Properties

Label 2-20-20.19-c8-0-0
Degree $2$
Conductor $20$
Sign $-0.184 - 0.982i$
Analytic cond. $8.14757$
Root an. cond. $2.85439$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.05 − 15.4i)2-s + 75.2·3-s + (−223. + 125. i)4-s + (−200. + 591. i)5-s + (−305. − 1.16e3i)6-s − 4.41e3·7-s + (2.85e3 + 2.94e3i)8-s − 900.·9-s + (9.97e3 + 706. i)10-s − 330. i·11-s + (−1.67e4 + 9.45e3i)12-s + 2.67e4i·13-s + (1.79e4 + 6.82e4i)14-s + (−1.51e4 + 4.45e4i)15-s + (3.39e4 − 5.60e4i)16-s + 6.86e4i·17-s + ⋯
L(s)  = 1  + (−0.253 − 0.967i)2-s + 0.928·3-s + (−0.871 + 0.490i)4-s + (−0.321 + 0.946i)5-s + (−0.235 − 0.898i)6-s − 1.83·7-s + (0.695 + 0.718i)8-s − 0.137·9-s + (0.997 + 0.0706i)10-s − 0.0225i·11-s + (−0.809 + 0.455i)12-s + 0.935i·13-s + (0.466 + 1.77i)14-s + (−0.298 + 0.879i)15-s + (0.518 − 0.855i)16-s + 0.822i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.184 - 0.982i$
Analytic conductor: \(8.14757\)
Root analytic conductor: \(2.85439\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :4),\ -0.184 - 0.982i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.346886 + 0.418166i\)
\(L(\frac12)\) \(\approx\) \(0.346886 + 0.418166i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.05 + 15.4i)T \)
5 \( 1 + (200. - 591. i)T \)
good3 \( 1 - 75.2T + 6.56e3T^{2} \)
7 \( 1 + 4.41e3T + 5.76e6T^{2} \)
11 \( 1 + 330. iT - 2.14e8T^{2} \)
13 \( 1 - 2.67e4iT - 8.15e8T^{2} \)
17 \( 1 - 6.86e4iT - 6.97e9T^{2} \)
19 \( 1 + 1.94e5iT - 1.69e10T^{2} \)
23 \( 1 - 2.48e5T + 7.83e10T^{2} \)
29 \( 1 + 4.96e5T + 5.00e11T^{2} \)
31 \( 1 - 7.34e5iT - 8.52e11T^{2} \)
37 \( 1 - 1.92e6iT - 3.51e12T^{2} \)
41 \( 1 + 8.79e5T + 7.98e12T^{2} \)
43 \( 1 + 1.87e6T + 1.16e13T^{2} \)
47 \( 1 - 5.21e5T + 2.38e13T^{2} \)
53 \( 1 + 8.82e5iT - 6.22e13T^{2} \)
59 \( 1 + 3.24e6iT - 1.46e14T^{2} \)
61 \( 1 + 9.76e5T + 1.91e14T^{2} \)
67 \( 1 - 1.58e7T + 4.06e14T^{2} \)
71 \( 1 + 1.19e7iT - 6.45e14T^{2} \)
73 \( 1 - 4.92e7iT - 8.06e14T^{2} \)
79 \( 1 - 5.67e7iT - 1.51e15T^{2} \)
83 \( 1 + 3.12e7T + 2.25e15T^{2} \)
89 \( 1 + 4.74e7T + 3.93e15T^{2} \)
97 \( 1 + 2.28e7iT - 7.83e15T^{2} \)
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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

−16.96348729529571984194256305167, −15.33761219173127129751610100751, −13.92559678482826492492454489531, −12.92176027598462821312171265749, −11.29729721351060247136674429222, −9.850763663587515888564863136569, −8.795118778316551151316277823943, −6.87595400631418981730090806975, −3.59561074113976697561166499226, −2.66456566062306442112993230518, 0.26049630563938995218920607552, 3.53075838706728455187174832398, 5.74625963894324724843658260020, 7.59814362846174515755917191941, 8.909399630128673725610622771734, 9.808969810215638791379965635059, 12.68392152991986388093296747074, 13.54000828728433560385165860154, 15.04922682795661274782868038630, 16.10262694293482943351536375253

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