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Properties

Label 2-20-20.3-c5-0-1
Degree $2$
Conductor $20$
Sign $-0.891 - 0.453i$
Analytic cond. $3.20767$
Root an. cond. $1.79099$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.53 + 5.05i)2-s + (−10.7 + 10.7i)3-s + (−19.1 + 25.6i)4-s + (−36.9 − 41.9i)5-s + (−81.4 − 26.9i)6-s + (129. + 129. i)7-s + (−178. − 31.4i)8-s + 13.1i·9-s + (118. − 293. i)10-s + 299. i·11-s + (−70.2 − 480. i)12-s + (370. + 370. i)13-s + (−326. + 984. i)14-s + (845. + 53.1i)15-s + (−293. − 981. i)16-s + (1.42e3 − 1.42e3i)17-s + ⋯
L(s)  = 1  + (0.448 + 0.893i)2-s + (−0.687 + 0.687i)3-s + (−0.597 + 0.802i)4-s + (−0.661 − 0.750i)5-s + (−0.923 − 0.306i)6-s + (0.999 + 0.999i)7-s + (−0.984 − 0.173i)8-s + 0.0541i·9-s + (0.373 − 0.927i)10-s + 0.746i·11-s + (−0.140 − 0.962i)12-s + (0.608 + 0.608i)13-s + (−0.444 + 1.34i)14-s + (0.970 + 0.0610i)15-s + (−0.286 − 0.958i)16-s + (1.19 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.453i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.891 - 0.453i$
Analytic conductor: \(3.20767\)
Root analytic conductor: \(1.79099\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5/2),\ -0.891 - 0.453i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.275742 + 1.14881i\)
\(L(\frac12)\) \(\approx\) \(0.275742 + 1.14881i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.53 - 5.05i)T \)
5 \( 1 + (36.9 + 41.9i)T \)
good3 \( 1 + (10.7 - 10.7i)T - 243iT^{2} \)
7 \( 1 + (-129. - 129. i)T + 1.68e4iT^{2} \)
11 \( 1 - 299. iT - 1.61e5T^{2} \)
13 \( 1 + (-370. - 370. i)T + 3.71e5iT^{2} \)
17 \( 1 + (-1.42e3 + 1.42e3i)T - 1.41e6iT^{2} \)
19 \( 1 + 540.T + 2.47e6T^{2} \)
23 \( 1 + (167. - 167. i)T - 6.43e6iT^{2} \)
29 \( 1 - 973. iT - 2.05e7T^{2} \)
31 \( 1 - 2.92e3iT - 2.86e7T^{2} \)
37 \( 1 + (-1.64e3 + 1.64e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 1.01e4T + 1.15e8T^{2} \)
43 \( 1 + (-4.73e3 + 4.73e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (-1.60e4 - 1.60e4i)T + 2.29e8iT^{2} \)
53 \( 1 + (1.44e4 + 1.44e4i)T + 4.18e8iT^{2} \)
59 \( 1 + 2.29e4T + 7.14e8T^{2} \)
61 \( 1 + 8.05e3T + 8.44e8T^{2} \)
67 \( 1 + (-2.71e4 - 2.71e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 5.71e4iT - 1.80e9T^{2} \)
73 \( 1 + (-5.29e4 - 5.29e4i)T + 2.07e9iT^{2} \)
79 \( 1 + 8.73e4T + 3.07e9T^{2} \)
83 \( 1 + (-6.66e4 + 6.66e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 3.17e4iT - 5.58e9T^{2} \)
97 \( 1 + (-6.23e4 + 6.23e4i)T - 8.58e9iT^{2} \)
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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

−17.39376553137897130299368105739, −16.26401615979874865682819030206, −15.57288593855662959999998170434, −14.28099127381182207307518848860, −12.39982415616193445768786665888, −11.47396805960901407154003476606, −9.160792779613959049739814752885, −7.77144232293427149420603171845, −5.45758786422751244861395420013, −4.50931672679728796403460654020, 0.930934854795047407102165621760, 3.78785164199027427430978307989, 6.02901003016506348305563693221, 7.934131589810684985794372488701, 10.59062005657630801456952526591, 11.27396559150651201460618258982, 12.50468989395186897045109323329, 13.93513069248169229148794671972, 15.03410891379632783269379966626, 17.15056149844793718388084536527

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