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Properties

Label 2-40-40.13-c2-0-6
Degree $2$
Conductor $40$
Sign $0.899 - 0.436i$
Analytic cond. $1.08992$
Root an. cond. $1.04399$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.18 + 1.60i)2-s + (2.52 − 2.52i)3-s + (−1.17 + 3.82i)4-s + (−3.09 − 3.92i)5-s + (7.06 + 1.05i)6-s + (−5.20 + 5.20i)7-s + (−7.54 + 2.66i)8-s − 3.76i·9-s + (2.64 − 9.64i)10-s + 2.49i·11-s + (6.69 + 12.6i)12-s + (6.65 − 6.65i)13-s + (−14.5 − 2.18i)14-s + (−17.7 − 2.11i)15-s + (−13.2 − 8.97i)16-s + (21.9 − 21.9i)17-s + ⋯
L(s)  = 1  + (0.594 + 0.804i)2-s + (0.842 − 0.842i)3-s + (−0.293 + 0.956i)4-s + (−0.618 − 0.785i)5-s + (1.17 + 0.176i)6-s + (−0.743 + 0.743i)7-s + (−0.943 + 0.332i)8-s − 0.418i·9-s + (0.264 − 0.964i)10-s + 0.226i·11-s + (0.558 + 1.05i)12-s + (0.512 − 0.512i)13-s + (−1.03 − 0.155i)14-s + (−1.18 − 0.141i)15-s + (−0.828 − 0.560i)16-s + (1.29 − 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.899 - 0.436i$
Analytic conductor: \(1.08992\)
Root analytic conductor: \(1.04399\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :1),\ 0.899 - 0.436i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.43893 + 0.330861i\)
\(L(\frac12)\) \(\approx\) \(1.43893 + 0.330861i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.18 - 1.60i)T \)
5 \( 1 + (3.09 + 3.92i)T \)
good3 \( 1 + (-2.52 + 2.52i)T - 9iT^{2} \)
7 \( 1 + (5.20 - 5.20i)T - 49iT^{2} \)
11 \( 1 - 2.49iT - 121T^{2} \)
13 \( 1 + (-6.65 + 6.65i)T - 169iT^{2} \)
17 \( 1 + (-21.9 + 21.9i)T - 289iT^{2} \)
19 \( 1 + 17.5T + 361T^{2} \)
23 \( 1 + (-20.1 - 20.1i)T + 529iT^{2} \)
29 \( 1 + 1.04T + 841T^{2} \)
31 \( 1 + 2.47T + 961T^{2} \)
37 \( 1 + (19.2 + 19.2i)T + 1.36e3iT^{2} \)
41 \( 1 + 43.9T + 1.68e3T^{2} \)
43 \( 1 + (32.9 - 32.9i)T - 1.84e3iT^{2} \)
47 \( 1 + (-33.7 + 33.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (-36.4 + 36.4i)T - 2.80e3iT^{2} \)
59 \( 1 + 30.5T + 3.48e3T^{2} \)
61 \( 1 - 23.8iT - 3.72e3T^{2} \)
67 \( 1 + (19.9 + 19.9i)T + 4.48e3iT^{2} \)
71 \( 1 + 21.8T + 5.04e3T^{2} \)
73 \( 1 + (-32.6 - 32.6i)T + 5.32e3iT^{2} \)
79 \( 1 - 16.4iT - 6.24e3T^{2} \)
83 \( 1 + (-0.343 + 0.343i)T - 6.88e3iT^{2} \)
89 \( 1 + 84.4iT - 7.92e3T^{2} \)
97 \( 1 + (24.1 - 24.1i)T - 9.40e3iT^{2} \)
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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

−15.84030202047066593324507110573, −14.90730142230624860600724053332, −13.55085052537022983237498768690, −12.80884054286239106871934186677, −11.93756454305683503936701996746, −9.173246654598166850083509066744, −8.198928091179992570222205544863, −7.11946518784197467704180179058, −5.34796711684067009201396636869, −3.21761200666604236228724004921, 3.25704489048730550819895918833, 4.06677324646513843167584780809, 6.51513774141682723798911451245, 8.623700625933676433389950330639, 10.12089521834353267763258307248, 10.72799434176907634526547858019, 12.31413818100791891379532097919, 13.68397210284349973270825164561, 14.67422448835610977084634099546, 15.34852385474063100873253512070

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