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Properties

Label 2-40-5.3-c6-0-3
Degree $2$
Conductor $40$
Sign $0.992 - 0.122i$
Analytic cond. $9.20216$
Root an. cond. $3.03350$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.14 − 4.14i)3-s + (124. + 13.6i)5-s + (−263. + 263. i)7-s − 694. i·9-s + 2.39e3·11-s + (2.40e3 + 2.40e3i)13-s + (−458. − 571. i)15-s + (4.43e3 − 4.43e3i)17-s + 5.08e3i·19-s + 2.18e3·21-s + (2.05e3 + 2.05e3i)23-s + (1.52e4 + 3.39e3i)25-s + (−5.90e3 + 5.90e3i)27-s − 2.73e4i·29-s − 4.30e4·31-s + ⋯
L(s)  = 1  + (−0.153 − 0.153i)3-s + (0.994 + 0.109i)5-s + (−0.767 + 0.767i)7-s − 0.952i·9-s + 1.79·11-s + (1.09 + 1.09i)13-s + (−0.135 − 0.169i)15-s + (0.903 − 0.903i)17-s + 0.741i·19-s + 0.235·21-s + (0.168 + 0.168i)23-s + (0.976 + 0.217i)25-s + (−0.299 + 0.299i)27-s − 1.12i·29-s − 1.44·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.992 - 0.122i$
Analytic conductor: \(9.20216\)
Root analytic conductor: \(3.03350\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :3),\ 0.992 - 0.122i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.95751 + 0.119899i\)
\(L(\frac12)\) \(\approx\) \(1.95751 + 0.119899i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-124. - 13.6i)T \)
good3 \( 1 + (4.14 + 4.14i)T + 729iT^{2} \)
7 \( 1 + (263. - 263. i)T - 1.17e5iT^{2} \)
11 \( 1 - 2.39e3T + 1.77e6T^{2} \)
13 \( 1 + (-2.40e3 - 2.40e3i)T + 4.82e6iT^{2} \)
17 \( 1 + (-4.43e3 + 4.43e3i)T - 2.41e7iT^{2} \)
19 \( 1 - 5.08e3iT - 4.70e7T^{2} \)
23 \( 1 + (-2.05e3 - 2.05e3i)T + 1.48e8iT^{2} \)
29 \( 1 + 2.73e4iT - 5.94e8T^{2} \)
31 \( 1 + 4.30e4T + 8.87e8T^{2} \)
37 \( 1 + (1.49e4 - 1.49e4i)T - 2.56e9iT^{2} \)
41 \( 1 + 1.91e3T + 4.75e9T^{2} \)
43 \( 1 + (1.19e4 + 1.19e4i)T + 6.32e9iT^{2} \)
47 \( 1 + (9.65e4 - 9.65e4i)T - 1.07e10iT^{2} \)
53 \( 1 + (-1.02e5 - 1.02e5i)T + 2.21e10iT^{2} \)
59 \( 1 + 3.06e5iT - 4.21e10T^{2} \)
61 \( 1 + 1.15e5T + 5.15e10T^{2} \)
67 \( 1 + (-1.40e5 + 1.40e5i)T - 9.04e10iT^{2} \)
71 \( 1 - 2.23e5T + 1.28e11T^{2} \)
73 \( 1 + (2.29e5 + 2.29e5i)T + 1.51e11iT^{2} \)
79 \( 1 + 3.85e5iT - 2.43e11T^{2} \)
83 \( 1 + (-2.23e5 - 2.23e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 5.60e5iT - 4.96e11T^{2} \)
97 \( 1 + (9.06e5 - 9.06e5i)T - 8.32e11iT^{2} \)
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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

−14.70831837499956756293149177636, −13.84418929129084663457048660079, −12.42135616410774541450407858789, −11.52675556168102835535422417335, −9.524728484646437336885592105623, −9.142503666570827743794392427177, −6.63080349088727485268236001378, −5.96469506289053131605639818878, −3.56573962604115306973527779447, −1.44380228216493497486693399322, 1.29636525998342777331845641115, 3.62655481126818886374780886628, 5.58907335206264251031681121048, 6.84966718025210249653495222213, 8.731198885725918659326871826202, 10.04410100439890009472367478546, 10.94604340092764703486944018587, 12.77099266535819321558330813306, 13.60084477998087357723346861983, 14.69299694865555531044094679440

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