Properties

Label 2-20-4.3-c8-0-1
Degree $2$
Conductor $20$
Sign $-0.831 + 0.555i$
Analytic cond. $8.14757$
Root an. cond. $2.85439$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.64 + 15.3i)2-s + 75.7i·3-s + (−212. + 142. i)4-s − 279.·5-s + (−1.15e3 + 351. i)6-s + 210. i·7-s + (−3.16e3 − 2.60e3i)8-s + 823.·9-s + (−1.29e3 − 4.27e3i)10-s + 141. i·11-s + (−1.07e4 − 1.61e4i)12-s − 1.76e4·13-s + (−3.22e3 + 976. i)14-s − 2.11e4i·15-s + (2.51e4 − 6.05e4i)16-s − 1.02e5·17-s + ⋯
L(s)  = 1  + (0.290 + 0.957i)2-s + 0.935i·3-s + (−0.831 + 0.555i)4-s − 0.447·5-s + (−0.894 + 0.271i)6-s + 0.0876i·7-s + (−0.772 − 0.635i)8-s + 0.125·9-s + (−0.129 − 0.427i)10-s + 0.00967i·11-s + (−0.519 − 0.777i)12-s − 0.618·13-s + (−0.0838 + 0.0254i)14-s − 0.418i·15-s + (0.383 − 0.923i)16-s − 1.22·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 + 0.555i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.831 + 0.555i$
Analytic conductor: \(8.14757\)
Root analytic conductor: \(2.85439\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :4),\ -0.831 + 0.555i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.317248 - 1.04690i\)
\(L(\frac12)\) \(\approx\) \(0.317248 - 1.04690i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.64 - 15.3i)T \)
5 \( 1 + 279.T \)
good3 \( 1 - 75.7iT - 6.56e3T^{2} \)
7 \( 1 - 210. iT - 5.76e6T^{2} \)
11 \( 1 - 141. iT - 2.14e8T^{2} \)
13 \( 1 + 1.76e4T + 8.15e8T^{2} \)
17 \( 1 + 1.02e5T + 6.97e9T^{2} \)
19 \( 1 - 1.07e5iT - 1.69e10T^{2} \)
23 \( 1 - 4.55e5iT - 7.83e10T^{2} \)
29 \( 1 - 8.65e5T + 5.00e11T^{2} \)
31 \( 1 - 4.29e5iT - 8.52e11T^{2} \)
37 \( 1 + 2.51e6T + 3.51e12T^{2} \)
41 \( 1 + 2.98e6T + 7.98e12T^{2} \)
43 \( 1 - 2.22e6iT - 1.16e13T^{2} \)
47 \( 1 + 7.63e6iT - 2.38e13T^{2} \)
53 \( 1 - 1.55e7T + 6.22e13T^{2} \)
59 \( 1 - 6.38e6iT - 1.46e14T^{2} \)
61 \( 1 - 2.03e6T + 1.91e14T^{2} \)
67 \( 1 - 2.03e7iT - 4.06e14T^{2} \)
71 \( 1 + 4.44e7iT - 6.45e14T^{2} \)
73 \( 1 - 1.79e7T + 8.06e14T^{2} \)
79 \( 1 - 2.03e7iT - 1.51e15T^{2} \)
83 \( 1 - 5.09e7iT - 2.25e15T^{2} \)
89 \( 1 + 2.68e7T + 3.93e15T^{2} \)
97 \( 1 - 3.38e7T + 7.83e15T^{2} \)
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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

−16.82505056881827895822532004236, −15.70539279619719810350025928704, −15.09349371216850227744702508251, −13.61194151266394672360615038596, −12.04107935202346541955117258646, −10.12396805961180226232645205038, −8.722489714291405577321319520713, −7.08207217789547570048866157648, −5.11341761834526078775171912163, −3.78346484632197283963486440159, 0.51280250130057676707105705305, 2.36301827240988936462333585362, 4.52563296332139867404795177551, 6.80601837928475869919397065408, 8.637286438227922853592375218164, 10.44671025903907878716636230989, 11.88157918035344887496121095072, 12.82474536826887540880631708740, 13.92402162098547023411917041904, 15.42633654764620041575424717724

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