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Properties

Label 2-20-20.7-c9-0-3
Degree $2$
Conductor $20$
Sign $0.982 + 0.184i$
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  + (−20.8 − 8.80i)2-s + (−99.4 − 99.4i)3-s + (356. + 367. i)4-s + (−1.39e3 + 84.3i)5-s + (1.19e3 + 2.94e3i)6-s + (−5.88e3 + 5.88e3i)7-s + (−4.20e3 − 1.07e4i)8-s + 86.8i·9-s + (2.98e4 + 1.05e4i)10-s + 3.23e4i·11-s + (1.00e3 − 7.19e4i)12-s + (1.17e5 − 1.17e5i)13-s + (1.74e5 − 7.08e4i)14-s + (1.47e5 + 1.30e5i)15-s + (−7.32e3 + 2.62e5i)16-s + (−9.11e4 − 9.11e4i)17-s + ⋯
L(s)  = 1  + (−0.921 − 0.389i)2-s + (−0.708 − 0.708i)3-s + (0.697 + 0.716i)4-s + (−0.998 + 0.0603i)5-s + (0.377 + 0.928i)6-s + (−0.925 + 0.925i)7-s + (−0.363 − 0.931i)8-s + 0.00441i·9-s + (0.942 + 0.332i)10-s + 0.665i·11-s + (0.0139 − 1.00i)12-s + (1.14 − 1.14i)13-s + (1.21 − 0.492i)14-s + (0.750 + 0.664i)15-s + (−0.0279 + 0.999i)16-s + (−0.264 − 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.184i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.982 + 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.982 + 0.184i$
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: \((2,\ 20,\ (\ :9/2),\ 0.982 + 0.184i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.524532 - 0.0487686i\)
\(L(\frac12)\) \(\approx\) \(0.524532 - 0.0487686i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (20.8 + 8.80i)T \)
5 \( 1 + (1.39e3 - 84.3i)T \)
good3 \( 1 + (99.4 + 99.4i)T + 1.96e4iT^{2} \)
7 \( 1 + (5.88e3 - 5.88e3i)T - 4.03e7iT^{2} \)
11 \( 1 - 3.23e4iT - 2.35e9T^{2} \)
13 \( 1 + (-1.17e5 + 1.17e5i)T - 1.06e10iT^{2} \)
17 \( 1 + (9.11e4 + 9.11e4i)T + 1.18e11iT^{2} \)
19 \( 1 - 6.11e5T + 3.22e11T^{2} \)
23 \( 1 + (6.78e5 + 6.78e5i)T + 1.80e12iT^{2} \)
29 \( 1 - 2.06e6iT - 1.45e13T^{2} \)
31 \( 1 - 6.14e6iT - 2.64e13T^{2} \)
37 \( 1 + (-6.20e5 - 6.20e5i)T + 1.29e14iT^{2} \)
41 \( 1 - 2.64e7T + 3.27e14T^{2} \)
43 \( 1 + (-2.48e7 - 2.48e7i)T + 5.02e14iT^{2} \)
47 \( 1 + (-2.21e7 + 2.21e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (4.15e7 - 4.15e7i)T - 3.29e15iT^{2} \)
59 \( 1 + 6.89e7T + 8.66e15T^{2} \)
61 \( 1 + 5.81e7T + 1.16e16T^{2} \)
67 \( 1 + (4.07e7 - 4.07e7i)T - 2.72e16iT^{2} \)
71 \( 1 + 2.26e8iT - 4.58e16T^{2} \)
73 \( 1 + (-7.83e7 + 7.83e7i)T - 5.88e16iT^{2} \)
79 \( 1 - 2.64e8T + 1.19e17T^{2} \)
83 \( 1 + (-4.17e8 - 4.17e8i)T + 1.86e17iT^{2} \)
89 \( 1 + 5.60e8iT - 3.50e17T^{2} \)
97 \( 1 + (-8.29e8 - 8.29e8i)T + 7.60e17iT^{2} \)
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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

−16.17353733211853942215074235066, −15.48798569734832716381664153494, −12.68006972750903831480070948490, −12.15450481960711101417420587644, −10.86628986191405033737921068540, −9.119860798398106754818423825740, −7.58392075829682528974984117145, −6.22163288716456753340708471345, −3.16114624935727659631964229081, −0.820264971000783105520440981937, 0.56776969791979139787618589361, 3.96410883323173930238248437035, 6.10901892390176373340017928936, 7.64630984134144694829108146366, 9.344441969208689954946894038298, 10.75894979211471554557707236277, 11.52907894238256092584248370291, 13.80810109984734012910605279772, 15.84148750134045071478799548970, 16.12594196946666105811257580093

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