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Properties

Label 2-40-40.13-c2-0-4
Degree $2$
Conductor $40$
Sign $0.988 - 0.152i$
Analytic cond. $1.08992$
Root an. cond. $1.04399$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.35 + 1.46i)2-s + (2.57 − 2.57i)3-s + (−0.312 − 3.98i)4-s + (4.90 + 0.973i)5-s + (0.284 + 7.27i)6-s + (−4.07 + 4.07i)7-s + (6.27 + 4.95i)8-s − 4.26i·9-s + (−8.08 + 5.87i)10-s − 16.0i·11-s + (−11.0 − 9.46i)12-s + (−9.77 + 9.77i)13-s + (−0.450 − 11.5i)14-s + (15.1 − 10.1i)15-s + (−15.8 + 2.49i)16-s + (−12.0 + 12.0i)17-s + ⋯
L(s)  = 1  + (−0.678 + 0.734i)2-s + (0.858 − 0.858i)3-s + (−0.0780 − 0.996i)4-s + (0.980 + 0.194i)5-s + (0.0474 + 1.21i)6-s + (−0.582 + 0.582i)7-s + (0.784 + 0.619i)8-s − 0.473i·9-s + (−0.808 + 0.587i)10-s − 1.46i·11-s + (−0.922 − 0.788i)12-s + (−0.752 + 0.752i)13-s + (−0.0321 − 0.822i)14-s + (1.00 − 0.674i)15-s + (−0.987 + 0.155i)16-s + (−0.709 + 0.709i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.988 - 0.152i$
Analytic conductor: \(1.08992\)
Root analytic conductor: \(1.04399\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :1),\ 0.988 - 0.152i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.02536 + 0.0786097i\)
\(L(\frac12)\) \(\approx\) \(1.02536 + 0.0786097i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 - 1.46i)T \)
5 \( 1 + (-4.90 - 0.973i)T \)
good3 \( 1 + (-2.57 + 2.57i)T - 9iT^{2} \)
7 \( 1 + (4.07 - 4.07i)T - 49iT^{2} \)
11 \( 1 + 16.0iT - 121T^{2} \)
13 \( 1 + (9.77 - 9.77i)T - 169iT^{2} \)
17 \( 1 + (12.0 - 12.0i)T - 289iT^{2} \)
19 \( 1 + 9.55T + 361T^{2} \)
23 \( 1 + (2.56 + 2.56i)T + 529iT^{2} \)
29 \( 1 - 10.1T + 841T^{2} \)
31 \( 1 + 24.9T + 961T^{2} \)
37 \( 1 + (26.8 + 26.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 62.0T + 1.68e3T^{2} \)
43 \( 1 + (-13.5 + 13.5i)T - 1.84e3iT^{2} \)
47 \( 1 + (-11.9 + 11.9i)T - 2.20e3iT^{2} \)
53 \( 1 + (-45.8 + 45.8i)T - 2.80e3iT^{2} \)
59 \( 1 - 28.9T + 3.48e3T^{2} \)
61 \( 1 + 68.7iT - 3.72e3T^{2} \)
67 \( 1 + (14.7 + 14.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 31.1T + 5.04e3T^{2} \)
73 \( 1 + (25.6 + 25.6i)T + 5.32e3iT^{2} \)
79 \( 1 - 70.6iT - 6.24e3T^{2} \)
83 \( 1 + (96.2 - 96.2i)T - 6.88e3iT^{2} \)
89 \( 1 + 103. iT - 7.92e3T^{2} \)
97 \( 1 + (-30.8 + 30.8i)T - 9.40e3iT^{2} \)
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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

−16.09923683701859522983583428334, −14.60008033671059505845584864703, −13.89768387346785327586442514220, −12.87541869135759938993375928642, −10.79354551042613257972325442109, −9.264148998683405047537979065512, −8.512376268204970759160115699578, −6.92879278322110787222341737698, −5.85075482152898559292262139826, −2.23883836550138875138379171732, 2.60137938308334966390968803633, 4.42956072479530774731511465404, 7.20051507116895812980747232981, 8.958234822761843519723468499427, 9.806766450035355236721764977206, 10.39490892855276845285827903406, 12.43166600441357647980089077966, 13.41921343428878828504107430383, 14.76020027359316297242168230873, 16.02952202708695720131488427787

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