↑

Properties

Label 2-20-20.19-c4-0-0
Degree $2$
Conductor $20$
Sign $-0.722 - 0.691i$
Analytic cond. $2.06739$
Root an. cond. $1.43784$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.85 − 2.79i)2-s − 6.64·3-s + (0.324 + 15.9i)4-s + (−17.6 + 17.7i)5-s + (18.9 + 18.5i)6-s − 39.0·7-s + (43.8 − 46.6i)8-s − 36.8·9-s + (99.9 − 1.17i)10-s − 138. i·11-s + (−2.15 − 106. i)12-s + 124. i·13-s + (111. + 109. i)14-s + (117. − 117. i)15-s + (−255. + 10.3i)16-s − 160. i·17-s + ⋯
L(s)  = 1  + (−0.714 − 0.699i)2-s − 0.737·3-s + (0.0202 + 0.999i)4-s + (−0.705 + 0.708i)5-s + (0.527 + 0.516i)6-s − 0.797·7-s + (0.685 − 0.728i)8-s − 0.455·9-s + (0.999 − 0.0117i)10-s − 1.14i·11-s + (−0.0149 − 0.737i)12-s + 0.739i·13-s + (0.569 + 0.558i)14-s + (0.520 − 0.522i)15-s + (−0.999 + 0.0405i)16-s − 0.555i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.722 - 0.691i$
Analytic conductor: \(2.06739\)
Root analytic conductor: \(1.43784\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :2),\ -0.722 - 0.691i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0392905 + 0.0978735i\)
\(L(\frac12)\) \(\approx\) \(0.0392905 + 0.0978735i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.85 + 2.79i)T \)
5 \( 1 + (17.6 - 17.7i)T \)
good3 \( 1 + 6.64T + 81T^{2} \)
7 \( 1 + 39.0T + 2.40e3T^{2} \)
11 \( 1 + 138. iT - 1.46e4T^{2} \)
13 \( 1 - 124. iT - 2.85e4T^{2} \)
17 \( 1 + 160. iT - 8.35e4T^{2} \)
19 \( 1 - 650. iT - 1.30e5T^{2} \)
23 \( 1 + 416.T + 2.79e5T^{2} \)
29 \( 1 - 236.T + 7.07e5T^{2} \)
31 \( 1 - 41.5iT - 9.23e5T^{2} \)
37 \( 1 - 206. iT - 1.87e6T^{2} \)
41 \( 1 + 1.81e3T + 2.82e6T^{2} \)
43 \( 1 + 3.16e3T + 3.41e6T^{2} \)
47 \( 1 + 823.T + 4.87e6T^{2} \)
53 \( 1 - 4.86e3iT - 7.89e6T^{2} \)
59 \( 1 + 3.63e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.13e3T + 1.38e7T^{2} \)
67 \( 1 - 3.20e3T + 2.01e7T^{2} \)
71 \( 1 - 456. iT - 2.54e7T^{2} \)
73 \( 1 + 5.90e3iT - 2.83e7T^{2} \)
79 \( 1 - 2.80e3iT - 3.89e7T^{2} \)
83 \( 1 + 3.04e3T + 4.74e7T^{2} \)
89 \( 1 + 5.14e3T + 6.27e7T^{2} \)
97 \( 1 - 4.50e3iT - 8.85e7T^{2} \)
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

−18.37868500609161538765048465906, −16.74389093083177795672315822550, −16.11585173767681955417669275597, −13.99446951213448126690560807975, −12.13437849913603196735575248873, −11.34757779371856277813639932040, −10.10540088649622426295744926933, −8.281178987279263171910779556090, −6.46338426504014591594221909146, −3.41056851681730072765891112072, 0.12384975899804383246254529761, 5.08271671197240105232596430376, 6.79071511726732262945812171517, 8.451907611045335728797936035910, 9.980896325735731593354135051502, 11.54938853350470945999263035905, 13.00887886046472521987828928432, 15.08538823744293884184113885280, 16.00151315583166038008256044236, 17.09736264201702190614935757230

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