Properties

Label 2-21e2-7.6-c6-0-7
Degree $2$
Conductor $441$
Sign $-0.755 - 0.654i$
Analytic cond. $101.453$
Root an. cond. $10.0724$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5.52·2-s − 33.4·4-s + 66.9i·5-s + 538.·8-s − 370. i·10-s − 1.72e3·11-s − 2.80e3i·13-s − 836.·16-s − 6.14e3i·17-s + 8.93e3i·19-s − 2.24e3i·20-s + 9.53e3·22-s − 9.90e3·23-s + 1.11e4·25-s + 1.55e4i·26-s + ⋯
L(s)  = 1  − 0.690·2-s − 0.522·4-s + 0.535i·5-s + 1.05·8-s − 0.370i·10-s − 1.29·11-s − 1.27i·13-s − 0.204·16-s − 1.25i·17-s + 1.30i·19-s − 0.280i·20-s + 0.895·22-s − 0.813·23-s + 0.712·25-s + 0.882i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.755 - 0.654i$
Analytic conductor: \(101.453\)
Root analytic conductor: \(10.0724\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3),\ -0.755 - 0.654i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.2501384725\)
\(L(\frac12)\) \(\approx\) \(0.2501384725\)
\(L(4)\) 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 \( 1 \)
good2 \( 1 + 5.52T + 64T^{2} \)
5 \( 1 - 66.9iT - 1.56e4T^{2} \)
11 \( 1 + 1.72e3T + 1.77e6T^{2} \)
13 \( 1 + 2.80e3iT - 4.82e6T^{2} \)
17 \( 1 + 6.14e3iT - 2.41e7T^{2} \)
19 \( 1 - 8.93e3iT - 4.70e7T^{2} \)
23 \( 1 + 9.90e3T + 1.48e8T^{2} \)
29 \( 1 - 1.36e4T + 5.94e8T^{2} \)
31 \( 1 - 2.52e4iT - 8.87e8T^{2} \)
37 \( 1 - 2.27e4T + 2.56e9T^{2} \)
41 \( 1 + 3.78e4iT - 4.75e9T^{2} \)
43 \( 1 - 7.36e4T + 6.32e9T^{2} \)
47 \( 1 + 1.39e5iT - 1.07e10T^{2} \)
53 \( 1 - 1.65e4T + 2.21e10T^{2} \)
59 \( 1 + 7.15e4iT - 4.21e10T^{2} \)
61 \( 1 + 2.63e5iT - 5.15e10T^{2} \)
67 \( 1 - 5.48e5T + 9.04e10T^{2} \)
71 \( 1 + 4.65e5T + 1.28e11T^{2} \)
73 \( 1 - 3.33e5iT - 1.51e11T^{2} \)
79 \( 1 + 4.10e5T + 2.43e11T^{2} \)
83 \( 1 - 7.19e4iT - 3.26e11T^{2} \)
89 \( 1 - 1.63e5iT - 4.96e11T^{2} \)
97 \( 1 + 6.51e5iT - 8.32e11T^{2} \)
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

−10.28870531225850647549522800382, −9.803861337783901162448935992209, −8.522952644149935630873294313783, −7.913917969274699670415356786626, −7.13036281821090326283140953749, −5.66135791702873662506763551042, −4.90175715715742628419258325473, −3.49598085262665379735333673294, −2.42266389989539049553370128369, −0.852885890635526741297725348350, 0.098251772568345648408968453292, 1.24028169590796622065971850454, 2.47550019288027967251432069286, 4.18957584990310788497538438967, 4.79954621459457778812657645849, 6.01606326240294202404619462883, 7.30009253981536906897842352054, 8.188934449913571942986164438888, 8.858800344845030099564916299063, 9.648106740153551553895772508940

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