Properties

Label 2-40-5.3-c4-0-5
Degree $2$
Conductor $40$
Sign $-0.999 + 0.0398i$
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 − 3i)3-s + (−24.5 − 4.77i)5-s + (−42.8 + 42.8i)7-s − 63i·9-s − 181.·11-s + (25.1 + 25.1i)13-s + (59.3 + 87.9i)15-s + (160. − 160. i)17-s − 80.2i·19-s + 257.·21-s + (324. + 324. i)23-s + (579. + 234. i)25-s + (−432 + 432i)27-s − 1.37e3i·29-s − 151.·31-s + ⋯
L(s)  = 1  + (−0.333 − 0.333i)3-s + (−0.981 − 0.190i)5-s + (−0.874 + 0.874i)7-s − 0.777i·9-s − 1.50·11-s + (0.148 + 0.148i)13-s + (0.263 + 0.390i)15-s + (0.555 − 0.555i)17-s − 0.222i·19-s + 0.583·21-s + (0.613 + 0.613i)23-s + (0.927 + 0.374i)25-s + (−0.592 + 0.592i)27-s − 1.63i·29-s − 0.157·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0398i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.999 + 0.0398i$
Analytic conductor: \(4.13479\)
Root analytic conductor: \(2.03342\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :2),\ -0.999 + 0.0398i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.00305317 - 0.153277i\)
\(L(\frac12)\) \(\approx\) \(0.00305317 - 0.153277i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (24.5 + 4.77i)T \)
good3 \( 1 + (3 + 3i)T + 81iT^{2} \)
7 \( 1 + (42.8 - 42.8i)T - 2.40e3iT^{2} \)
11 \( 1 + 181.T + 1.46e4T^{2} \)
13 \( 1 + (-25.1 - 25.1i)T + 2.85e4iT^{2} \)
17 \( 1 + (-160. + 160. i)T - 8.35e4iT^{2} \)
19 \( 1 + 80.2iT - 1.30e5T^{2} \)
23 \( 1 + (-324. - 324. i)T + 2.79e5iT^{2} \)
29 \( 1 + 1.37e3iT - 7.07e5T^{2} \)
31 \( 1 + 151.T + 9.23e5T^{2} \)
37 \( 1 + (1.42e3 - 1.42e3i)T - 1.87e6iT^{2} \)
41 \( 1 + 3.01e3T + 2.82e6T^{2} \)
43 \( 1 + (1.61e3 + 1.61e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (397. - 397. i)T - 4.87e6iT^{2} \)
53 \( 1 + (-923. - 923. i)T + 7.89e6iT^{2} \)
59 \( 1 + 4.54e3iT - 1.21e7T^{2} \)
61 \( 1 - 631.T + 1.38e7T^{2} \)
67 \( 1 + (2.68e3 - 2.68e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 4.54e3T + 2.54e7T^{2} \)
73 \( 1 + (-5.85e3 - 5.85e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 9.42e3iT - 3.89e7T^{2} \)
83 \( 1 + (8.39e3 + 8.39e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 8.36e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.01e3 + 1.01e3i)T - 8.85e7iT^{2} \)
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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

−15.36870719810951495415703578582, −13.33131846953000880109084325293, −12.34098334776695619825621744517, −11.51460292565724222219108446617, −9.806615495525444337359550466908, −8.378993031342201575205071941172, −6.93762712238470241735905450364, −5.36999767415097759849494969725, −3.19811716933352371559581142205, −0.10183671525259519435567250948, 3.39775794188556722192905451134, 5.08147379778681274403972326026, 7.06822993089736014769140693933, 8.221926532255516463626822504672, 10.33476274392271626657055836712, 10.79385868558341383901081913603, 12.48818341693679943901071384733, 13.49253299819088164243605540717, 15.04889298108895116932348428794, 16.19779527696963708557387102417

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