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Properties

Label 2-40-8.5-c5-0-19
Degree $2$
Conductor $40$
Sign $-0.918 - 0.395i$
Analytic cond. $6.41535$
Root an. cond. $2.53285$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.765 − 5.60i)2-s − 17.3i·3-s + (−30.8 − 8.57i)4-s − 25i·5-s + (−97.0 − 13.2i)6-s − 9.19·7-s + (−71.6 + 166. i)8-s − 56.8·9-s + (−140. − 19.1i)10-s − 160. i·11-s + (−148. + 533. i)12-s + 368. i·13-s + (−7.03 + 51.5i)14-s − 432.·15-s + (876. + 528. i)16-s − 1.26e3·17-s + ⋯
L(s)  = 1  + (0.135 − 0.990i)2-s − 1.11i·3-s + (−0.963 − 0.268i)4-s − 0.447i·5-s + (−1.10 − 0.150i)6-s − 0.0708·7-s + (−0.395 + 0.918i)8-s − 0.233·9-s + (−0.443 − 0.0604i)10-s − 0.399i·11-s + (−0.297 + 1.07i)12-s + 0.604i·13-s + (−0.00958 + 0.0702i)14-s − 0.496·15-s + (0.856 + 0.516i)16-s − 1.05·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.918 - 0.395i$
Analytic conductor: \(6.41535\)
Root analytic conductor: \(2.53285\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :5/2),\ -0.918 - 0.395i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.251481 + 1.21852i\)
\(L(\frac12)\) \(\approx\) \(0.251481 + 1.21852i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.765 + 5.60i)T \)
5 \( 1 + 25iT \)
good3 \( 1 + 17.3iT - 243T^{2} \)
7 \( 1 + 9.19T + 1.68e4T^{2} \)
11 \( 1 + 160. iT - 1.61e5T^{2} \)
13 \( 1 - 368. iT - 3.71e5T^{2} \)
17 \( 1 + 1.26e3T + 1.41e6T^{2} \)
19 \( 1 + 2.48e3iT - 2.47e6T^{2} \)
23 \( 1 + 422.T + 6.43e6T^{2} \)
29 \( 1 + 5.66e3iT - 2.05e7T^{2} \)
31 \( 1 - 9.38e3T + 2.86e7T^{2} \)
37 \( 1 + 3.56e3iT - 6.93e7T^{2} \)
41 \( 1 + 5.94e3T + 1.15e8T^{2} \)
43 \( 1 - 1.06e4iT - 1.47e8T^{2} \)
47 \( 1 - 9.24e3T + 2.29e8T^{2} \)
53 \( 1 - 8.97e3iT - 4.18e8T^{2} \)
59 \( 1 + 2.74e4iT - 7.14e8T^{2} \)
61 \( 1 - 5.05e4iT - 8.44e8T^{2} \)
67 \( 1 + 5.96e3iT - 1.35e9T^{2} \)
71 \( 1 - 6.72e4T + 1.80e9T^{2} \)
73 \( 1 - 8.57e4T + 2.07e9T^{2} \)
79 \( 1 - 5.65e4T + 3.07e9T^{2} \)
83 \( 1 + 3.02e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.13e5T + 5.58e9T^{2} \)
97 \( 1 + 1.38e5T + 8.58e9T^{2} \)
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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

−13.74034248867147023610426526881, −13.28226565804638980831587232350, −12.12388599050166477140121135457, −11.18023072865742854293285037914, −9.490705227205113866218250110034, −8.251764503294293200993432746839, −6.48948822270616495069229571163, −4.53643788785416897609963132828, −2.32198114007091358806659784116, −0.69280420922858683866645964031, 3.68257727971470362261367562087, 5.03310369004182162750710361338, 6.61428942665751244713049649166, 8.188494609838226486908530402234, 9.626090344103047718901483784114, 10.54459785363828731243801764390, 12.44368354577939447796673037125, 13.87257186900404214230338944171, 15.01040813284074318794588460803, 15.62491546472039742523793433575

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