Properties

Label 2-40-40.29-c5-0-5
Degree $2$
Conductor $40$
Sign $-0.990 - 0.136i$
Analytic cond. $6.41535$
Root an. cond. $2.53285$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.25 + 4.62i)2-s − 1.29·3-s + (−10.8 + 30.1i)4-s + (−51.3 − 21.9i)5-s + (−4.22 − 6.00i)6-s + 170. i·7-s + (−174. + 47.7i)8-s − 241.·9-s + (−65.4 − 309. i)10-s − 39.8i·11-s + (14.0 − 39.0i)12-s + 537.·13-s + (−788. + 554. i)14-s + (66.7 + 28.5i)15-s + (−789. − 652. i)16-s + 1.35e3i·17-s + ⋯
L(s)  = 1  + (0.575 + 0.818i)2-s − 0.0832·3-s + (−0.338 + 0.940i)4-s + (−0.919 − 0.393i)5-s + (−0.0478 − 0.0681i)6-s + 1.31i·7-s + (−0.964 + 0.264i)8-s − 0.993·9-s + (−0.206 − 0.978i)10-s − 0.0992i·11-s + (0.0282 − 0.0783i)12-s + 0.881·13-s + (−1.07 + 0.755i)14-s + (0.0765 + 0.0327i)15-s + (−0.770 − 0.637i)16-s + 1.13i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.136i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.990 - 0.136i$
Analytic conductor: \(6.41535\)
Root analytic conductor: \(2.53285\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :5/2),\ -0.990 - 0.136i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0782754 + 1.13985i\)
\(L(\frac12)\) \(\approx\) \(0.0782754 + 1.13985i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.25 - 4.62i)T \)
5 \( 1 + (51.3 + 21.9i)T \)
good3 \( 1 + 1.29T + 243T^{2} \)
7 \( 1 - 170. iT - 1.68e4T^{2} \)
11 \( 1 + 39.8iT - 1.61e5T^{2} \)
13 \( 1 - 537.T + 3.71e5T^{2} \)
17 \( 1 - 1.35e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.03e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.61e3iT - 6.43e6T^{2} \)
29 \( 1 - 5.91e3iT - 2.05e7T^{2} \)
31 \( 1 - 9.61e3T + 2.86e7T^{2} \)
37 \( 1 - 5.29e3T + 6.93e7T^{2} \)
41 \( 1 + 1.21e4T + 1.15e8T^{2} \)
43 \( 1 + 1.93e4T + 1.47e8T^{2} \)
47 \( 1 + 7.89e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.89e4T + 4.18e8T^{2} \)
59 \( 1 + 3.23e4iT - 7.14e8T^{2} \)
61 \( 1 - 4.88e4iT - 8.44e8T^{2} \)
67 \( 1 - 882.T + 1.35e9T^{2} \)
71 \( 1 + 8.81e3T + 1.80e9T^{2} \)
73 \( 1 + 1.43e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.98e4T + 3.07e9T^{2} \)
83 \( 1 + 1.39e4T + 3.93e9T^{2} \)
89 \( 1 - 7.58e3T + 5.58e9T^{2} \)
97 \( 1 - 6.67e4iT - 8.58e9T^{2} \)
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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

−15.51710426898917526744989440754, −14.89388929034968764078270123311, −13.39172977071340185557618198790, −12.19481426796955848466757806014, −11.39811904243063739376943910417, −8.788970013452624653688526125717, −8.260200957242783895807526977916, −6.33528862834168971237106335048, −5.10068830901643426837830582629, −3.29020830296107357514029930785, 0.54161663991036587798235947678, 3.17671049346862884973479420855, 4.48259969964408069295435128585, 6.46643541267537917551314523457, 8.195689399627203890436463479205, 10.07358223969028083824596944115, 11.15985648678301127157693059355, 11.89862380621203462896492517680, 13.53107280115793823366085160697, 14.24640713146177594601730572879

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