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Properties

Label 2-40-40.37-c4-0-8
Degree $2$
Conductor $40$
Sign $0.993 + 0.111i$
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  + (0.134 − 3.99i)2-s + (9.27 + 9.27i)3-s + (−15.9 − 1.07i)4-s + (21.1 + 13.3i)5-s + (38.3 − 35.8i)6-s + (26.2 + 26.2i)7-s + (−6.46 + 63.6i)8-s + 90.9i·9-s + (56.2 − 82.6i)10-s − 212. i·11-s + (−138. − 158. i)12-s + (35.2 + 35.2i)13-s + (108. − 101. i)14-s + (72.1 + 319. i)15-s + (253. + 34.4i)16-s + (−128. − 128. i)17-s + ⋯
L(s)  = 1  + (0.0337 − 0.999i)2-s + (1.03 + 1.03i)3-s + (−0.997 − 0.0674i)4-s + (0.845 + 0.534i)5-s + (1.06 − 0.995i)6-s + (0.536 + 0.536i)7-s + (−0.101 + 0.994i)8-s + 1.12i·9-s + (0.562 − 0.826i)10-s − 1.75i·11-s + (−0.958 − 1.09i)12-s + (0.208 + 0.208i)13-s + (0.553 − 0.517i)14-s + (0.320 + 1.42i)15-s + (0.990 + 0.134i)16-s + (−0.444 − 0.444i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.993 + 0.111i$
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: \((2,\ 40,\ (\ :2),\ 0.993 + 0.111i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.05042 - 0.114210i\)
\(L(\frac12)\) \(\approx\) \(2.05042 - 0.114210i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.134 + 3.99i)T \)
5 \( 1 + (-21.1 - 13.3i)T \)
good3 \( 1 + (-9.27 - 9.27i)T + 81iT^{2} \)
7 \( 1 + (-26.2 - 26.2i)T + 2.40e3iT^{2} \)
11 \( 1 + 212. iT - 1.46e4T^{2} \)
13 \( 1 + (-35.2 - 35.2i)T + 2.85e4iT^{2} \)
17 \( 1 + (128. + 128. i)T + 8.35e4iT^{2} \)
19 \( 1 + 283.T + 1.30e5T^{2} \)
23 \( 1 + (219. - 219. i)T - 2.79e5iT^{2} \)
29 \( 1 - 304.T + 7.07e5T^{2} \)
31 \( 1 + 1.69e3T + 9.23e5T^{2} \)
37 \( 1 + (-1.55e3 + 1.55e3i)T - 1.87e6iT^{2} \)
41 \( 1 + 247.T + 2.82e6T^{2} \)
43 \( 1 + (2.28e3 + 2.28e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-510. - 510. i)T + 4.87e6iT^{2} \)
53 \( 1 + (-1.01e3 - 1.01e3i)T + 7.89e6iT^{2} \)
59 \( 1 - 3.14e3T + 1.21e7T^{2} \)
61 \( 1 - 3.79e3iT - 1.38e7T^{2} \)
67 \( 1 + (413. - 413. i)T - 2.01e7iT^{2} \)
71 \( 1 - 937.T + 2.54e7T^{2} \)
73 \( 1 + (4.21e3 - 4.21e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 719. iT - 3.89e7T^{2} \)
83 \( 1 + (-726. - 726. i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.16e4iT - 6.27e7T^{2} \)
97 \( 1 + (-1.56e3 - 1.56e3i)T + 8.85e7iT^{2} \)
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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

−14.93692118222737884329113880587, −14.15171586113279085630841963450, −13.34103845078999396796384680809, −11.36682039731463242378898941644, −10.48557375442547635265176976463, −9.204051078658884998275501394718, −8.538687624037375172322024990792, −5.56160629507116507654517216870, −3.71202478829666236667575899170, −2.38157833328713891074636289669, 1.74763420893932003897608441522, 4.60432872539886069948841696515, 6.54841016058065918357259120413, 7.70026757408127275859911659167, 8.720408026574279157530531280435, 9.991185172025438248287834553568, 12.70060086187679959635418646067, 13.17599493543944254273027275767, 14.34000306692426687713760348542, 15.00796099701452730592073961606

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