 5.82i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.00656 + 0.0113i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.717 + 1.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.90iT - 53T^{2} \)
59 \( 1 + (6.10 + 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.73 + 5.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.57 - 4.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.0iT - 71T^{2} \)
73 \( 1 - 8.67iT - 73T^{2} \)
79 \( 1 + (2.74 - 4.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.60 - 2.78i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.96T + 89T^{2} \)
97 \( 1 + (2.06 + 1.19i)T + (48.5 + 84.0i)T^{2} \)
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

−10.59517843802814732707970634789, −9.724760291597002538287020239119, −8.958965942981902521139351999706, −8.412527646767739523219788435075, −7.57772335926843833189420052425, −6.80236757410148037607580856582, −4.52198435748711658838655609613, −3.11310583334642588491770131619, −2.32363252830386459483798135773, −0.75599547181153374188854862580, 1.66824162998996908676256243262, 3.08380611491542591317778759471, 4.98908809093423224449008983758, 6.25258218908911562006669407392, 7.44380656627752733742191058682, 7.71659512380946810677185062896, 8.608690951076860721757551894352, 9.540627734614620985365598702067, 10.19991679579272204252741469069, 10.76972712702847685585138985820

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