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Properties

Label 2-40-8.5-c5-0-12
Degree $2$
Conductor $40$
Sign $0.945 - 0.326i$
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  + (5.18 + 2.26i)2-s − 10.8i·3-s + (21.7 + 23.5i)4-s + 25i·5-s + (24.5 − 56.0i)6-s + 163.·7-s + (59.0 + 171. i)8-s + 125.·9-s + (−56.7 + 129. i)10-s − 321. i·11-s + (254. − 234. i)12-s − 128. i·13-s + (848. + 371. i)14-s + 270.·15-s + (−82.2 + 1.02e3i)16-s − 2.11e3·17-s + ⋯
L(s)  = 1  + (0.916 + 0.401i)2-s − 0.694i·3-s + (0.678 + 0.734i)4-s + 0.447i·5-s + (0.278 − 0.636i)6-s + 1.26·7-s + (0.326 + 0.945i)8-s + 0.517·9-s + (−0.179 + 0.409i)10-s − 0.801i·11-s + (0.510 − 0.470i)12-s − 0.210i·13-s + (1.15 + 0.506i)14-s + 0.310·15-s + (−0.0802 + 0.996i)16-s − 1.77·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.945 - 0.326i$
Analytic conductor: \(6.41535\)
Root analytic conductor: \(2.53285\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :5/2),\ 0.945 - 0.326i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.89643 + 0.485904i\)
\(L(\frac12)\) \(\approx\) \(2.89643 + 0.485904i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.18 - 2.26i)T \)
5 \( 1 - 25iT \)
good3 \( 1 + 10.8iT - 243T^{2} \)
7 \( 1 - 163.T + 1.68e4T^{2} \)
11 \( 1 + 321. iT - 1.61e5T^{2} \)
13 \( 1 + 128. iT - 3.71e5T^{2} \)
17 \( 1 + 2.11e3T + 1.41e6T^{2} \)
19 \( 1 - 1.45e3iT - 2.47e6T^{2} \)
23 \( 1 + 1.23e3T + 6.43e6T^{2} \)
29 \( 1 + 4.07e3iT - 2.05e7T^{2} \)
31 \( 1 + 3.95e3T + 2.86e7T^{2} \)
37 \( 1 + 1.06e4iT - 6.93e7T^{2} \)
41 \( 1 + 5.90e3T + 1.15e8T^{2} \)
43 \( 1 - 1.64e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.32e4T + 2.29e8T^{2} \)
53 \( 1 + 3.06e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.52e4iT - 7.14e8T^{2} \)
61 \( 1 - 3.91e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.08e4iT - 1.35e9T^{2} \)
71 \( 1 - 1.38e4T + 1.80e9T^{2} \)
73 \( 1 + 4.34e4T + 2.07e9T^{2} \)
79 \( 1 + 1.25e4T + 3.07e9T^{2} \)
83 \( 1 - 6.68e3iT - 3.93e9T^{2} \)
89 \( 1 + 9.04e4T + 5.58e9T^{2} \)
97 \( 1 - 1.49e5T + 8.58e9T^{2} \)
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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

−14.97851372459763404962210368324, −13.98339746519964938259100307875, −13.08657516563342074336351834029, −11.76936898636663582347080464838, −10.82725405574746679456243909915, −8.333160160210301216437431003078, −7.28810874557694058423842226091, −5.94865955057077434870041459250, −4.22625133240152247630448011255, −2.05107904257099029181520697953, 1.85198993067242620407059917037, 4.29779487810107692870137103651, 4.98114516788141715119206867846, 7.06141438047157465259765250740, 9.049492131689540386901768102978, 10.49629057476465266884988488523, 11.45969168745184979157955690864, 12.72957821055764041579730998104, 13.90234733526450390365291262893, 15.16865175963323760124973902657

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