↑

Properties

Label 2-20-20.19-c8-0-14
Degree $2$
Conductor $20$
Sign $-0.0769 + 0.997i$
Analytic cond. $8.14757$
Root an. cond. $2.85439$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−7.23 − 14.2i)2-s + 44.3·3-s + (−151. + 206. i)4-s + (530. − 329. i)5-s + (−320. − 632. i)6-s + 2.82e3·7-s + (4.04e3 + 667. i)8-s − 4.59e3·9-s + (−8.54e3 − 5.19e3i)10-s − 1.83e4i·11-s + (−6.71e3 + 9.15e3i)12-s − 1.39e4i·13-s + (−2.04e4 − 4.03e4i)14-s + (2.35e4 − 1.46e4i)15-s + (−1.97e4 − 6.25e4i)16-s − 9.61e4i·17-s + ⋯
L(s)  = 1  + (−0.452 − 0.892i)2-s + 0.547·3-s + (−0.591 + 0.806i)4-s + (0.849 − 0.527i)5-s + (−0.247 − 0.488i)6-s + 1.17·7-s + (0.986 + 0.162i)8-s − 0.700·9-s + (−0.854 − 0.519i)10-s − 1.25i·11-s + (−0.323 + 0.441i)12-s − 0.488i·13-s + (−0.532 − 1.05i)14-s + (0.464 − 0.288i)15-s + (−0.300 − 0.953i)16-s − 1.15i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0769 + 0.997i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0769 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.0769 + 0.997i$
Analytic conductor: \(8.14757\)
Root analytic conductor: \(2.85439\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :4),\ -0.0769 + 0.997i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.20986 - 1.30685i\)
\(L(\frac12)\) \(\approx\) \(1.20986 - 1.30685i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.23 + 14.2i)T \)
5 \( 1 + (-530. + 329. i)T \)
good3 \( 1 - 44.3T + 6.56e3T^{2} \)
7 \( 1 - 2.82e3T + 5.76e6T^{2} \)
11 \( 1 + 1.83e4iT - 2.14e8T^{2} \)
13 \( 1 + 1.39e4iT - 8.15e8T^{2} \)
17 \( 1 + 9.61e4iT - 6.97e9T^{2} \)
19 \( 1 - 1.53e5iT - 1.69e10T^{2} \)
23 \( 1 - 3.39e5T + 7.83e10T^{2} \)
29 \( 1 - 7.58e5T + 5.00e11T^{2} \)
31 \( 1 - 1.01e6iT - 8.52e11T^{2} \)
37 \( 1 + 1.35e5iT - 3.51e12T^{2} \)
41 \( 1 + 2.56e6T + 7.98e12T^{2} \)
43 \( 1 + 8.52e5T + 1.16e13T^{2} \)
47 \( 1 + 1.25e6T + 2.38e13T^{2} \)
53 \( 1 - 1.00e7iT - 6.22e13T^{2} \)
59 \( 1 - 2.35e7iT - 1.46e14T^{2} \)
61 \( 1 + 5.69e6T + 1.91e14T^{2} \)
67 \( 1 - 1.76e7T + 4.06e14T^{2} \)
71 \( 1 + 3.89e6iT - 6.45e14T^{2} \)
73 \( 1 - 9.62e6iT - 8.06e14T^{2} \)
79 \( 1 + 2.34e6iT - 1.51e15T^{2} \)
83 \( 1 - 1.26e7T + 2.25e15T^{2} \)
89 \( 1 - 6.42e7T + 3.93e15T^{2} \)
97 \( 1 + 6.62e6iT - 7.83e15T^{2} \)
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

−16.56425490006688448439106726026, −14.28466061471291228410947708079, −13.56469340029325163379640496934, −11.90275797206907919648382687364, −10.62305372627577555968212197983, −8.983720420703210176958804040069, −8.175277119364508733743419601341, −5.19599112767326393038960201071, −2.89210090164555921704616742053, −1.14902901108809527324791755760, 1.92146554945788057348779046654, 4.97118430848246867359507344925, 6.75098502057753260648888650884, 8.271852516995340304925274934627, 9.537951152899666339614789316349, 11.02322566672747850023653047422, 13.41993602160970889835296031252, 14.60699641533324346502947820171, 15.06972618855206029589204549916, 17.27558766252016500238126064592

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