↑

Properties

Label 2-40-40.13-c4-0-13
Degree $2$
Conductor $40$
Sign $0.816 + 0.577i$
Analytic cond. $4.13479$
Root an. cond. $2.03342$
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  + (3.46 − 2.00i)2-s + (−4.26 + 4.26i)3-s + (7.96 − 13.8i)4-s + (24.9 + 1.64i)5-s + (−6.21 + 23.3i)6-s + (50.3 − 50.3i)7-s + (−0.223 − 63.9i)8-s + 44.5i·9-s + (89.6 − 44.2i)10-s − 40.0i·11-s + (25.2 + 93.1i)12-s + (−198. + 198. i)13-s + (73.4 − 275. i)14-s + (−113. + 99.3i)15-s + (−129. − 221. i)16-s + (−45.5 + 45.5i)17-s + ⋯
L(s)  = 1  + (0.865 − 0.501i)2-s + (−0.474 + 0.474i)3-s + (0.497 − 0.867i)4-s + (0.997 + 0.0659i)5-s + (−0.172 + 0.647i)6-s + (1.02 − 1.02i)7-s + (−0.00348 − 0.999i)8-s + 0.550i·9-s + (0.896 − 0.442i)10-s − 0.330i·11-s + (0.175 + 0.647i)12-s + (−1.17 + 1.17i)13-s + (0.374 − 1.40i)14-s + (−0.504 + 0.441i)15-s + (−0.504 − 0.863i)16-s + (−0.157 + 0.157i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(4.13479\)
Root analytic conductor: \(2.03342\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :2),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.25022 - 0.715969i\)
\(L(\frac12)\) \(\approx\) \(2.25022 - 0.715969i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.46 + 2.00i)T \)
5 \( 1 + (-24.9 - 1.64i)T \)
good3 \( 1 + (4.26 - 4.26i)T - 81iT^{2} \)
7 \( 1 + (-50.3 + 50.3i)T - 2.40e3iT^{2} \)
11 \( 1 + 40.0iT - 1.46e4T^{2} \)
13 \( 1 + (198. - 198. i)T - 2.85e4iT^{2} \)
17 \( 1 + (45.5 - 45.5i)T - 8.35e4iT^{2} \)
19 \( 1 + 184.T + 1.30e5T^{2} \)
23 \( 1 + (-409. - 409. i)T + 2.79e5iT^{2} \)
29 \( 1 + 925.T + 7.07e5T^{2} \)
31 \( 1 + 1.25e3T + 9.23e5T^{2} \)
37 \( 1 + (-1.47e3 - 1.47e3i)T + 1.87e6iT^{2} \)
41 \( 1 + 405.T + 2.82e6T^{2} \)
43 \( 1 + (-1.54e3 + 1.54e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (402. - 402. i)T - 4.87e6iT^{2} \)
53 \( 1 + (-1.79e3 + 1.79e3i)T - 7.89e6iT^{2} \)
59 \( 1 + 840.T + 1.21e7T^{2} \)
61 \( 1 + 224. iT - 1.38e7T^{2} \)
67 \( 1 + (278. + 278. i)T + 2.01e7iT^{2} \)
71 \( 1 + 1.20e3T + 2.54e7T^{2} \)
73 \( 1 + (-4.35e3 - 4.35e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 1.83e3iT - 3.89e7T^{2} \)
83 \( 1 + (2.67e3 - 2.67e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 9.97e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.65e3 + 1.65e3i)T - 8.85e7iT^{2} \)
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

−14.85685994927990090554205900002, −14.04604762494784542644746824028, −13.13870208694788733053917680697, −11.41387463155512379817490319057, −10.72882739869360299064493946402, −9.585831733448751171624840761144, −7.14495491743405525713236473833, −5.42699210193773678703486849599, −4.39578557131910090023251472539, −1.88383696289884341101180639943, 2.32796926524308499322028503351, 5.07119687557906040278004547461, 5.95487297915740074065776038473, 7.45384456034101257497544640045, 9.077760481882279790115874465876, 11.05453141465311668214300241653, 12.39437882870811813635921069310, 12.91584847698391125348522679063, 14.70905551339780525331312744330, 14.95078791300481707093483355810

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