Properties

Label 2-40-5.2-c4-0-2
Degree $2$
Conductor $40$
Sign $0.875 - 0.482i$
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 + (18.5 − 16.7i)5-s + (64.8 + 64.8i)7-s + 63i·9-s + 33.7·11-s + (132. − 132. i)13-s + (−5.31 + 105. i)15-s + (−270. − 270. i)17-s + 295. i·19-s − 389.·21-s + (1.44 − 1.44i)23-s + (62.5 − 621. i)25-s + (−432 − 432i)27-s + 297. i·29-s − 1.22e3·31-s + ⋯
L(s)  = 1  + (−0.333 + 0.333i)3-s + (0.741 − 0.670i)5-s + (1.32 + 1.32i)7-s + 0.777i·9-s + 0.278·11-s + (0.786 − 0.786i)13-s + (−0.0236 + 0.470i)15-s + (−0.935 − 0.935i)17-s + 0.819i·19-s − 0.882·21-s + (0.00273 − 0.00273i)23-s + (0.100 − 0.994i)25-s + (−0.592 − 0.592i)27-s + 0.353i·29-s − 1.27·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.482i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.875 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.875 - 0.482i$
Analytic conductor: \(4.13479\)
Root analytic conductor: \(2.03342\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :2),\ 0.875 - 0.482i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.56959 + 0.403694i\)
\(L(\frac12)\) \(\approx\) \(1.56959 + 0.403694i\)
\(L(3)\) 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 \( 1 + (-18.5 + 16.7i)T \)
good3 \( 1 + (3 - 3i)T - 81iT^{2} \)
7 \( 1 + (-64.8 - 64.8i)T + 2.40e3iT^{2} \)
11 \( 1 - 33.7T + 1.46e4T^{2} \)
13 \( 1 + (-132. + 132. i)T - 2.85e4iT^{2} \)
17 \( 1 + (270. + 270. i)T + 8.35e4iT^{2} \)
19 \( 1 - 295. iT - 1.30e5T^{2} \)
23 \( 1 + (-1.44 + 1.44i)T - 2.79e5iT^{2} \)
29 \( 1 - 297. iT - 7.07e5T^{2} \)
31 \( 1 + 1.22e3T + 9.23e5T^{2} \)
37 \( 1 + (-402. - 402. i)T + 1.87e6iT^{2} \)
41 \( 1 - 860.T + 2.82e6T^{2} \)
43 \( 1 + (-753. + 753. i)T - 3.41e6iT^{2} \)
47 \( 1 + (2.22e3 + 2.22e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (-2.53e3 + 2.53e3i)T - 7.89e6iT^{2} \)
59 \( 1 + 3.42e3iT - 1.21e7T^{2} \)
61 \( 1 + 6.69e3T + 1.38e7T^{2} \)
67 \( 1 + (318. + 318. i)T + 2.01e7iT^{2} \)
71 \( 1 + 3.42e3T + 2.54e7T^{2} \)
73 \( 1 + (-3.05e3 + 3.05e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 2.63e3iT - 3.89e7T^{2} \)
83 \( 1 + (-225. + 225. i)T - 4.74e7iT^{2} \)
89 \( 1 + 4.98e3iT - 6.27e7T^{2} \)
97 \( 1 + (-4.89e3 - 4.89e3i)T + 8.85e7iT^{2} \)
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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

−15.59722169789954559326571312925, −14.35589274219989844086552555985, −13.13475530384620722794169789116, −11.79760149885992364698411041259, −10.74467976594632696587611396444, −9.141931480489875094216659074786, −8.145413864702020628307011276755, −5.74324186231419842167196673709, −4.90734721876261123874445241170, −1.94705735032545744072602092616, 1.47994745097478402845102598562, 4.16454323899122216656106174179, 6.22141577204021419068447328967, 7.30230905447808388992626924352, 9.077800964433028957343356334296, 10.75797221882905724367195367605, 11.35561803397674411099398316399, 13.16816156033721220578621088745, 14.11997932156369572017914000293, 15.01402937983055434311282623346

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