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Properties

Label 2-40-40.13-c4-0-10
Degree $2$
Conductor $40$
Sign $-0.0910 + 0.995i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.99 − 0.134i)2-s + (−9.27 + 9.27i)3-s + (15.9 + 1.07i)4-s + (−21.1 + 13.3i)5-s + (38.3 − 35.8i)6-s + (26.2 − 26.2i)7-s + (−63.6 − 6.46i)8-s − 90.9i·9-s + (86.2 − 50.5i)10-s − 212. i·11-s + (−158. + 138. i)12-s + (−35.2 + 35.2i)13-s + (−108. + 101. i)14-s + (72.1 − 319. i)15-s + (253. + 34.4i)16-s + (−128. + 128. i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0337i)2-s + (−1.03 + 1.03i)3-s + (0.997 + 0.0674i)4-s + (−0.845 + 0.534i)5-s + (1.06 − 0.995i)6-s + (0.536 − 0.536i)7-s + (−0.994 − 0.101i)8-s − 1.12i·9-s + (0.862 − 0.505i)10-s − 1.75i·11-s + (−1.09 + 0.958i)12-s + (−0.208 + 0.208i)13-s + (−0.553 + 0.517i)14-s + (0.320 − 1.42i)15-s + (0.990 + 0.134i)16-s + (−0.444 + 0.444i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0910 + 0.995i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0910 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.0910 + 0.995i$
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: \((2,\ 40,\ (\ :2),\ -0.0910 + 0.995i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.136672 - 0.149734i\)
\(L(\frac12)\) \(\approx\) \(0.136672 - 0.149734i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.99 + 0.134i)T \)
5 \( 1 + (21.1 - 13.3i)T \)
good3 \( 1 + (9.27 - 9.27i)T - 81iT^{2} \)
7 \( 1 + (-26.2 + 26.2i)T - 2.40e3iT^{2} \)
11 \( 1 + 212. iT - 1.46e4T^{2} \)
13 \( 1 + (35.2 - 35.2i)T - 2.85e4iT^{2} \)
17 \( 1 + (128. - 128. i)T - 8.35e4iT^{2} \)
19 \( 1 - 283.T + 1.30e5T^{2} \)
23 \( 1 + (219. + 219. i)T + 2.79e5iT^{2} \)
29 \( 1 + 304.T + 7.07e5T^{2} \)
31 \( 1 + 1.69e3T + 9.23e5T^{2} \)
37 \( 1 + (1.55e3 + 1.55e3i)T + 1.87e6iT^{2} \)
41 \( 1 + 247.T + 2.82e6T^{2} \)
43 \( 1 + (-2.28e3 + 2.28e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (-510. + 510. i)T - 4.87e6iT^{2} \)
53 \( 1 + (1.01e3 - 1.01e3i)T - 7.89e6iT^{2} \)
59 \( 1 + 3.14e3T + 1.21e7T^{2} \)
61 \( 1 - 3.79e3iT - 1.38e7T^{2} \)
67 \( 1 + (-413. - 413. i)T + 2.01e7iT^{2} \)
71 \( 1 - 937.T + 2.54e7T^{2} \)
73 \( 1 + (4.21e3 + 4.21e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 719. iT - 3.89e7T^{2} \)
83 \( 1 + (726. - 726. i)T - 4.74e7iT^{2} \)
89 \( 1 + 1.16e4iT - 6.27e7T^{2} \)
97 \( 1 + (-1.56e3 + 1.56e3i)T - 8.85e7iT^{2} \)
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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

−15.69491561460518528041806508984, −14.33246246006511173565822205497, −11.93155629904522437742345400600, −10.91659381642705538849924263155, −10.68656489794872266850780529519, −8.907379593587068917293262703846, −7.44733602274037644363129616591, −5.80640860110236153745780709225, −3.74680541221044248161576108011, −0.21109625022054977733164345960, 1.63546638651334089849965974635, 5.24675511171944734828177453675, 7.03236052920150939094493373684, 7.79863865741997605090710675741, 9.412812485803889216276862576792, 11.17159507230076068769591793209, 12.01165508443045836175599853848, 12.65181655319374587653702728519, 14.97731587962303932189264077077, 15.98015102670642265368220378944

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