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Properties

Label 2-40-40.13-c4-0-5
Degree $2$
Conductor $40$
Sign $-0.438 - 0.898i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.98 + 0.381i)2-s + (−3.10 + 3.10i)3-s + (15.7 − 3.04i)4-s + (18.7 − 16.4i)5-s + (11.1 − 13.5i)6-s + (−52.9 + 52.9i)7-s + (−61.3 + 18.1i)8-s + 61.7i·9-s + (−68.4 + 72.8i)10-s + 95.9i·11-s + (−39.3 + 58.1i)12-s + (−23.7 + 23.7i)13-s + (190. − 231. i)14-s + (−7.09 + 109. i)15-s + (237. − 95.5i)16-s + (−187. + 187. i)17-s + ⋯
L(s)  = 1  + (−0.995 + 0.0954i)2-s + (−0.344 + 0.344i)3-s + (0.981 − 0.190i)4-s + (0.751 − 0.659i)5-s + (0.310 − 0.376i)6-s + (−1.08 + 1.08i)7-s + (−0.959 + 0.282i)8-s + 0.762i·9-s + (−0.684 + 0.728i)10-s + 0.793i·11-s + (−0.272 + 0.404i)12-s + (−0.140 + 0.140i)13-s + (0.973 − 1.17i)14-s + (−0.0315 + 0.486i)15-s + (0.927 − 0.373i)16-s + (−0.650 + 0.650i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.438 - 0.898i$
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: \((2,\ 40,\ (\ :2),\ -0.438 - 0.898i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.338026 + 0.540986i\)
\(L(\frac12)\) \(\approx\) \(0.338026 + 0.540986i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.98 - 0.381i)T \)
5 \( 1 + (-18.7 + 16.4i)T \)
good3 \( 1 + (3.10 - 3.10i)T - 81iT^{2} \)
7 \( 1 + (52.9 - 52.9i)T - 2.40e3iT^{2} \)
11 \( 1 - 95.9iT - 1.46e4T^{2} \)
13 \( 1 + (23.7 - 23.7i)T - 2.85e4iT^{2} \)
17 \( 1 + (187. - 187. i)T - 8.35e4iT^{2} \)
19 \( 1 + 312.T + 1.30e5T^{2} \)
23 \( 1 + (-658. - 658. i)T + 2.79e5iT^{2} \)
29 \( 1 - 1.30e3T + 7.07e5T^{2} \)
31 \( 1 + 215.T + 9.23e5T^{2} \)
37 \( 1 + (886. + 886. i)T + 1.87e6iT^{2} \)
41 \( 1 + 1.20e3T + 2.82e6T^{2} \)
43 \( 1 + (413. - 413. i)T - 3.41e6iT^{2} \)
47 \( 1 + (-738. + 738. i)T - 4.87e6iT^{2} \)
53 \( 1 + (-2.78e3 + 2.78e3i)T - 7.89e6iT^{2} \)
59 \( 1 + 1.38e3T + 1.21e7T^{2} \)
61 \( 1 - 2.79e3iT - 1.38e7T^{2} \)
67 \( 1 + (-436. - 436. i)T + 2.01e7iT^{2} \)
71 \( 1 - 1.91e3T + 2.54e7T^{2} \)
73 \( 1 + (-5.03e3 - 5.03e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 1.41e3iT - 3.89e7T^{2} \)
83 \( 1 + (-7.06e3 + 7.06e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.28e4iT - 6.27e7T^{2} \)
97 \( 1 + (1.95e3 - 1.95e3i)T - 8.85e7iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.99523764209238704330679153666, −15.16367535122219486063578673199, −13.16702548765672696083428426576, −12.10779850054071480187210986906, −10.53176601670256888483067655385, −9.542104841195658655425069440897, −8.601388341292721325862026988931, −6.63014094914311643389232623750, −5.32609159648351758804583415461, −2.19133180463632419555736541333, 0.58198796417165842930661877817, 3.05045050770971953877263433330, 6.43354217510283096881606481521, 6.88124225006920546690960916710, 8.922803449427600220201087445591, 10.15786909766299248936054317047, 10.97333635702867442279940250851, 12.53728192716359767634949248498, 13.74210990131374779017170910566, 15.26374060584070047496190679273

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