Properties

Label 2-20-20.7-c9-0-8
Degree $2$
Conductor $20$
Sign $0.747 - 0.664i$
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  + (−0.858 − 22.6i)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 + (1.31e3 + 1.15e4i)8-s + 3.37e4i·9-s + (−4.25e3 − 3.13e4i)10-s + 3.58e4i·11-s + (−8.97e4 − 7.70e4i)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  + (−0.0379 − 0.999i)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 + (0.113 + 0.993i)8-s + 1.71i·9-s + (−0.134 − 0.990i)10-s + 0.738i·11-s + (−1.24 − 1.07i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.747 - 0.664i$
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: \((2,\ 20,\ (\ :9/2),\ 0.747 - 0.664i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.10350 + 0.799579i\)
\(L(\frac12)\) \(\approx\) \(2.10350 + 0.799579i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.858 + 22.6i)T \)
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

−16.08737421725467345894675783949, −14.91241756683159934832388007517, −13.62106061720724793429127294520, −12.52465587715066895058635464609, −10.34507916713463641318488047610, −9.490991923312710150349640400629, −8.790229202572619711475127042814, −5.38994706103301350337768536721, −3.41720775356439162912415574286, −2.31866815231112966897229699954, 1.01534834711203043486275870186, 3.36980360966244101058956830031, 6.30682913752403635926104946860, 7.17049414458420301399963099423, 8.718708378876080369468213388793, 9.914768456648063776003472957309, 13.01541269217136085855677006520, 13.64298251541006510125420068662, 14.22250104925066848048499376161, 16.12153551504380796581448623628

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