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Properties

Label 2-20-20.19-c8-0-1
Degree $2$
Conductor $20$
Sign $0.997 - 0.0723i$
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  + (−7.06 − 14.3i)2-s − 134.·3-s + (−156. + 202. i)4-s + (−416. − 466. i)5-s + (953. + 1.93e3i)6-s − 1.86e3·7-s + (4.01e3 + 810. i)8-s + 1.16e4·9-s + (−3.75e3 + 9.26e3i)10-s − 5.27e3i·11-s + (2.10e4 − 2.73e4i)12-s − 3.33e4i·13-s + (1.31e4 + 2.67e4i)14-s + (5.61e4 + 6.29e4i)15-s + (−1.67e4 − 6.33e4i)16-s + 1.42e5i·17-s + ⋯
L(s)  = 1  + (−0.441 − 0.897i)2-s − 1.66·3-s + (−0.610 + 0.792i)4-s + (−0.665 − 0.745i)5-s + (0.735 + 1.49i)6-s − 0.776·7-s + (0.980 + 0.197i)8-s + 1.77·9-s + (−0.375 + 0.926i)10-s − 0.360i·11-s + (1.01 − 1.32i)12-s − 1.16i·13-s + (0.342 + 0.696i)14-s + (1.10 + 1.24i)15-s + (−0.255 − 0.966i)16-s + 1.70i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0723i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.997 - 0.0723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.997 - 0.0723i$
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: \((2,\ 20,\ (\ :4),\ 0.997 - 0.0723i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.279300 + 0.0101214i\)
\(L(\frac12)\) \(\approx\) \(0.279300 + 0.0101214i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.06 + 14.3i)T \)
5 \( 1 + (416. + 466. i)T \)
good3 \( 1 + 134.T + 6.56e3T^{2} \)
7 \( 1 + 1.86e3T + 5.76e6T^{2} \)
11 \( 1 + 5.27e3iT - 2.14e8T^{2} \)
13 \( 1 + 3.33e4iT - 8.15e8T^{2} \)
17 \( 1 - 1.42e5iT - 6.97e9T^{2} \)
19 \( 1 - 7.03e4iT - 1.69e10T^{2} \)
23 \( 1 + 2.55e5T + 7.83e10T^{2} \)
29 \( 1 - 5.34e5T + 5.00e11T^{2} \)
31 \( 1 + 1.24e6iT - 8.52e11T^{2} \)
37 \( 1 - 1.69e5iT - 3.51e12T^{2} \)
41 \( 1 - 2.36e6T + 7.98e12T^{2} \)
43 \( 1 - 4.03e6T + 1.16e13T^{2} \)
47 \( 1 + 1.65e6T + 2.38e13T^{2} \)
53 \( 1 - 9.81e5iT - 6.22e13T^{2} \)
59 \( 1 - 7.47e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.41e7T + 1.91e14T^{2} \)
67 \( 1 - 3.75e7T + 4.06e14T^{2} \)
71 \( 1 - 3.57e7iT - 6.45e14T^{2} \)
73 \( 1 - 7.26e6iT - 8.06e14T^{2} \)
79 \( 1 - 3.72e7iT - 1.51e15T^{2} \)
83 \( 1 + 2.09e7T + 2.25e15T^{2} \)
89 \( 1 - 2.52e7T + 3.93e15T^{2} \)
97 \( 1 + 1.35e6iT - 7.83e15T^{2} \)
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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

−16.78542353511000964839105972432, −15.81634725881027717721795021842, −12.88066242512891725493922393892, −12.36017103705621220013075733795, −11.09739997216664163933997981553, −10.02370128888717719359421804771, −8.065069854559902026071884892135, −5.83239621976836055354369482398, −4.02983357155486815663226214693, −0.842024870488518937037620577093, 0.31157285065325777959832878730, 4.67470868915454102252757149857, 6.40388312615203701492511882214, 7.16512683522297741423224825925, 9.620093039707112028392883970157, 10.94730036830264410010074805487, 12.14118264822588125960833947337, 14.07252387442949300104341481791, 15.91913975543598319755248540373, 16.18216492728079655814216289069

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