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Properties

Label 2-40-40.3-c1-0-1
Degree $2$
Conductor $40$
Sign $0.453 - 0.891i$
Analytic cond. $0.319401$
Root an. cond. $0.565156$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.221 + 1.39i)2-s + (0.618 + 0.618i)3-s + (−1.90 − 0.618i)4-s + (−1.17 − 1.90i)5-s + (−1 + 0.726i)6-s + (1.90 + 1.90i)7-s + (1.28 − 2.52i)8-s − 2.23i·9-s + (2.91 − 1.22i)10-s − 3.23·11-s + (−0.793 − 1.55i)12-s + (−0.726 + 0.726i)13-s + (−3.07 + 2.23i)14-s + (0.449 − 1.90i)15-s + (3.23 + 2.35i)16-s + (−1 + i)17-s + ⋯
L(s)  = 1  + (−0.156 + 0.987i)2-s + (0.356 + 0.356i)3-s + (−0.951 − 0.309i)4-s + (−0.525 − 0.850i)5-s + (−0.408 + 0.296i)6-s + (0.718 + 0.718i)7-s + (0.453 − 0.891i)8-s − 0.745i·9-s + (0.922 − 0.386i)10-s − 0.975·11-s + (−0.229 − 0.449i)12-s + (−0.201 + 0.201i)13-s + (−0.822 + 0.597i)14-s + (0.115 − 0.491i)15-s + (0.809 + 0.587i)16-s + (−0.242 + 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.453 - 0.891i$
Analytic conductor: \(0.319401\)
Root analytic conductor: \(0.565156\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :1/2),\ 0.453 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.606098 + 0.371417i\)
\(L(\frac12)\) \(\approx\) \(0.606098 + 0.371417i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.221 - 1.39i)T \)
5 \( 1 + (1.17 + 1.90i)T \)
good3 \( 1 + (-0.618 - 0.618i)T + 3iT^{2} \)
7 \( 1 + (-1.90 - 1.90i)T + 7iT^{2} \)
11 \( 1 + 3.23T + 11T^{2} \)
13 \( 1 + (0.726 - 0.726i)T - 13iT^{2} \)
17 \( 1 + (1 - i)T - 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (4.25 - 4.25i)T - 23iT^{2} \)
29 \( 1 - 6.15T + 29T^{2} \)
31 \( 1 + 8.50iT - 31T^{2} \)
37 \( 1 + (-0.726 - 0.726i)T + 37iT^{2} \)
41 \( 1 - 5.70T + 41T^{2} \)
43 \( 1 + (-4.61 - 4.61i)T + 43iT^{2} \)
47 \( 1 + (3.35 + 3.35i)T + 47iT^{2} \)
53 \( 1 + (-3.07 + 3.07i)T - 53iT^{2} \)
59 \( 1 - 0.472iT - 59T^{2} \)
61 \( 1 - 0.898iT - 61T^{2} \)
67 \( 1 + (4.61 - 4.61i)T - 67iT^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (4.70 + 4.70i)T + 73iT^{2} \)
79 \( 1 + 2.90T + 79T^{2} \)
83 \( 1 + (6.61 + 6.61i)T + 83iT^{2} \)
89 \( 1 + 2.47iT - 89T^{2} \)
97 \( 1 + (-4.23 + 4.23i)T - 97iT^{2} \)
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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

−16.09119167977129046229863959410, −15.42432532938327960209002284268, −14.51818335758886467702151179283, −13.07571988894713913965088476824, −11.82593680175864417049077554792, −9.817936982215229398636201059864, −8.663720283320876697815894009651, −7.81073265124145177978484073132, −5.72626834946188853059300205419, −4.32096800543823176313844239520, 2.65222453251457968239482593243, 4.63270301901924420749172501689, 7.44565328305927917049725714879, 8.327322170605221076301393296150, 10.41005753674729245719995953943, 10.92197376614155181270660541103, 12.36288778288061812781763556467, 13.70873965934288771198747065524, 14.36996157508253928053987855012, 16.02921739806235586052959539089

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