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Properties

Label 2-20-20.19-c10-0-10
Degree $2$
Conductor $20$
Sign $0.220 - 0.975i$
Analytic cond. $12.7071$
Root an. cond. $3.56470$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−31.7 − 4.07i)2-s + 190.·3-s + (990. + 258. i)4-s + (101. + 3.12e3i)5-s + (−6.03e3 − 773. i)6-s + 1.40e4·7-s + (−3.03e4 − 1.22e4i)8-s − 2.29e4·9-s + (9.48e3 − 9.95e4i)10-s − 1.73e5i·11-s + (1.88e5 + 4.91e4i)12-s + 6.64e5i·13-s + (−4.46e5 − 5.72e4i)14-s + (1.93e4 + 5.93e5i)15-s + (9.14e5 + 5.12e5i)16-s + 1.08e6i·17-s + ⋯
L(s)  = 1  + (−0.991 − 0.127i)2-s + 0.782·3-s + (0.967 + 0.252i)4-s + (0.0325 + 0.999i)5-s + (−0.775 − 0.0995i)6-s + 0.836·7-s + (−0.927 − 0.373i)8-s − 0.388·9-s + (0.0948 − 0.995i)10-s − 1.07i·11-s + (0.756 + 0.197i)12-s + 1.78i·13-s + (−0.829 − 0.106i)14-s + (0.0254 + 0.781i)15-s + (0.872 + 0.488i)16-s + 0.765i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.220 - 0.975i$
Analytic conductor: \(12.7071\)
Root analytic conductor: \(3.56470\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5),\ 0.220 - 0.975i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.10162 + 0.880159i\)
\(L(\frac12)\) \(\approx\) \(1.10162 + 0.880159i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (31.7 + 4.07i)T \)
5 \( 1 + (-101. - 3.12e3i)T \)
good3 \( 1 - 190.T + 5.90e4T^{2} \)
7 \( 1 - 1.40e4T + 2.82e8T^{2} \)
11 \( 1 + 1.73e5iT - 2.59e10T^{2} \)
13 \( 1 - 6.64e5iT - 1.37e11T^{2} \)
17 \( 1 - 1.08e6iT - 2.01e12T^{2} \)
19 \( 1 - 2.14e6iT - 6.13e12T^{2} \)
23 \( 1 - 6.85e6T + 4.14e13T^{2} \)
29 \( 1 - 1.60e7T + 4.20e14T^{2} \)
31 \( 1 - 3.87e7iT - 8.19e14T^{2} \)
37 \( 1 - 1.29e7iT - 4.80e15T^{2} \)
41 \( 1 - 4.48e7T + 1.34e16T^{2} \)
43 \( 1 - 8.66e7T + 2.16e16T^{2} \)
47 \( 1 + 2.64e8T + 5.25e16T^{2} \)
53 \( 1 - 2.29e8iT - 1.74e17T^{2} \)
59 \( 1 + 5.54e8iT - 5.11e17T^{2} \)
61 \( 1 - 8.72e8T + 7.13e17T^{2} \)
67 \( 1 - 1.53e8T + 1.82e18T^{2} \)
71 \( 1 + 3.28e9iT - 3.25e18T^{2} \)
73 \( 1 + 1.61e9iT - 4.29e18T^{2} \)
79 \( 1 + 3.08e9iT - 9.46e18T^{2} \)
83 \( 1 - 1.89e9T + 1.55e19T^{2} \)
89 \( 1 - 8.69e9T + 3.11e19T^{2} \)
97 \( 1 - 5.78e9iT - 7.37e19T^{2} \)
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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

−16.38385873487814724268306356520, −14.77173905257544367897795898221, −14.05993195788993560986106358535, −11.65845947399342616662880785818, −10.74446489839902425479483779876, −9.050008980853144620214678984534, −8.016129706088691695357190528533, −6.45595414671210832752316139757, −3.29457990740397674839040838364, −1.79776183170362224865440117398, 0.77119563886594737481136995654, 2.51285389392919036411237669588, 5.20767684406801861682320413893, 7.62008198264242464736493913788, 8.545213048426971723455871061740, 9.715543034795727857839688053912, 11.40563616827032940388060875759, 12.95546721791366480660903265049, 14.73290237391862795446086836746, 15.63329610056056013020874453149

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