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Properties

Label 2-20-20.19-c10-0-26
Degree $2$
Conductor $20$
Sign $-0.870 - 0.492i$
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  + (17.2 − 26.9i)2-s + 146.·3-s + (−429. − 929. i)4-s + (−2.53e3 + 1.82e3i)5-s + (2.52e3 − 3.94e3i)6-s − 1.39e4·7-s + (−3.24e4 − 4.42e3i)8-s − 3.76e4·9-s + (5.40e3 + 9.98e4i)10-s − 8.83e4i·11-s + (−6.29e4 − 1.36e5i)12-s − 2.49e5i·13-s + (−2.40e5 + 3.76e5i)14-s + (−3.71e5 + 2.66e5i)15-s + (−6.78e5 + 7.99e5i)16-s − 4.10e5i·17-s + ⋯
L(s)  = 1  + (0.538 − 0.842i)2-s + 0.602·3-s + (−0.419 − 0.907i)4-s + (−0.812 + 0.583i)5-s + (0.324 − 0.507i)6-s − 0.831·7-s + (−0.990 − 0.135i)8-s − 0.636·9-s + (0.0540 + 0.998i)10-s − 0.548i·11-s + (−0.252 − 0.546i)12-s − 0.671i·13-s + (−0.447 + 0.700i)14-s + (−0.489 + 0.351i)15-s + (−0.647 + 0.762i)16-s − 0.289i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.870 - 0.492i$
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: \((2,\ 20,\ (\ :5),\ -0.870 - 0.492i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.160117 + 0.608408i\)
\(L(\frac12)\) \(\approx\) \(0.160117 + 0.608408i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-17.2 + 26.9i)T \)
5 \( 1 + (2.53e3 - 1.82e3i)T \)
good3 \( 1 - 146.T + 5.90e4T^{2} \)
7 \( 1 + 1.39e4T + 2.82e8T^{2} \)
11 \( 1 + 8.83e4iT - 2.59e10T^{2} \)
13 \( 1 + 2.49e5iT - 1.37e11T^{2} \)
17 \( 1 + 4.10e5iT - 2.01e12T^{2} \)
19 \( 1 - 3.01e6iT - 6.13e12T^{2} \)
23 \( 1 + 4.53e6T + 4.14e13T^{2} \)
29 \( 1 - 2.62e7T + 4.20e14T^{2} \)
31 \( 1 + 3.62e7iT - 8.19e14T^{2} \)
37 \( 1 + 8.29e7iT - 4.80e15T^{2} \)
41 \( 1 - 1.07e8T + 1.34e16T^{2} \)
43 \( 1 - 1.58e6T + 2.16e16T^{2} \)
47 \( 1 + 4.34e8T + 5.25e16T^{2} \)
53 \( 1 + 3.42e8iT - 1.74e17T^{2} \)
59 \( 1 - 8.22e8iT - 5.11e17T^{2} \)
61 \( 1 + 9.43e8T + 7.13e17T^{2} \)
67 \( 1 - 1.13e9T + 1.82e18T^{2} \)
71 \( 1 + 2.91e9iT - 3.25e18T^{2} \)
73 \( 1 - 2.08e9iT - 4.29e18T^{2} \)
79 \( 1 - 5.83e9iT - 9.46e18T^{2} \)
83 \( 1 + 9.99e8T + 1.55e19T^{2} \)
89 \( 1 + 3.97e9T + 3.11e19T^{2} \)
97 \( 1 + 6.44e9iT - 7.37e19T^{2} \)
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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

−14.82543065523460575137387761709, −13.87430174144772788645706437562, −12.46117783482532968542064975654, −11.21205378737679976439936165774, −9.854760044285357181381523263814, −8.186953128628968237602597554410, −6.03169209565323173787291224983, −3.73558036920819620148271657422, −2.75331712377197278487800580734, −0.20046755535396165577957155767, 3.14487326613963406100047470556, 4.65971548426877117853361389678, 6.64548539670450413583718617056, 8.156697885900449628086415653469, 9.240682777757166359061642607405, 11.80783534011247628595279881190, 12.97064333206976700650596186030, 14.19902540494286529742388251319, 15.44718731576266469683259597910, 16.30560378889626923099628536340

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