↑

Properties

Label 2-40-40.13-c4-0-12
Degree $2$
Conductor $40$
Sign $0.789 + 0.613i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.32 + 3.25i)2-s + (−1.65 + 1.65i)3-s + (−5.19 − 15.1i)4-s + (−4.54 − 24.5i)5-s + (−1.53 − 9.21i)6-s + (28.3 − 28.3i)7-s + (61.3 + 18.2i)8-s + 75.5i·9-s + (90.5 + 42.3i)10-s − 206. i·11-s + (33.5 + 16.4i)12-s + (90.6 − 90.6i)13-s + (26.4 + 158. i)14-s + (48.1 + 33.0i)15-s + (−202. + 157. i)16-s + (258. − 258. i)17-s + ⋯
L(s)  = 1  + (−0.581 + 0.813i)2-s + (−0.183 + 0.183i)3-s + (−0.324 − 0.945i)4-s + (−0.181 − 0.983i)5-s + (−0.0426 − 0.255i)6-s + (0.579 − 0.579i)7-s + (0.958 + 0.285i)8-s + 0.932i·9-s + (0.905 + 0.423i)10-s − 1.70i·11-s + (0.233 + 0.114i)12-s + (0.536 − 0.536i)13-s + (0.134 + 0.808i)14-s + (0.213 + 0.147i)15-s + (−0.789 + 0.613i)16-s + (0.894 − 0.894i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.789 + 0.613i$
Analytic conductor: \(4.13479\)
Root analytic conductor: \(2.03342\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :2),\ 0.789 + 0.613i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.865200 - 0.296590i\)
\(L(\frac12)\) \(\approx\) \(0.865200 - 0.296590i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.32 - 3.25i)T \)
5 \( 1 + (4.54 + 24.5i)T \)
good3 \( 1 + (1.65 - 1.65i)T - 81iT^{2} \)
7 \( 1 + (-28.3 + 28.3i)T - 2.40e3iT^{2} \)
11 \( 1 + 206. iT - 1.46e4T^{2} \)
13 \( 1 + (-90.6 + 90.6i)T - 2.85e4iT^{2} \)
17 \( 1 + (-258. + 258. i)T - 8.35e4iT^{2} \)
19 \( 1 + 555.T + 1.30e5T^{2} \)
23 \( 1 + (193. + 193. i)T + 2.79e5iT^{2} \)
29 \( 1 - 533.T + 7.07e5T^{2} \)
31 \( 1 - 424.T + 9.23e5T^{2} \)
37 \( 1 + (-1.52e3 - 1.52e3i)T + 1.87e6iT^{2} \)
41 \( 1 + 480.T + 2.82e6T^{2} \)
43 \( 1 + (366. - 366. i)T - 3.41e6iT^{2} \)
47 \( 1 + (-1.30e3 + 1.30e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (990. - 990. i)T - 7.89e6iT^{2} \)
59 \( 1 + 800.T + 1.21e7T^{2} \)
61 \( 1 - 2.31e3iT - 1.38e7T^{2} \)
67 \( 1 + (-2.20e3 - 2.20e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 2.33e3T + 2.54e7T^{2} \)
73 \( 1 + (-4.39e3 - 4.39e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 8.85e3iT - 3.89e7T^{2} \)
83 \( 1 + (-643. + 643. i)T - 4.74e7iT^{2} \)
89 \( 1 + 3.67e3iT - 6.27e7T^{2} \)
97 \( 1 + (-9.02e3 + 9.02e3i)T - 8.85e7iT^{2} \)
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

−15.71152017065769703729206318959, −14.12297329606386277864648094057, −13.31691869797466559479540125968, −11.31451390080559671564109163098, −10.29685110108021243282363070336, −8.549611863277529561086018732589, −7.954114608705740004739996033515, −5.90354032069087283939779562002, −4.63872376892692820743030756563, −0.805647708571844790246091932056, 2.01643504435024427029840125590, 4.01392122405310830887960613681, 6.58945609199297528758429869519, 8.021920619703170578356810864700, 9.530650245495941749957967270353, 10.69268621551321407569026693012, 11.88626144715171000029023838903, 12.62962659825662814235991716398, 14.48604972481559400366889303939, 15.35722946500770533740219442326

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