↑

Properties

Label 2-21e2-63.41-c1-0-9
Degree $2$
Conductor $441$
Sign $0.630 - 0.775i$
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  + (1.5 − 0.866i)2-s + (−1.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (−1.5 + 2.59i)5-s − 3·6-s + 1.73i·8-s + (1.5 + 2.59i)9-s + 5.19i·10-s + (−1.5 + 0.866i)11-s + (−1.5 + 0.866i)12-s + (1.5 + 0.866i)13-s + (4.5 − 2.59i)15-s + (2.49 + 4.33i)16-s + 3·17-s + (4.5 + 2.59i)18-s + 5.19i·19-s + ⋯
L(s)  = 1  + (1.06 − 0.612i)2-s + (−0.866 − 0.499i)3-s + (0.250 − 0.433i)4-s + (−0.670 + 1.16i)5-s − 1.22·6-s + 0.612i·8-s + (0.5 + 0.866i)9-s + 1.64i·10-s + (−0.452 + 0.261i)11-s + (−0.433 + 0.250i)12-s + (0.416 + 0.240i)13-s + (1.16 − 0.670i)15-s + (0.624 + 1.08i)16-s + 0.727·17-s + (1.06 + 0.612i)18-s + 1.19i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.630 - 0.775i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.630 - 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20594 + 0.573829i\)
\(L(\frac12)\) \(\approx\) \(1.20594 + 0.573829i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (-1.5 + 0.866i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 5.19iT - 19T^{2} \)
23 \( 1 + (4.5 + 2.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.5 - 2.59i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7T + 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.66iT - 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-12 + 6.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + 5.19iT - 73T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.5 - 12.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (-1.5 + 0.866i)T + (48.5 - 84.0i)T^{2} \)
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

−11.34309667794627545208757575007, −10.84449289716475491813986113804, −10.00111177048033179770062021191, −8.105906143421949373308181723334, −7.48253917747855247628383853242, −6.30248501849053917088018744441, −5.52116323346776438286267520855, −4.27247618503601042088236419458, −3.35712317757571173717209091884, −2.02913079154226870435598697461, 0.68326145947227450555186667382, 3.59954221768073163717725153600, 4.39129839933754766901679782866, 5.28165064771684952727449187367, 5.80582302645882997174266365702, 7.02652294076935970066588016532, 8.080598341922261260261085829153, 9.245591015880795978741399114923, 10.12063969445282115975338871024, 11.32378838568453523619681595055

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