↑

Properties

Label 2-20-5.3-c6-0-0
Degree $2$
Conductor $20$
Sign $-0.743 - 0.668i$
Analytic cond. $4.60108$
Root an. cond. $2.14501$
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  + (−16.7 − 16.7i)3-s + (−22.1 + 123. i)5-s + (−422. + 422. i)7-s − 169. i·9-s − 1.35e3·11-s + (1.24e3 + 1.24e3i)13-s + (2.42e3 − 1.68e3i)15-s + (−648. + 648. i)17-s − 9.63e3i·19-s + 1.41e4·21-s + (886. + 886. i)23-s + (−1.46e4 − 5.46e3i)25-s + (−1.50e4 + 1.50e4i)27-s + 2.67e4i·29-s − 6.88e3·31-s + ⋯
L(s)  = 1  + (−0.619 − 0.619i)3-s + (−0.177 + 0.984i)5-s + (−1.23 + 1.23i)7-s − 0.232i·9-s − 1.02·11-s + (0.568 + 0.568i)13-s + (0.719 − 0.499i)15-s + (−0.131 + 0.131i)17-s − 1.40i·19-s + 1.52·21-s + (0.0728 + 0.0728i)23-s + (−0.936 − 0.349i)25-s + (−0.763 + 0.763i)27-s + 1.09i·29-s − 0.230·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.743 - 0.668i$
Analytic conductor: \(4.60108\)
Root analytic conductor: \(2.14501\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :3),\ -0.743 - 0.668i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.148100 + 0.386369i\)
\(L(\frac12)\) \(\approx\) \(0.148100 + 0.386369i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (22.1 - 123. i)T \)
good3 \( 1 + (16.7 + 16.7i)T + 729iT^{2} \)
7 \( 1 + (422. - 422. i)T - 1.17e5iT^{2} \)
11 \( 1 + 1.35e3T + 1.77e6T^{2} \)
13 \( 1 + (-1.24e3 - 1.24e3i)T + 4.82e6iT^{2} \)
17 \( 1 + (648. - 648. i)T - 2.41e7iT^{2} \)
19 \( 1 + 9.63e3iT - 4.70e7T^{2} \)
23 \( 1 + (-886. - 886. i)T + 1.48e8iT^{2} \)
29 \( 1 - 2.67e4iT - 5.94e8T^{2} \)
31 \( 1 + 6.88e3T + 8.87e8T^{2} \)
37 \( 1 + (4.23e4 - 4.23e4i)T - 2.56e9iT^{2} \)
41 \( 1 - 1.32e5T + 4.75e9T^{2} \)
43 \( 1 + (-2.08e4 - 2.08e4i)T + 6.32e9iT^{2} \)
47 \( 1 + (3.16e4 - 3.16e4i)T - 1.07e10iT^{2} \)
53 \( 1 + (-8.01e4 - 8.01e4i)T + 2.21e10iT^{2} \)
59 \( 1 - 2.90e5iT - 4.21e10T^{2} \)
61 \( 1 + 2.53e5T + 5.15e10T^{2} \)
67 \( 1 + (-1.13e4 + 1.13e4i)T - 9.04e10iT^{2} \)
71 \( 1 - 1.79e5T + 1.28e11T^{2} \)
73 \( 1 + (4.14e5 + 4.14e5i)T + 1.51e11iT^{2} \)
79 \( 1 - 1.45e5iT - 2.43e11T^{2} \)
83 \( 1 + (6.31e4 + 6.31e4i)T + 3.26e11iT^{2} \)
89 \( 1 - 3.33e5iT - 4.96e11T^{2} \)
97 \( 1 + (-5.19e5 + 5.19e5i)T - 8.32e11iT^{2} \)
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

−17.90610113069689817035881000154, −16.04112704298128067585001987171, −15.15533760388501473674945538843, −13.29275611286430748848142474169, −12.19132359826948902107552982145, −10.92195156438794204391059210241, −9.170338664068122482181094191590, −6.98485549389751587989967109293, −5.93590423495439834479120060935, −2.87013618533827115854548418431, 0.27452828787882270211525242084, 4.00989951124966273499554036944, 5.68294929008872680421598238461, 7.83826793311872947758391779672, 9.831003535769792820642266267023, 10.78127475180349642864845892078, 12.64833279849642477146546836804, 13.57934320140161118879857349755, 15.89123710086312055116480365944, 16.29865372306267756418706209132

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