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Properties

Label 2-40-40.37-c4-0-3
Degree $2$
Conductor $40$
Sign $-0.194 - 0.980i$
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  + (−3.82 − 1.16i)2-s + (12.1 + 12.1i)3-s + (13.3 + 8.88i)4-s + (−21.6 + 12.5i)5-s + (−32.4 − 60.7i)6-s + (−17.4 − 17.4i)7-s + (−40.6 − 49.4i)8-s + 215. i·9-s + (97.3 − 22.8i)10-s + 32.1i·11-s + (53.8 + 270. i)12-s + (56.8 + 56.8i)13-s + (46.4 + 86.8i)14-s + (−416. − 110. i)15-s + (98.0 + 236. i)16-s + (−59.1 − 59.1i)17-s + ⋯
L(s)  = 1  + (−0.956 − 0.290i)2-s + (1.35 + 1.35i)3-s + (0.831 + 0.555i)4-s + (−0.865 + 0.501i)5-s + (−0.902 − 1.68i)6-s + (−0.355 − 0.355i)7-s + (−0.634 − 0.772i)8-s + 2.66i·9-s + (0.973 − 0.228i)10-s + 0.265i·11-s + (0.373 + 1.87i)12-s + (0.336 + 0.336i)13-s + (0.236 + 0.442i)14-s + (−1.84 − 0.492i)15-s + (0.382 + 0.923i)16-s + (−0.204 − 0.204i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.194 - 0.980i$
Analytic conductor: \(4.13479\)
Root analytic conductor: \(2.03342\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :2),\ -0.194 - 0.980i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.756275 + 0.920780i\)
\(L(\frac12)\) \(\approx\) \(0.756275 + 0.920780i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.82 + 1.16i)T \)
5 \( 1 + (21.6 - 12.5i)T \)
good3 \( 1 + (-12.1 - 12.1i)T + 81iT^{2} \)
7 \( 1 + (17.4 + 17.4i)T + 2.40e3iT^{2} \)
11 \( 1 - 32.1iT - 1.46e4T^{2} \)
13 \( 1 + (-56.8 - 56.8i)T + 2.85e4iT^{2} \)
17 \( 1 + (59.1 + 59.1i)T + 8.35e4iT^{2} \)
19 \( 1 - 463.T + 1.30e5T^{2} \)
23 \( 1 + (222. - 222. i)T - 2.79e5iT^{2} \)
29 \( 1 - 617.T + 7.07e5T^{2} \)
31 \( 1 - 997.T + 9.23e5T^{2} \)
37 \( 1 + (-991. + 991. i)T - 1.87e6iT^{2} \)
41 \( 1 - 1.13e3T + 2.82e6T^{2} \)
43 \( 1 + (72.8 + 72.8i)T + 3.41e6iT^{2} \)
47 \( 1 + (2.43e3 + 2.43e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (-2.16e3 - 2.16e3i)T + 7.89e6iT^{2} \)
59 \( 1 + 56.0T + 1.21e7T^{2} \)
61 \( 1 - 4.46e3iT - 1.38e7T^{2} \)
67 \( 1 + (3.20e3 - 3.20e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 1.80e3T + 2.54e7T^{2} \)
73 \( 1 + (-4.80e3 + 4.80e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 3.85e3iT - 3.89e7T^{2} \)
83 \( 1 + (3.90e3 + 3.90e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 419. iT - 6.27e7T^{2} \)
97 \( 1 + (6.70e3 + 6.70e3i)T + 8.85e7iT^{2} \)
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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

−15.86266979795938705436357857518, −14.94139438322165993217173365635, −13.67123063721139706841734771616, −11.64186170412263159692241560677, −10.43665101768674302800175173563, −9.589374990259677728219987374556, −8.409378757625731618087671214658, −7.34037282972919079640435972199, −4.03742303951400183321634320772, −2.88510141718386803145177847388, 0.995807661706927715993716055030, 2.96650894307755817128954600838, 6.42953810416136875155841256742, 7.77073726071624941853093867140, 8.404463609666367524458307088459, 9.497218865940438244267469807543, 11.66658606663945732535345467860, 12.66756500010985791562383831818, 13.99087965206748945563261029901, 15.20069800154143124163914940107

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