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Properties

Label 2-20-20.19-c8-0-20
Degree $2$
Conductor $20$
Sign $-0.891 + 0.453i$
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  + (14.3 − 7.17i)2-s − 42.6·3-s + (153. − 205. i)4-s + (−560. − 276. i)5-s + (−610. + 306. i)6-s − 869.·7-s + (717. − 4.03e3i)8-s − 4.74e3·9-s + (−9.99e3 + 58.9i)10-s − 2.47e4i·11-s + (−6.53e3 + 8.75e3i)12-s + 4.11e4i·13-s + (−1.24e4 + 6.23e3i)14-s + (2.39e4 + 1.18e4i)15-s + (−1.86e4 − 6.28e4i)16-s − 4.09e4i·17-s + ⋯
L(s)  = 1  + (0.893 − 0.448i)2-s − 0.526·3-s + (0.597 − 0.801i)4-s + (−0.896 − 0.443i)5-s + (−0.470 + 0.236i)6-s − 0.362·7-s + (0.175 − 0.984i)8-s − 0.722·9-s + (−0.999 + 0.00589i)10-s − 1.68i·11-s + (−0.314 + 0.422i)12-s + 1.44i·13-s + (−0.323 + 0.162i)14-s + (0.472 + 0.233i)15-s + (−0.284 − 0.958i)16-s − 0.489i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.891 + 0.453i$
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: \((2,\ 20,\ (\ :4),\ -0.891 + 0.453i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.313501 - 1.30697i\)
\(L(\frac12)\) \(\approx\) \(0.313501 - 1.30697i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-14.3 + 7.17i)T \)
5 \( 1 + (560. + 276. i)T \)
good3 \( 1 + 42.6T + 6.56e3T^{2} \)
7 \( 1 + 869.T + 5.76e6T^{2} \)
11 \( 1 + 2.47e4iT - 2.14e8T^{2} \)
13 \( 1 - 4.11e4iT - 8.15e8T^{2} \)
17 \( 1 + 4.09e4iT - 6.97e9T^{2} \)
19 \( 1 + 7.95e4iT - 1.69e10T^{2} \)
23 \( 1 - 4.28e5T + 7.83e10T^{2} \)
29 \( 1 - 1.84e5T + 5.00e11T^{2} \)
31 \( 1 + 1.54e5iT - 8.52e11T^{2} \)
37 \( 1 + 1.72e6iT - 3.51e12T^{2} \)
41 \( 1 + 2.00e6T + 7.98e12T^{2} \)
43 \( 1 + 1.43e6T + 1.16e13T^{2} \)
47 \( 1 + 4.98e6T + 2.38e13T^{2} \)
53 \( 1 + 6.97e6iT - 6.22e13T^{2} \)
59 \( 1 + 5.65e6iT - 1.46e14T^{2} \)
61 \( 1 - 6.73e6T + 1.91e14T^{2} \)
67 \( 1 - 4.99e6T + 4.06e14T^{2} \)
71 \( 1 - 2.07e7iT - 6.45e14T^{2} \)
73 \( 1 - 3.31e7iT - 8.06e14T^{2} \)
79 \( 1 + 7.49e7iT - 1.51e15T^{2} \)
83 \( 1 - 8.75e7T + 2.25e15T^{2} \)
89 \( 1 + 1.14e7T + 3.93e15T^{2} \)
97 \( 1 + 8.74e7iT - 7.83e15T^{2} \)
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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

−16.00560214018151243184797635520, −14.40049189858113761609722748841, −13.16327594177554044216873972621, −11.61137524453344339996848442985, −11.20648902957750739083654780527, −8.913782079523590907197765541108, −6.61818956584262513490283067313, −5.03505537818715756457221236376, −3.29721319497889540923557189002, −0.54167392194165738326450557691, 3.14997832508320920874127144432, 4.97529068023140413031194532468, 6.65956559340336628873115516837, 8.003962206027240221222289126223, 10.59325221070473278310463984029, 11.96388409971325882698234330201, 12.86985689891248566267347600426, 14.78720303568959040108433691419, 15.37506341351897301368131129405, 16.82798058467715506895382371896

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