Properties

Label 2-20-5.4-c9-0-3
Degree $2$
Conductor $20$
Sign $-0.805 + 0.592i$
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  − 188. i·3-s + (1.12e3 − 827. i)5-s − 1.84e3i·7-s − 1.57e4·9-s − 5.86e4·11-s − 1.98e4i·13-s + (−1.55e5 − 2.11e5i)15-s − 4.41e5i·17-s − 5.52e5·19-s − 3.46e5·21-s + 2.52e6i·23-s + (5.83e5 − 1.86e6i)25-s − 7.45e5i·27-s + 5.88e6·29-s + 3.66e6·31-s + ⋯
L(s)  = 1  − 1.34i·3-s + (0.805 − 0.592i)5-s − 0.290i·7-s − 0.798·9-s − 1.20·11-s − 0.192i·13-s + (−0.793 − 1.08i)15-s − 1.28i·17-s − 0.973·19-s − 0.389·21-s + 1.87i·23-s + (0.298 − 0.954i)25-s − 0.270i·27-s + 1.54·29-s + 0.713·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 + 0.592i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.805 + 0.592i$
Analytic conductor: \(10.3007\)
Root analytic conductor: \(3.20947\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :9/2),\ -0.805 + 0.592i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.500865 - 1.52782i\)
\(L(\frac12)\) \(\approx\) \(0.500865 - 1.52782i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.12e3 + 827. i)T \)
good3 \( 1 + 188. iT - 1.96e4T^{2} \)
7 \( 1 + 1.84e3iT - 4.03e7T^{2} \)
11 \( 1 + 5.86e4T + 2.35e9T^{2} \)
13 \( 1 + 1.98e4iT - 1.06e10T^{2} \)
17 \( 1 + 4.41e5iT - 1.18e11T^{2} \)
19 \( 1 + 5.52e5T + 3.22e11T^{2} \)
23 \( 1 - 2.52e6iT - 1.80e12T^{2} \)
29 \( 1 - 5.88e6T + 1.45e13T^{2} \)
31 \( 1 - 3.66e6T + 2.64e13T^{2} \)
37 \( 1 - 5.09e6iT - 1.29e14T^{2} \)
41 \( 1 + 6.04e6T + 3.27e14T^{2} \)
43 \( 1 + 3.72e7iT - 5.02e14T^{2} \)
47 \( 1 + 2.53e7iT - 1.11e15T^{2} \)
53 \( 1 + 6.60e7iT - 3.29e15T^{2} \)
59 \( 1 - 8.69e7T + 8.66e15T^{2} \)
61 \( 1 - 1.37e8T + 1.16e16T^{2} \)
67 \( 1 + 8.79e7iT - 2.72e16T^{2} \)
71 \( 1 + 2.06e8T + 4.58e16T^{2} \)
73 \( 1 - 3.67e8iT - 5.88e16T^{2} \)
79 \( 1 - 3.10e8T + 1.19e17T^{2} \)
83 \( 1 - 1.90e8iT - 1.86e17T^{2} \)
89 \( 1 - 1.15e8T + 3.50e17T^{2} \)
97 \( 1 - 1.45e9iT - 7.60e17T^{2} \)
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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

−15.78238154292439161159429292131, −13.73314688644223047834749510753, −13.24160099009959189204346377886, −11.98606079222486151300111152813, −10.09462104451817059545417671303, −8.306664121623483277076990165800, −6.93584124157412999645812275227, −5.31349708618594896827777726877, −2.30817244625680916472162502770, −0.74568161285112177365677040010, 2.62898624204244308987092990620, 4.57230197280096095532331445340, 6.18237397002615047002713785311, 8.579167606187041829929906428618, 10.18418884109282435000271511635, 10.67511409237740328380888084798, 12.83102210160281865391965084127, 14.45669354315919355761733823702, 15.36093858103003075654782957055, 16.53947329734860525332502351977

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