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Properties

Label 2-20-4.3-c8-0-14
Degree $2$
Conductor $20$
Sign $-0.648 + 0.760i$
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  + (14.5 − 6.70i)2-s − 150. i·3-s + (166. − 194. i)4-s + 279.·5-s + (−1.00e3 − 2.18e3i)6-s + 2.62e3i·7-s + (1.10e3 − 3.94e3i)8-s − 1.60e4·9-s + (4.06e3 − 1.87e3i)10-s + 2.30e3i·11-s + (−2.92e4 − 2.49e4i)12-s + 4.70e4·13-s + (1.76e4 + 3.81e4i)14-s − 4.19e4i·15-s + (−1.03e4 − 6.47e4i)16-s − 5.19e4·17-s + ⋯
L(s)  = 1  + (0.907 − 0.419i)2-s − 1.85i·3-s + (0.648 − 0.760i)4-s + 0.447·5-s + (−0.777 − 1.68i)6-s + 1.09i·7-s + (0.270 − 0.962i)8-s − 2.43·9-s + (0.406 − 0.187i)10-s + 0.157i·11-s + (−1.41 − 1.20i)12-s + 1.64·13-s + (0.458 + 0.993i)14-s − 0.829i·15-s + (−0.158 − 0.987i)16-s − 0.622·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 + 0.760i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.648 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.648 + 0.760i$
Analytic conductor: \(8.14757\)
Root analytic conductor: \(2.85439\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :4),\ -0.648 + 0.760i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.22152 - 2.64666i\)
\(L(\frac12)\) \(\approx\) \(1.22152 - 2.64666i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-14.5 + 6.70i)T \)
5 \( 1 - 279.T \)
good3 \( 1 + 150. iT - 6.56e3T^{2} \)
7 \( 1 - 2.62e3iT - 5.76e6T^{2} \)
11 \( 1 - 2.30e3iT - 2.14e8T^{2} \)
13 \( 1 - 4.70e4T + 8.15e8T^{2} \)
17 \( 1 + 5.19e4T + 6.97e9T^{2} \)
19 \( 1 + 5.95e4iT - 1.69e10T^{2} \)
23 \( 1 - 7.75e4iT - 7.83e10T^{2} \)
29 \( 1 - 9.02e5T + 5.00e11T^{2} \)
31 \( 1 - 3.40e5iT - 8.52e11T^{2} \)
37 \( 1 + 5.84e5T + 3.51e12T^{2} \)
41 \( 1 - 2.93e5T + 7.98e12T^{2} \)
43 \( 1 + 2.95e6iT - 1.16e13T^{2} \)
47 \( 1 - 5.03e6iT - 2.38e13T^{2} \)
53 \( 1 + 7.54e6T + 6.22e13T^{2} \)
59 \( 1 - 8.82e6iT - 1.46e14T^{2} \)
61 \( 1 - 1.08e7T + 1.91e14T^{2} \)
67 \( 1 - 1.44e7iT - 4.06e14T^{2} \)
71 \( 1 + 3.71e6iT - 6.45e14T^{2} \)
73 \( 1 + 3.62e7T + 8.06e14T^{2} \)
79 \( 1 - 4.88e7iT - 1.51e15T^{2} \)
83 \( 1 - 6.93e7iT - 2.25e15T^{2} \)
89 \( 1 + 1.05e8T + 3.93e15T^{2} \)
97 \( 1 - 1.33e8T + 7.83e15T^{2} \)
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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

−15.66560290665519186484929803359, −14.07511086121599701753966192058, −13.23528422310747722163467344254, −12.29203443980098206680380205905, −11.19730325812909987382950422848, −8.671603078449678144646382246232, −6.69365924149322481092786763346, −5.73478656668839146746396030333, −2.63512132988613350818261800083, −1.33712179568876155361826335193, 3.47245825117799841406362489030, 4.55936780330067536834850846953, 6.14530291547538057190344280552, 8.562130603951211080146939617194, 10.33845626394398019153541661234, 11.24095804188412135799724963172, 13.49392006347764136135385446220, 14.40435980169181561049877537078, 15.72863031433225763754543261831, 16.41616774221097918317991903882

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