↑

Properties

Label 2-40-40.27-c5-0-25
Degree $2$
Conductor $40$
Sign $-0.943 + 0.330i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−4.60 + 3.28i)2-s + (11.5 − 11.5i)3-s + (10.4 − 30.2i)4-s + (−48.9 − 26.9i)5-s + (−15.3 + 91.4i)6-s + (−75.8 + 75.8i)7-s + (51.2 + 173. i)8-s − 25.8i·9-s + (314. − 36.5i)10-s − 570.·11-s + (−229. − 471. i)12-s + (−373. − 373. i)13-s + (100. − 598. i)14-s + (−880. + 255. i)15-s + (−806. − 631. i)16-s + (120. + 120. i)17-s + ⋯
L(s)  = 1  + (−0.814 + 0.580i)2-s + (0.743 − 0.743i)3-s + (0.326 − 0.945i)4-s + (−0.875 − 0.482i)5-s + (−0.173 + 1.03i)6-s + (−0.585 + 0.585i)7-s + (0.282 + 0.959i)8-s − 0.106i·9-s + (0.993 − 0.115i)10-s − 1.42·11-s + (−0.460 − 0.945i)12-s + (−0.613 − 0.613i)13-s + (0.136 − 0.816i)14-s + (−1.01 + 0.292i)15-s + (−0.787 − 0.616i)16-s + (0.100 + 0.100i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.943 + 0.330i$
Analytic conductor: \(6.41535\)
Root analytic conductor: \(2.53285\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :5/2),\ -0.943 + 0.330i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0417076 - 0.245081i\)
\(L(\frac12)\) \(\approx\) \(0.0417076 - 0.245081i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.60 - 3.28i)T \)
5 \( 1 + (48.9 + 26.9i)T \)
good3 \( 1 + (-11.5 + 11.5i)T - 243iT^{2} \)
7 \( 1 + (75.8 - 75.8i)T - 1.68e4iT^{2} \)
11 \( 1 + 570.T + 1.61e5T^{2} \)
13 \( 1 + (373. + 373. i)T + 3.71e5iT^{2} \)
17 \( 1 + (-120. - 120. i)T + 1.41e6iT^{2} \)
19 \( 1 + 1.01e3iT - 2.47e6T^{2} \)
23 \( 1 + (785. + 785. i)T + 6.43e6iT^{2} \)
29 \( 1 + 6.33e3T + 2.05e7T^{2} \)
31 \( 1 + 2.42e3iT - 2.86e7T^{2} \)
37 \( 1 + (-6.30e3 + 6.30e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 1.84e4T + 1.15e8T^{2} \)
43 \( 1 + (-1.59e4 + 1.59e4i)T - 1.47e8iT^{2} \)
47 \( 1 + (-1.02e3 + 1.02e3i)T - 2.29e8iT^{2} \)
53 \( 1 + (-2.33e4 - 2.33e4i)T + 4.18e8iT^{2} \)
59 \( 1 + 2.55e4iT - 7.14e8T^{2} \)
61 \( 1 - 561. iT - 8.44e8T^{2} \)
67 \( 1 + (2.43e4 + 2.43e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 6.37e4iT - 1.80e9T^{2} \)
73 \( 1 + (7.25e3 - 7.25e3i)T - 2.07e9iT^{2} \)
79 \( 1 - 2.81e4T + 3.07e9T^{2} \)
83 \( 1 + (7.17e4 - 7.17e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 2.59e4iT - 5.58e9T^{2} \)
97 \( 1 + (7.28e4 + 7.28e4i)T + 8.58e9iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.03058946319430983245756003221, −13.42405747063475098185209242291, −12.41522529482283230242436483679, −10.72362826310070424210314654094, −9.159575281038821906025895690078, −8.040287678366261528225445321791, −7.34585623540190646571934212566, −5.37510613620764585729195143484, −2.49726658127808647664741146386, −0.15435856216792749272513597383, 2.88565796087569863794817223949, 4.02706905909611864868533450743, 7.17400567844135278209782385353, 8.249436216006066485428230642837, 9.704775588863941443512031096710, 10.45981979738330472680696960338, 11.79838213700117877101964136827, 13.17188964571886934526418886957, 14.75314877325995723877528372496, 15.82572991087802890657911248228

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