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Properties

Label 2-20-4.3-c6-0-9
Degree $2$
Conductor $20$
Sign $-0.972 - 0.233i$
Analytic cond. $4.60108$
Root an. cond. $2.14501$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.94 − 0.938i)2-s − 50.7i·3-s + (62.2 + 14.9i)4-s − 55.9·5-s + (−47.6 + 403. i)6-s − 91.7i·7-s + (−480. − 176. i)8-s − 1.84e3·9-s + (444. + 52.4i)10-s + 1.26e3i·11-s + (757. − 3.16e3i)12-s + 802.·13-s + (−86.1 + 729. i)14-s + 2.83e3i·15-s + (3.65e3 + 1.85e3i)16-s − 3.53e3·17-s + ⋯
L(s)  = 1  + (−0.993 − 0.117i)2-s − 1.88i·3-s + (0.972 + 0.233i)4-s − 0.447·5-s + (−0.220 + 1.86i)6-s − 0.267i·7-s + (−0.938 − 0.345i)8-s − 2.53·9-s + (0.444 + 0.0524i)10-s + 0.950i·11-s + (0.438 − 1.82i)12-s + 0.365·13-s + (−0.0313 + 0.265i)14-s + 0.841i·15-s + (0.891 + 0.453i)16-s − 0.719·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.972 - 0.233i$
Analytic conductor: \(4.60108\)
Root analytic conductor: \(2.14501\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :3),\ -0.972 - 0.233i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0648325 + 0.548744i\)
\(L(\frac12)\) \(\approx\) \(0.0648325 + 0.548744i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.94 + 0.938i)T \)
5 \( 1 + 55.9T \)
good3 \( 1 + 50.7iT - 729T^{2} \)
7 \( 1 + 91.7iT - 1.17e5T^{2} \)
11 \( 1 - 1.26e3iT - 1.77e6T^{2} \)
13 \( 1 - 802.T + 4.82e6T^{2} \)
17 \( 1 + 3.53e3T + 2.41e7T^{2} \)
19 \( 1 + 9.42e3iT - 4.70e7T^{2} \)
23 \( 1 + 1.59e4iT - 1.48e8T^{2} \)
29 \( 1 + 1.97e4T + 5.94e8T^{2} \)
31 \( 1 + 2.59e4iT - 8.87e8T^{2} \)
37 \( 1 + 8.26e3T + 2.56e9T^{2} \)
41 \( 1 - 1.55e3T + 4.75e9T^{2} \)
43 \( 1 + 5.73e3iT - 6.32e9T^{2} \)
47 \( 1 + 1.15e5iT - 1.07e10T^{2} \)
53 \( 1 + 7.67e4T + 2.21e10T^{2} \)
59 \( 1 - 2.77e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.43e5T + 5.15e10T^{2} \)
67 \( 1 + 3.32e5iT - 9.04e10T^{2} \)
71 \( 1 + 4.72e5iT - 1.28e11T^{2} \)
73 \( 1 - 1.68e5T + 1.51e11T^{2} \)
79 \( 1 - 4.39e5iT - 2.43e11T^{2} \)
83 \( 1 + 9.92e4iT - 3.26e11T^{2} \)
89 \( 1 - 9.01e5T + 4.96e11T^{2} \)
97 \( 1 + 5.40e5T + 8.32e11T^{2} \)
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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

−16.89752296827185099373675868210, −15.09444829808345976997320082252, −13.28507402776705198228729935091, −12.19651243767437272850850026097, −11.06045148017470034271274582871, −8.806400364475396229768288223458, −7.50944499986664308623620380872, −6.61514905317232533540325570860, −2.24156613668624243008794522486, −0.46932507345757774429255973184, 3.50123336864796534788260189966, 5.70260997797306783608385044695, 8.327241060537961903275326443638, 9.391648968029572900617957186765, 10.67881747187530730061059851160, 11.55590727583846394760980001403, 14.46307893082061408027035669218, 15.66409690848948322881941803410, 16.19310357844578200655663814293, 17.33278578834702700743772055650

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