↑

Properties

Label 2-21e2-21.20-c3-0-17
Degree $2$
Conductor $441$
Sign $-0.860 - 0.508i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5.38i·2-s − 20.9·4-s + 18.3·5-s − 69.8i·8-s + 98.6i·10-s − 23.7i·11-s − 11.7i·13-s + 208.·16-s + 96.4·17-s + 21.0i·19-s − 384.·20-s + 127.·22-s + 152. i·23-s + 210.·25-s + 63.5·26-s + ⋯
L(s)  = 1  + 1.90i·2-s − 2.62·4-s + 1.63·5-s − 3.08i·8-s + 3.11i·10-s − 0.651i·11-s − 0.251i·13-s + 3.25·16-s + 1.37·17-s + 0.253i·19-s − 4.29·20-s + 1.23·22-s + 1.38i·23-s + 1.68·25-s + 0.478·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.860 - 0.508i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -0.860 - 0.508i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.230985411\)
\(L(\frac12)\) \(\approx\) \(2.230985411\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2 \( 1 - 5.38iT - 8T^{2} \)
5 \( 1 - 18.3T + 125T^{2} \)
11 \( 1 + 23.7iT - 1.33e3T^{2} \)
13 \( 1 + 11.7iT - 2.19e3T^{2} \)
17 \( 1 - 96.4T + 4.91e3T^{2} \)
19 \( 1 - 21.0iT - 6.85e3T^{2} \)
23 \( 1 - 152. iT - 1.21e4T^{2} \)
29 \( 1 - 77.5iT - 2.43e4T^{2} \)
31 \( 1 - 199. iT - 2.97e4T^{2} \)
37 \( 1 - 164.T + 5.06e4T^{2} \)
41 \( 1 - 435.T + 6.89e4T^{2} \)
43 \( 1 + 106.T + 7.95e4T^{2} \)
47 \( 1 + 11.0T + 1.03e5T^{2} \)
53 \( 1 - 110. iT - 1.48e5T^{2} \)
59 \( 1 + 829.T + 2.05e5T^{2} \)
61 \( 1 - 22.5iT - 2.26e5T^{2} \)
67 \( 1 - 398.T + 3.00e5T^{2} \)
71 \( 1 + 228. iT - 3.57e5T^{2} \)
73 \( 1 + 266. iT - 3.89e5T^{2} \)
79 \( 1 - 920.T + 4.93e5T^{2} \)
83 \( 1 - 642.T + 5.71e5T^{2} \)
89 \( 1 + 445.T + 7.04e5T^{2} \)
97 \( 1 + 1.61e3iT - 9.12e5T^{2} \)
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

−10.65083018185004374764442066165, −9.664165223878738476040842693390, −9.221676211772745079177347178792, −8.149435214148263380483037025141, −7.30238956857326301774803247416, −6.16105409171811129869150630925, −5.72684778510862084568352702838, −4.97085634546339219121970888933, −3.34260202917264647420360070126, −1.18092459516633979698686635137, 0.906913387282169388124961461772, 2.04337524185623659892171199095, 2.75801251616512297653789340342, 4.21309953167558614283516603672, 5.20591295101513901272244844916, 6.21186295072738644975726626000, 7.961886978184062316140520855961, 9.232792085404135022003514854391, 9.644693140489639657163361759044, 10.30666406255617673251906548831

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