↑

Properties

Label 2-21e2-147.104-c1-0-6
Degree $2$
Conductor $441$
Sign $-0.721 + 0.692i$
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  + (−2.13 − 1.70i)2-s + (1.21 + 5.30i)4-s + (−1.31 − 0.635i)5-s + (−0.795 − 2.52i)7-s + (4.07 − 8.46i)8-s + (1.73 + 3.60i)10-s + (2.75 + 2.19i)11-s + (4.51 + 3.60i)13-s + (−2.59 + 6.73i)14-s + (−13.2 + 6.40i)16-s + (0.0162 − 0.0710i)17-s − 4.52i·19-s + (1.77 − 7.77i)20-s + (−2.14 − 9.37i)22-s + (1.76 − 0.402i)23-s + ⋯
L(s)  = 1  + (−1.50 − 1.20i)2-s + (0.605 + 2.65i)4-s + (−0.590 − 0.284i)5-s + (−0.300 − 0.953i)7-s + (1.44 − 2.99i)8-s + (0.548 + 1.13i)10-s + (0.831 + 0.662i)11-s + (1.25 + 0.999i)13-s + (−0.693 + 1.80i)14-s + (−3.32 + 1.60i)16-s + (0.00393 − 0.0172i)17-s − 1.03i·19-s + (0.396 − 1.73i)20-s + (−0.456 − 1.99i)22-s + (0.367 − 0.0839i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.721 + 0.692i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.721 + 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.202105 - 0.502425i\)
\(L(\frac12)\) \(\approx\) \(0.202105 - 0.502425i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.795 + 2.52i)T \)
good2 \( 1 + (2.13 + 1.70i)T + (0.445 + 1.94i)T^{2} \)
5 \( 1 + (1.31 + 0.635i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-2.75 - 2.19i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-4.51 - 3.60i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-0.0162 + 0.0710i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 4.52iT - 19T^{2} \)
23 \( 1 + (-1.76 + 0.402i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (2.61 + 0.597i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 7.13iT - 31T^{2} \)
37 \( 1 + (-1.33 + 5.82i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (11.3 + 5.46i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (1.31 - 0.632i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-5.84 + 7.32i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-9.10 + 2.07i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-0.107 + 0.0516i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-3.20 - 0.730i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 5.78T + 67T^{2} \)
71 \( 1 + (0.0872 - 0.0199i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-3.43 + 2.73i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 2.89T + 79T^{2} \)
83 \( 1 + (0.251 + 0.315i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-6.88 - 8.63i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 14.7iT - 97T^{2} \)
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

−10.79701431209698342964898020113, −9.846597011043275743511836462604, −9.102149002509192138697288274021, −8.418085884718935188498157543756, −7.30745914391814967504131795304, −6.72599293742677066035208253793, −4.16455535147291344191720669269, −3.72155563524845878071907735293, −1.99938665570570351284501745107, −0.64598822766616234224132375197, 1.31897313458667193726908934653, 3.37454305591819473941221194651, 5.41551673343039942995506093373, 6.10481068662067171794907203737, 6.91109586710629505278645194257, 8.121768198703796857762555773668, 8.502356900976752995034670831041, 9.352802208759215719305604630997, 10.32690035937678978248500176506, 11.13300287313568765555931155444

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