↑

Properties

Label 2-21-21.5-c3-0-4
Degree $2$
Conductor $21$
Sign $0.996 - 0.0860i$
Analytic cond. $1.23904$
Root an. cond. $1.11312$
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  + (2.24 + 1.29i)2-s + (2.59 − 4.50i)3-s + (−0.652 − 1.13i)4-s + (−8.05 + 13.9i)5-s + (11.6 − 6.73i)6-s + (−5.67 + 17.6i)7-s − 24.0i·8-s + (−13.5 − 23.3i)9-s + (−36.1 + 20.8i)10-s + (30.8 − 17.7i)11-s + (−6.78 + 0.00678i)12-s + 7.40i·13-s + (−35.5 + 32.1i)14-s + (41.9 + 72.4i)15-s + (25.9 − 44.9i)16-s + (−14.4 − 25.0i)17-s + ⋯
L(s)  = 1  + (0.792 + 0.457i)2-s + (0.499 − 0.866i)3-s + (−0.0815 − 0.141i)4-s + (−0.720 + 1.24i)5-s + (0.791 − 0.458i)6-s + (−0.306 + 0.951i)7-s − 1.06i·8-s + (−0.501 − 0.865i)9-s + (−1.14 + 0.659i)10-s + (0.845 − 0.487i)11-s + (−0.163 + 0.000163i)12-s + 0.158i·13-s + (−0.678 + 0.613i)14-s + (0.722 + 1.24i)15-s + (0.405 − 0.701i)16-s + (−0.206 − 0.357i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0860i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.996 - 0.0860i$
Analytic conductor: \(1.23904\)
Root analytic conductor: \(1.11312\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3/2),\ 0.996 - 0.0860i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.51639 + 0.0653859i\)
\(L(\frac12)\) \(\approx\) \(1.51639 + 0.0653859i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.59 + 4.50i)T \)
7 \( 1 + (5.67 - 17.6i)T \)
good2 \( 1 + (-2.24 - 1.29i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (8.05 - 13.9i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-30.8 + 17.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 7.40iT - 2.19e3T^{2} \)
17 \( 1 + (14.4 + 25.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-30.4 - 17.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-48.0 - 27.7i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 68.1iT - 2.43e4T^{2} \)
31 \( 1 + (154. - 89.3i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-116. + 202. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 370.T + 6.89e4T^{2} \)
43 \( 1 + 187.T + 7.95e4T^{2} \)
47 \( 1 + (87.3 - 151. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (235. - 136. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (48.4 + 83.8i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (333. + 192. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-509. - 881. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 125. iT - 3.57e5T^{2} \)
73 \( 1 + (-195. + 112. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-532. + 921. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 601.T + 5.71e5T^{2} \)
89 \( 1 + (-752. + 1.30e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 327. iT - 9.12e5T^{2} \)
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

−18.17102844078361268984224483734, −15.90419587498655417440088997415, −14.76188163716022629172173296935, −14.19795231784813987074326261520, −12.71182978358503022126117452056, −11.41159518534847792023898221125, −9.184084484417770851324161699070, −7.22625062570993616495898070397, −6.11556571519401314145195555042, −3.34255824695526558205209022964, 3.79934788836565004940873003757, 4.72408387646424694577468886017, 8.018000716602660433023255949952, 9.360422430136782602000207801174, 11.21069259791649465507169360859, 12.57963182715146617467837929733, 13.64408075249968156650970310990, 14.96773990184310035400081775796, 16.43348753529220482977047916113, 17.15382722201457553847212661659

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