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Properties

Label 2-20-20.7-c9-0-16
Degree $2$
Conductor $20$
Sign $-0.839 + 0.543i$
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  + (−22.6 − 0.858i)2-s + (−163. − 163. i)3-s + (510. + 38.8i)4-s + (1.39e3 − 135. i)5-s + (3.55e3 + 3.83e3i)6-s + (7.88e3 − 7.88e3i)7-s + (−1.15e4 − 1.31e3i)8-s + 3.37e4i·9-s + (−3.15e4 + 1.86e3i)10-s − 3.58e4i·11-s + (−7.70e4 − 8.97e4i)12-s + (6.21e4 − 6.21e4i)13-s + (−1.85e5 + 1.71e5i)14-s + (−2.49e5 − 2.05e5i)15-s + (2.59e5 + 3.96e4i)16-s + (1.77e4 + 1.77e4i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0379i)2-s + (−1.16 − 1.16i)3-s + (0.997 + 0.0758i)4-s + (0.995 − 0.0968i)5-s + (1.11 + 1.20i)6-s + (1.24 − 1.24i)7-s + (−0.993 − 0.113i)8-s + 1.71i·9-s + (−0.998 + 0.0589i)10-s − 0.738i·11-s + (−1.07 − 1.24i)12-s + (0.603 − 0.603i)13-s + (−1.28 + 1.19i)14-s + (−1.27 − 1.04i)15-s + (0.988 + 0.151i)16-s + (0.0514 + 0.0514i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.839 + 0.543i$
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: \((2,\ 20,\ (\ :9/2),\ -0.839 + 0.543i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.255845 - 0.865848i\)
\(L(\frac12)\) \(\approx\) \(0.255845 - 0.865848i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (22.6 + 0.858i)T \)
5 \( 1 + (-1.39e3 + 135. i)T \)
good3 \( 1 + (163. + 163. i)T + 1.96e4iT^{2} \)
7 \( 1 + (-7.88e3 + 7.88e3i)T - 4.03e7iT^{2} \)
11 \( 1 + 3.58e4iT - 2.35e9T^{2} \)
13 \( 1 + (-6.21e4 + 6.21e4i)T - 1.06e10iT^{2} \)
17 \( 1 + (-1.77e4 - 1.77e4i)T + 1.18e11iT^{2} \)
19 \( 1 + 3.81e5T + 3.22e11T^{2} \)
23 \( 1 + (1.41e5 + 1.41e5i)T + 1.80e12iT^{2} \)
29 \( 1 - 1.44e6iT - 1.45e13T^{2} \)
31 \( 1 - 4.97e5iT - 2.64e13T^{2} \)
37 \( 1 + (1.34e7 + 1.34e7i)T + 1.29e14iT^{2} \)
41 \( 1 - 2.65e6T + 3.27e14T^{2} \)
43 \( 1 + (-2.77e6 - 2.77e6i)T + 5.02e14iT^{2} \)
47 \( 1 + (1.64e7 - 1.64e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (2.01e7 - 2.01e7i)T - 3.29e15iT^{2} \)
59 \( 1 + 7.03e7T + 8.66e15T^{2} \)
61 \( 1 - 1.52e8T + 1.16e16T^{2} \)
67 \( 1 + (-9.27e7 + 9.27e7i)T - 2.72e16iT^{2} \)
71 \( 1 - 4.19e8iT - 4.58e16T^{2} \)
73 \( 1 + (7.79e7 - 7.79e7i)T - 5.88e16iT^{2} \)
79 \( 1 - 3.89e5T + 1.19e17T^{2} \)
83 \( 1 + (2.65e8 + 2.65e8i)T + 1.86e17iT^{2} \)
89 \( 1 - 4.33e8iT - 3.50e17T^{2} \)
97 \( 1 + (-5.75e8 - 5.75e8i)T + 7.60e17iT^{2} \)
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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

−16.46402430108382312877107620859, −14.11582591502254162159961464569, −12.80095864559116819923661695778, −11.18565627823759740019523034538, −10.56265002643734746940236178367, −8.277614187944972261417051352262, −6.95255039314133631632812862800, −5.66547089369097064873081872243, −1.64453806292625854495821806210, −0.70678062973494155389182807602, 1.85118121252833437948191203223, 5.06955579672048281607984414287, 6.25962032583413097302019948068, 8.757641346039095765033059613419, 9.911095735572942552772969530632, 11.05057696428510205477740162231, 12.00941470045197707253986758487, 14.75707434053004992342891452985, 15.68308224482277976888592679001, 17.01906905726615903254273972833

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