↑

Properties

Label 2-21e2-63.25-c1-0-21
Degree $2$
Conductor $441$
Sign $0.676 + 0.736i$
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  − 1.72·2-s + (1.70 − 0.276i)3-s + 0.981·4-s + (1.75 − 3.04i)5-s + (−2.95 + 0.477i)6-s + 1.75·8-s + (2.84 − 0.946i)9-s + (−3.03 + 5.25i)10-s + (3.04 + 5.27i)11-s + (1.67 − 0.271i)12-s + (−0.560 − 0.970i)13-s + (2.16 − 5.68i)15-s − 4.99·16-s + (0.601 − 1.04i)17-s + (−4.91 + 1.63i)18-s + (−1.10 − 1.90i)19-s + ⋯
L(s)  = 1  − 1.22·2-s + (0.987 − 0.159i)3-s + 0.490·4-s + (0.785 − 1.36i)5-s + (−1.20 + 0.195i)6-s + 0.621·8-s + (0.948 − 0.315i)9-s + (−0.958 + 1.66i)10-s + (0.918 + 1.59i)11-s + (0.484 − 0.0783i)12-s + (−0.155 − 0.269i)13-s + (0.557 − 1.46i)15-s − 1.24·16-s + (0.146 − 0.252i)17-s + (−1.15 + 0.385i)18-s + (−0.252 − 0.438i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.676 + 0.736i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.676 + 0.736i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11510 - 0.490128i\)
\(L(\frac12)\) \(\approx\) \(1.11510 - 0.490128i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 + 0.276i)T \)
7 \( 1 \)
good2 \( 1 + 1.72T + 2T^{2} \)
5 \( 1 + (-1.75 + 3.04i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.04 - 5.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.560 + 0.970i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.601 + 1.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.10 + 1.90i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.636 + 1.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.10 - 5.37i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.188T + 31T^{2} \)
37 \( 1 + (1.78 + 3.09i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.68 + 2.91i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.90 - 3.29i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.72T + 47T^{2} \)
53 \( 1 + (-4.16 + 7.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + 12.0T + 61T^{2} \)
67 \( 1 + 7.91T + 67T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 + (2.65 - 4.60i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 9.21T + 79T^{2} \)
83 \( 1 + (0.624 - 1.08i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.77 - 4.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.24 - 14.2i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.42447285834240564037227922527, −9.687687383431100068154525063935, −9.117952594969150756488634254436, −8.688044697248268995364302802078, −7.58314183362128595583487539090, −6.82652404492832411933383019954, −5.06966483939550489750650681756, −4.18934238189869810439273422296, −2.12222469742924856834312068527, −1.25797198420444690898557637027, 1.63419229794690583245975520751, 2.89888794660481138047594584900, 4.00262965122752021848445105309, 5.95242722075579879807240426552, 6.88318288189651326226102116462, 7.79688426317774559486256066322, 8.729716759602114485794168508288, 9.348477904486125710309084930069, 10.21400889657296181025737014537, 10.72929610139815865148811145796

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