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Properties

Label 2-20-20.7-c9-0-7
Degree $2$
Conductor $20$
Sign $-0.227 - 0.973i$
Analytic cond. $10.3007$
Root an. cond. $3.20947$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−21.5 − 6.87i)2-s + (110. + 110. i)3-s + (417. + 296. i)4-s + (404. + 1.33e3i)5-s + (−1.62e3 − 3.13e3i)6-s + (1.21e3 − 1.21e3i)7-s + (−6.96e3 − 9.25e3i)8-s + 4.67e3i·9-s + (478. − 3.16e4i)10-s + 3.90e4i·11-s + (1.33e4 + 7.87e4i)12-s + (−5.35e4 + 5.35e4i)13-s + (−3.45e4 + 1.78e4i)14-s + (−1.03e5 + 1.92e5i)15-s + (8.64e4 + 2.47e5i)16-s + (3.81e5 + 3.81e5i)17-s + ⋯
L(s)  = 1  + (−0.952 − 0.303i)2-s + (0.786 + 0.786i)3-s + (0.815 + 0.578i)4-s + (0.289 + 0.957i)5-s + (−0.510 − 0.988i)6-s + (0.191 − 0.191i)7-s + (−0.601 − 0.799i)8-s + 0.237i·9-s + (0.0151 − 0.999i)10-s + 0.803i·11-s + (0.186 + 1.09i)12-s + (−0.519 + 0.519i)13-s + (−0.240 + 0.124i)14-s + (−0.525 + 0.980i)15-s + (0.329 + 0.944i)16-s + (1.10 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.227 - 0.973i$
Analytic conductor: \(10.3007\)
Root analytic conductor: \(3.20947\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :9/2),\ -0.227 - 0.973i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.850893 + 1.07291i\)
\(L(\frac12)\) \(\approx\) \(0.850893 + 1.07291i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (21.5 + 6.87i)T \)
5 \( 1 + (-404. - 1.33e3i)T \)
good3 \( 1 + (-110. - 110. i)T + 1.96e4iT^{2} \)
7 \( 1 + (-1.21e3 + 1.21e3i)T - 4.03e7iT^{2} \)
11 \( 1 - 3.90e4iT - 2.35e9T^{2} \)
13 \( 1 + (5.35e4 - 5.35e4i)T - 1.06e10iT^{2} \)
17 \( 1 + (-3.81e5 - 3.81e5i)T + 1.18e11iT^{2} \)
19 \( 1 + 5.75e5T + 3.22e11T^{2} \)
23 \( 1 + (1.06e6 + 1.06e6i)T + 1.80e12iT^{2} \)
29 \( 1 - 3.51e6iT - 1.45e13T^{2} \)
31 \( 1 - 4.45e6iT - 2.64e13T^{2} \)
37 \( 1 + (-5.44e6 - 5.44e6i)T + 1.29e14iT^{2} \)
41 \( 1 - 2.53e6T + 3.27e14T^{2} \)
43 \( 1 + (2.19e7 + 2.19e7i)T + 5.02e14iT^{2} \)
47 \( 1 + (-1.86e7 + 1.86e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (-6.92e7 + 6.92e7i)T - 3.29e15iT^{2} \)
59 \( 1 - 1.64e8T + 8.66e15T^{2} \)
61 \( 1 + 1.57e6T + 1.16e16T^{2} \)
67 \( 1 + (-2.14e8 + 2.14e8i)T - 2.72e16iT^{2} \)
71 \( 1 + 5.04e7iT - 4.58e16T^{2} \)
73 \( 1 + (-4.03e7 + 4.03e7i)T - 5.88e16iT^{2} \)
79 \( 1 + 3.84e8T + 1.19e17T^{2} \)
83 \( 1 + (-5.30e8 - 5.30e8i)T + 1.86e17iT^{2} \)
89 \( 1 - 4.15e8iT - 3.50e17T^{2} \)
97 \( 1 + (-8.68e7 - 8.68e7i)T + 7.60e17iT^{2} \)
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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

−16.70321743808943892179136106008, −15.14655131886817942885778080459, −14.46440905960775706882971377141, −12.28722385541294626729366401816, −10.51513107020206054106829554663, −9.851340145196780510950977084723, −8.375865145453289922707549429835, −6.78749152015599844741902519451, −3.73114253878815602280291300400, −2.15178158214690551715036628464, 0.788996035825790303024120928124, 2.33579924150212106140358450573, 5.65340207665877733518854883249, 7.64607160692767361405550708149, 8.501006521315968924573057090493, 9.815534267702294585992736867949, 11.78132518378964373630554093708, 13.28491281237900526825042238190, 14.53261436924589015717968085001, 16.09993686747250397733104032972

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