Properties

Label 2-21e2-49.11-c1-0-10
Degree $2$
Conductor $441$
Sign $0.995 + 0.0927i$
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  + (1.29 − 1.19i)2-s + (0.0828 − 1.10i)4-s + (−1.35 + 3.45i)5-s + (1.47 − 2.19i)7-s + (0.979 + 1.22i)8-s + (2.38 + 6.08i)10-s + (1.46 + 0.451i)11-s + (0.545 + 2.39i)13-s + (−0.727 − 4.60i)14-s + (4.93 + 0.743i)16-s + (3.98 − 2.71i)17-s + (1.90 + 3.30i)19-s + (3.70 + 1.78i)20-s + (2.43 − 1.17i)22-s + (−3.98 − 2.71i)23-s + ⋯
L(s)  = 1  + (0.913 − 0.848i)2-s + (0.0414 − 0.552i)4-s + (−0.605 + 1.54i)5-s + (0.557 − 0.830i)7-s + (0.346 + 0.434i)8-s + (0.755 + 1.92i)10-s + (0.441 + 0.136i)11-s + (0.151 + 0.662i)13-s + (−0.194 − 1.23i)14-s + (1.23 + 0.185i)16-s + (0.967 − 0.659i)17-s + (0.437 + 0.757i)19-s + (0.828 + 0.398i)20-s + (0.518 − 0.249i)22-s + (−0.830 − 0.566i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.995 + 0.0927i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.995 + 0.0927i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22075 - 0.103205i\)
\(L(\frac12)\) \(\approx\) \(2.22075 - 0.103205i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.47 + 2.19i)T \)
good2 \( 1 + (-1.29 + 1.19i)T + (0.149 - 1.99i)T^{2} \)
5 \( 1 + (1.35 - 3.45i)T + (-3.66 - 3.40i)T^{2} \)
11 \( 1 + (-1.46 - 0.451i)T + (9.08 + 6.19i)T^{2} \)
13 \( 1 + (-0.545 - 2.39i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (-3.98 + 2.71i)T + (6.21 - 15.8i)T^{2} \)
19 \( 1 + (-1.90 - 3.30i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.98 + 2.71i)T + (8.40 + 21.4i)T^{2} \)
29 \( 1 + (0.292 + 0.140i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (3.32 - 5.75i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.669 + 8.92i)T + (-36.5 + 5.51i)T^{2} \)
41 \( 1 + (1.52 + 1.90i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (3.39 - 4.25i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (6.44 - 5.98i)T + (3.51 - 46.8i)T^{2} \)
53 \( 1 + (-0.754 + 10.0i)T + (-52.4 - 7.89i)T^{2} \)
59 \( 1 + (4.97 + 12.6i)T + (-43.2 + 40.1i)T^{2} \)
61 \( 1 + (-0.0620 - 0.827i)T + (-60.3 + 9.09i)T^{2} \)
67 \( 1 + (-6.41 + 11.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.993 - 0.478i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (4.14 + 3.84i)T + (5.45 + 72.7i)T^{2} \)
79 \( 1 + (-5.85 - 10.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.63 - 7.15i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-15.7 + 4.86i)T + (73.5 - 50.1i)T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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

−11.24805242069727474480021279808, −10.60959244047784975421762923768, −9.791500003595714227365353108155, −8.060382699765374613866891093950, −7.40703220431842486555900746150, −6.44238840807336843307391286382, −4.98854247115031328661482644156, −3.82378632560176619263782056977, −3.34908810952215361833649405175, −1.89304839272661931121185752582, 1.29560726730652720735994798453, 3.58896720534088614777732649863, 4.62701885476791253788298035700, 5.37653095321227707476968576592, 6.02399363204439064036465224013, 7.55073445940031935799569974135, 8.197602504166930329924625513192, 9.043625033109089744822863633066, 10.12463123568646318018255833219, 11.66198157522882744268336418931

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