↑

Properties

Label 2-20-20.3-c7-0-11
Degree $2$
Conductor $20$
Sign $0.146 + 0.989i$
Analytic cond. $6.24770$
Root an. cond. $2.49954$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−9.99 − 5.29i)2-s + (51.5 − 51.5i)3-s + (71.9 + 105. i)4-s + (278. − 28.9i)5-s + (−788. + 242. i)6-s + (645. + 645. i)7-s + (−159. − 1.43e3i)8-s − 3.13e3i·9-s + (−2.93e3 − 1.18e3i)10-s − 5.33e3i·11-s + (9.16e3 + 1.74e3i)12-s + (−149. − 149. i)13-s + (−3.03e3 − 9.86e3i)14-s + (1.28e4 − 1.58e4i)15-s + (−6.02e3 + 1.52e4i)16-s + (−2.08e4 + 2.08e4i)17-s + ⋯
L(s)  = 1  + (−0.883 − 0.467i)2-s + (1.10 − 1.10i)3-s + (0.562 + 0.827i)4-s + (0.994 − 0.103i)5-s + (−1.49 + 0.458i)6-s + (0.711 + 0.711i)7-s + (−0.109 − 0.993i)8-s − 1.43i·9-s + (−0.927 − 0.373i)10-s − 1.20i·11-s + (1.53 + 0.292i)12-s + (−0.0188 − 0.0188i)13-s + (−0.295 − 0.961i)14-s + (0.982 − 1.21i)15-s + (−0.367 + 0.929i)16-s + (−1.03 + 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.146 + 0.989i$
Analytic conductor: \(6.24770\)
Root analytic conductor: \(2.49954\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :7/2),\ 0.146 + 0.989i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.35250 - 1.16697i\)
\(L(\frac12)\) \(\approx\) \(1.35250 - 1.16697i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (9.99 + 5.29i)T \)
5 \( 1 + (-278. + 28.9i)T \)
good3 \( 1 + (-51.5 + 51.5i)T - 2.18e3iT^{2} \)
7 \( 1 + (-645. - 645. i)T + 8.23e5iT^{2} \)
11 \( 1 + 5.33e3iT - 1.94e7T^{2} \)
13 \( 1 + (149. + 149. i)T + 6.27e7iT^{2} \)
17 \( 1 + (2.08e4 - 2.08e4i)T - 4.10e8iT^{2} \)
19 \( 1 + 2.07e4T + 8.93e8T^{2} \)
23 \( 1 + (-4.47e4 + 4.47e4i)T - 3.40e9iT^{2} \)
29 \( 1 - 8.85e4iT - 1.72e10T^{2} \)
31 \( 1 + 4.24e4iT - 2.75e10T^{2} \)
37 \( 1 + (3.16e5 - 3.16e5i)T - 9.49e10iT^{2} \)
41 \( 1 - 4.88e5T + 1.94e11T^{2} \)
43 \( 1 + (4.56e5 - 4.56e5i)T - 2.71e11iT^{2} \)
47 \( 1 + (1.10e5 + 1.10e5i)T + 5.06e11iT^{2} \)
53 \( 1 + (-1.13e6 - 1.13e6i)T + 1.17e12iT^{2} \)
59 \( 1 + 1.06e6T + 2.48e12T^{2} \)
61 \( 1 + 1.50e6T + 3.14e12T^{2} \)
67 \( 1 + (-4.30e5 - 4.30e5i)T + 6.06e12iT^{2} \)
71 \( 1 + 1.74e6iT - 9.09e12T^{2} \)
73 \( 1 + (-2.65e6 - 2.65e6i)T + 1.10e13iT^{2} \)
79 \( 1 + 1.27e6T + 1.92e13T^{2} \)
83 \( 1 + (2.49e6 - 2.49e6i)T - 2.71e13iT^{2} \)
89 \( 1 - 2.86e6iT - 4.42e13T^{2} \)
97 \( 1 + (4.61e6 - 4.61e6i)T - 8.07e13iT^{2} \)
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

−16.94198941414618021740724605466, −14.94352607241694560123751893466, −13.55195949691150145768768339090, −12.58507950860317206075818923287, −10.86495346767927877957608662654, −8.876631485528745272235922003800, −8.377498831103336848244684224841, −6.49620957865429838449421536529, −2.65428261733280247927131277700, −1.48926594218548639416118400610, 2.13266878724663141377224138649, 4.79681108482066637774406322578, 7.21230728080593099533738921213, 8.885860831968617316687981236725, 9.790787959582156325290750209601, 10.82484513360242440325912934909, 13.74020581162231182138640977414, 14.69009258028442276458940963032, 15.61310227802065561731883925284, 17.10926622313787389182208186203

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