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Properties

Label 2-20-5.3-c8-0-0
Degree $2$
Conductor $20$
Sign $0.437 - 0.899i$
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  + (80.1 + 80.1i)3-s + (621. − 62.6i)5-s + (−1.04e3 + 1.04e3i)7-s + 6.27e3i·9-s − 1.09e3·11-s + (3.65e4 + 3.65e4i)13-s + (5.48e4 + 4.48e4i)15-s + (−6.72e4 + 6.72e4i)17-s − 1.77e5i·19-s − 1.67e5·21-s + (−1.10e5 − 1.10e5i)23-s + (3.82e5 − 7.79e4i)25-s + (2.25e4 − 2.25e4i)27-s − 9.03e5i·29-s − 1.11e6·31-s + ⋯
L(s)  = 1  + (0.989 + 0.989i)3-s + (0.994 − 0.100i)5-s + (−0.435 + 0.435i)7-s + 0.957i·9-s − 0.0749·11-s + (1.28 + 1.28i)13-s + (1.08 + 0.885i)15-s + (−0.804 + 0.804i)17-s − 1.36i·19-s − 0.861·21-s + (−0.396 − 0.396i)23-s + (0.979 − 0.199i)25-s + (0.0423 − 0.0423i)27-s − 1.27i·29-s − 1.21·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.437 - 0.899i$
Analytic conductor: \(8.14757\)
Root analytic conductor: \(2.85439\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :4),\ 0.437 - 0.899i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.12383 + 1.32808i\)
\(L(\frac12)\) \(\approx\) \(2.12383 + 1.32808i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-621. + 62.6i)T \)
good3 \( 1 + (-80.1 - 80.1i)T + 6.56e3iT^{2} \)
7 \( 1 + (1.04e3 - 1.04e3i)T - 5.76e6iT^{2} \)
11 \( 1 + 1.09e3T + 2.14e8T^{2} \)
13 \( 1 + (-3.65e4 - 3.65e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (6.72e4 - 6.72e4i)T - 6.97e9iT^{2} \)
19 \( 1 + 1.77e5iT - 1.69e10T^{2} \)
23 \( 1 + (1.10e5 + 1.10e5i)T + 7.83e10iT^{2} \)
29 \( 1 + 9.03e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.11e6T + 8.52e11T^{2} \)
37 \( 1 + (-1.48e6 + 1.48e6i)T - 3.51e12iT^{2} \)
41 \( 1 - 4.18e6T + 7.98e12T^{2} \)
43 \( 1 + (1.23e6 + 1.23e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (-3.39e6 + 3.39e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (6.26e6 + 6.26e6i)T + 6.22e13iT^{2} \)
59 \( 1 + 4.61e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.06e7T + 1.91e14T^{2} \)
67 \( 1 + (3.39e6 - 3.39e6i)T - 4.06e14iT^{2} \)
71 \( 1 + 4.31e7T + 6.45e14T^{2} \)
73 \( 1 + (9.30e6 + 9.30e6i)T + 8.06e14iT^{2} \)
79 \( 1 - 5.42e7iT - 1.51e15T^{2} \)
83 \( 1 + (-4.43e7 - 4.43e7i)T + 2.25e15iT^{2} \)
89 \( 1 - 2.12e7iT - 3.93e15T^{2} \)
97 \( 1 + (4.71e7 - 4.71e7i)T - 7.83e15iT^{2} \)
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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

−16.38782134657297832892327183769, −15.38477168648855534073922655044, −14.12904698428743573325271427026, −13.12744721099551365821635579861, −10.92942774413343907723118504140, −9.409896351362601471332231624102, −8.817924298035328812887307336259, −6.23750678637450605036127611333, −4.15715734147188816438990658587, −2.31964632469469384098061149790, 1.40176512064521413425336520465, 3.08519779852104617380974202390, 6.04940699757793439287152211671, 7.59316665787641425230654589331, 9.023269743451955318866213765715, 10.57682714010000478848168616280, 12.84256562951720178488936848485, 13.48116839905106484510619103884, 14.50568193322817963745611304381, 16.23015124672337795935200886529

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