↑

Properties

Label 2-40-8.5-c5-0-8
Degree $2$
Conductor $40$
Sign $0.113 - 0.993i$
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  + (5.65 + 0.214i)2-s + 18.7i·3-s + (31.9 + 2.42i)4-s + 25i·5-s + (−4.03 + 106. i)6-s − 107.·7-s + (179. + 20.5i)8-s − 109.·9-s + (−5.36 + 141. i)10-s + 272. i·11-s + (−45.5 + 599. i)12-s − 198. i·13-s + (−607. − 23.0i)14-s − 469.·15-s + (1.01e3 + 154. i)16-s + 2.06e3·17-s + ⋯
L(s)  = 1  + (0.999 + 0.0379i)2-s + 1.20i·3-s + (0.997 + 0.0757i)4-s + 0.447i·5-s + (−0.0457 + 1.20i)6-s − 0.829·7-s + (0.993 + 0.113i)8-s − 0.452·9-s + (−0.0169 + 0.446i)10-s + 0.678i·11-s + (−0.0913 + 1.20i)12-s − 0.325i·13-s + (−0.828 − 0.0314i)14-s − 0.538·15-s + (0.988 + 0.151i)16-s + 1.73·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.113 - 0.993i$
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: \((2,\ 40,\ (\ :5/2),\ 0.113 - 0.993i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.06815 + 1.84524i\)
\(L(\frac12)\) \(\approx\) \(2.06815 + 1.84524i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.65 - 0.214i)T \)
5 \( 1 - 25iT \)
good3 \( 1 - 18.7iT - 243T^{2} \)
7 \( 1 + 107.T + 1.68e4T^{2} \)
11 \( 1 - 272. iT - 1.61e5T^{2} \)
13 \( 1 + 198. iT - 3.71e5T^{2} \)
17 \( 1 - 2.06e3T + 1.41e6T^{2} \)
19 \( 1 + 1.89e3iT - 2.47e6T^{2} \)
23 \( 1 + 987.T + 6.43e6T^{2} \)
29 \( 1 + 8.01e3iT - 2.05e7T^{2} \)
31 \( 1 - 827.T + 2.86e7T^{2} \)
37 \( 1 + 9.42e3iT - 6.93e7T^{2} \)
41 \( 1 + 8.22e3T + 1.15e8T^{2} \)
43 \( 1 - 9.30e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.38e4T + 2.29e8T^{2} \)
53 \( 1 - 2.77e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.51e4iT - 7.14e8T^{2} \)
61 \( 1 + 2.64e4iT - 8.44e8T^{2} \)
67 \( 1 - 3.85e4iT - 1.35e9T^{2} \)
71 \( 1 + 7.10e4T + 1.80e9T^{2} \)
73 \( 1 - 1.86e4T + 2.07e9T^{2} \)
79 \( 1 + 7.55e4T + 3.07e9T^{2} \)
83 \( 1 + 1.25e5iT - 3.93e9T^{2} \)
89 \( 1 - 3.03e4T + 5.58e9T^{2} \)
97 \( 1 - 1.56e4T + 8.58e9T^{2} \)
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

−15.38937361700400351641546521128, −14.47783595777758704725784933950, −13.14254587119782950182781986059, −11.89088350378391249443911396928, −10.48101279123838092589358593120, −9.693708107433917985657486203320, −7.38725626505603359987221098558, −5.80543402466692591856624172192, −4.30648586362947197258287218510, −3.01074785981801008334792860425, 1.38196599346536565210767609909, 3.40461411700401652480889856081, 5.60900084173551514360781797109, 6.76632136243214221371148098716, 8.058028911296737497962023160205, 10.14749211548815230313873758640, 11.95840752507810315439241074446, 12.53545714483357837271779794177, 13.53101307125176870042151011827, 14.43686829562530717398079799775

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