↑

Properties

Label 2-20-4.3-c6-0-4
Degree $2$
Conductor $20$
Sign $-0.292 - 0.956i$
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  + (6.43 + 4.75i)2-s + 20.5i·3-s + (18.7 + 61.1i)4-s − 55.9·5-s + (−97.9 + 132. i)6-s + 24.2i·7-s + (−170. + 482. i)8-s + 305.·9-s + (−359. − 265. i)10-s + 40.6i·11-s + (−1.25e3 + 385. i)12-s + 1.93e3·13-s + (−115. + 156. i)14-s − 1.15e3i·15-s + (−3.39e3 + 2.29e3i)16-s + 5.83e3·17-s + ⋯
L(s)  = 1  + (0.803 + 0.594i)2-s + 0.762i·3-s + (0.292 + 0.956i)4-s − 0.447·5-s + (−0.453 + 0.612i)6-s + 0.0707i·7-s + (−0.333 + 0.942i)8-s + 0.419·9-s + (−0.359 − 0.265i)10-s + 0.0305i·11-s + (−0.728 + 0.223i)12-s + 0.882·13-s + (−0.0420 + 0.0568i)14-s − 0.340i·15-s + (−0.828 + 0.559i)16-s + 1.18·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 - 0.956i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.292 - 0.956i$
Analytic conductor: \(4.60108\)
Root analytic conductor: \(2.14501\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :3),\ -0.292 - 0.956i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.30059 + 1.75816i\)
\(L(\frac12)\) \(\approx\) \(1.30059 + 1.75816i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-6.43 - 4.75i)T \)
5 \( 1 + 55.9T \)
good3 \( 1 - 20.5iT - 729T^{2} \)
7 \( 1 - 24.2iT - 1.17e5T^{2} \)
11 \( 1 - 40.6iT - 1.77e6T^{2} \)
13 \( 1 - 1.93e3T + 4.82e6T^{2} \)
17 \( 1 - 5.83e3T + 2.41e7T^{2} \)
19 \( 1 + 1.20e4iT - 4.70e7T^{2} \)
23 \( 1 + 8.56e3iT - 1.48e8T^{2} \)
29 \( 1 + 1.90e4T + 5.94e8T^{2} \)
31 \( 1 - 4.83e4iT - 8.87e8T^{2} \)
37 \( 1 + 9.46e3T + 2.56e9T^{2} \)
41 \( 1 + 1.00e5T + 4.75e9T^{2} \)
43 \( 1 + 1.37e5iT - 6.32e9T^{2} \)
47 \( 1 + 3.58e4iT - 1.07e10T^{2} \)
53 \( 1 - 6.08e4T + 2.21e10T^{2} \)
59 \( 1 - 5.72e3iT - 4.21e10T^{2} \)
61 \( 1 + 1.89e5T + 5.15e10T^{2} \)
67 \( 1 + 4.35e5iT - 9.04e10T^{2} \)
71 \( 1 - 2.81e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.46e5T + 1.51e11T^{2} \)
79 \( 1 + 5.68e5iT - 2.43e11T^{2} \)
83 \( 1 - 8.77e5iT - 3.26e11T^{2} \)
89 \( 1 + 4.93e4T + 4.96e11T^{2} \)
97 \( 1 - 1.50e6T + 8.32e11T^{2} \)
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

−16.85586702497155325216973069462, −15.82391161269833772543556546739, −15.08608287700932181194312647376, −13.62821339478874987088711729837, −12.23187494043739026240066687655, −10.74616717596545727501573236367, −8.764644729671026402895836488235, −7.02968070731907796712985279183, −5.06150356408780285635382080587, −3.56670619744232069385394824590, 1.38975922603843618220019728918, 3.76188652382186830084515432730, 5.95527019775948961030723224673, 7.69179973917511558770761371694, 10.01459740005735717849016264475, 11.58552362516207092581200910982, 12.63083693464488434884553139507, 13.70959884734203351087168918208, 15.03899468324517906816771658559, 16.41544659953842100461627354392

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