↑

Properties

Label 2-20-20.7-c9-0-13
Degree $2$
Conductor $20$
Sign $-0.156 + 0.987i$
Analytic cond. $10.3007$
Root an. cond. $3.20947$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−8.80 − 20.8i)2-s + (99.4 + 99.4i)3-s + (−356. + 367. i)4-s + (−1.39e3 + 84.3i)5-s + (1.19e3 − 2.94e3i)6-s + (5.88e3 − 5.88e3i)7-s + (1.07e4 + 4.20e3i)8-s + 86.8i·9-s + (1.40e4 + 2.83e4i)10-s − 3.23e4i·11-s + (−7.19e4 + 1.00e3i)12-s + (1.17e5 − 1.17e5i)13-s + (−1.74e5 − 7.08e4i)14-s + (−1.47e5 − 1.30e5i)15-s + (−7.32e3 − 2.62e5i)16-s + (−9.11e4 − 9.11e4i)17-s + ⋯
L(s)  = 1  + (−0.389 − 0.921i)2-s + (0.708 + 0.708i)3-s + (−0.697 + 0.716i)4-s + (−0.998 + 0.0603i)5-s + (0.377 − 0.928i)6-s + (0.925 − 0.925i)7-s + (0.931 + 0.363i)8-s + 0.00441i·9-s + (0.443 + 0.896i)10-s − 0.665i·11-s + (−1.00 + 0.0139i)12-s + (1.14 − 1.14i)13-s + (−1.21 − 0.492i)14-s + (−0.750 − 0.664i)15-s + (−0.0279 − 0.999i)16-s + (−0.264 − 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(10.3007\)
Root analytic conductor: \(3.20947\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :9/2),\ -0.156 + 0.987i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.917445 - 1.07463i\)
\(L(\frac12)\) \(\approx\) \(0.917445 - 1.07463i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (8.80 + 20.8i)T \)
5 \( 1 + (1.39e3 - 84.3i)T \)
good3 \( 1 + (-99.4 - 99.4i)T + 1.96e4iT^{2} \)
7 \( 1 + (-5.88e3 + 5.88e3i)T - 4.03e7iT^{2} \)
11 \( 1 + 3.23e4iT - 2.35e9T^{2} \)
13 \( 1 + (-1.17e5 + 1.17e5i)T - 1.06e10iT^{2} \)
17 \( 1 + (9.11e4 + 9.11e4i)T + 1.18e11iT^{2} \)
19 \( 1 + 6.11e5T + 3.22e11T^{2} \)
23 \( 1 + (-6.78e5 - 6.78e5i)T + 1.80e12iT^{2} \)
29 \( 1 - 2.06e6iT - 1.45e13T^{2} \)
31 \( 1 + 6.14e6iT - 2.64e13T^{2} \)
37 \( 1 + (-6.20e5 - 6.20e5i)T + 1.29e14iT^{2} \)
41 \( 1 - 2.64e7T + 3.27e14T^{2} \)
43 \( 1 + (2.48e7 + 2.48e7i)T + 5.02e14iT^{2} \)
47 \( 1 + (2.21e7 - 2.21e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (4.15e7 - 4.15e7i)T - 3.29e15iT^{2} \)
59 \( 1 - 6.89e7T + 8.66e15T^{2} \)
61 \( 1 + 5.81e7T + 1.16e16T^{2} \)
67 \( 1 + (-4.07e7 + 4.07e7i)T - 2.72e16iT^{2} \)
71 \( 1 - 2.26e8iT - 4.58e16T^{2} \)
73 \( 1 + (-7.83e7 + 7.83e7i)T - 5.88e16iT^{2} \)
79 \( 1 + 2.64e8T + 1.19e17T^{2} \)
83 \( 1 + (4.17e8 + 4.17e8i)T + 1.86e17iT^{2} \)
89 \( 1 + 5.60e8iT - 3.50e17T^{2} \)
97 \( 1 + (-8.29e8 - 8.29e8i)T + 7.60e17iT^{2} \)
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

−15.78308387252664186875970858921, −14.49644099347399107842726725533, −13.12556634030394623268248305948, −11.31786520120403795128783718944, −10.56905805902972059854394787609, −8.770586784601726429841659850631, −7.87203474583206551748514718704, −4.30463011677857221863448620237, −3.30625148083287825238654203145, −0.76434795111585589981775498410, 1.71185962679133299608633141673, 4.57901685194872329231589227380, 6.74819477982195843968066537295, 8.173389790727521306948456030491, 8.761651592055546844894250566807, 11.15798040147645259682194698921, 12.83059758881825080435027853697, 14.33493513504160646339925370640, 15.15990906632735500821546912600, 16.37887962934979988076545353545

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