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Properties

Label 2-20-5.4-c11-0-4
Degree $2$
Conductor $20$
Sign $-0.412 + 0.911i$
Analytic cond. $15.3668$
Root an. cond. $3.92005$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 48.1i·3-s + (−2.87e3 + 6.36e3i)5-s − 3.79e4i·7-s + 1.74e5·9-s − 6.68e5·11-s − 5.15e5i·13-s + (3.06e5 + 1.38e5i)15-s − 1.14e7i·17-s − 1.10e7·19-s − 1.82e6·21-s − 4.67e7i·23-s + (−3.22e7 − 3.66e7i)25-s − 1.69e7i·27-s + 1.68e7·29-s − 7.80e7·31-s + ⋯
L(s)  = 1  − 0.114i·3-s + (−0.412 + 0.911i)5-s − 0.852i·7-s + 0.986·9-s − 1.25·11-s − 0.384i·13-s + (0.104 + 0.0471i)15-s − 1.96i·17-s − 1.02·19-s − 0.0974·21-s − 1.51i·23-s + (−0.660 − 0.750i)25-s − 0.227i·27-s + 0.152·29-s − 0.489·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.412 + 0.911i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.412 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.412 + 0.911i$
Analytic conductor: \(15.3668\)
Root analytic conductor: \(3.92005\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :11/2),\ -0.412 + 0.911i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.512282 - 0.793886i\)
\(L(\frac12)\) \(\approx\) \(0.512282 - 0.793886i\)
\(L(\frac{13}{2})\) 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 \( 1 + (2.87e3 - 6.36e3i)T \)
good3 \( 1 + 48.1iT - 1.77e5T^{2} \)
7 \( 1 + 3.79e4iT - 1.97e9T^{2} \)
11 \( 1 + 6.68e5T + 2.85e11T^{2} \)
13 \( 1 + 5.15e5iT - 1.79e12T^{2} \)
17 \( 1 + 1.14e7iT - 3.42e13T^{2} \)
19 \( 1 + 1.10e7T + 1.16e14T^{2} \)
23 \( 1 + 4.67e7iT - 9.52e14T^{2} \)
29 \( 1 - 1.68e7T + 1.22e16T^{2} \)
31 \( 1 + 7.80e7T + 2.54e16T^{2} \)
37 \( 1 - 4.07e8iT - 1.77e17T^{2} \)
41 \( 1 - 7.02e8T + 5.50e17T^{2} \)
43 \( 1 - 9.99e8iT - 9.29e17T^{2} \)
47 \( 1 + 4.52e8iT - 2.47e18T^{2} \)
53 \( 1 + 2.24e9iT - 9.26e18T^{2} \)
59 \( 1 + 8.02e9T + 3.01e19T^{2} \)
61 \( 1 + 2.42e9T + 4.35e19T^{2} \)
67 \( 1 + 1.19e10iT - 1.22e20T^{2} \)
71 \( 1 - 1.02e10T + 2.31e20T^{2} \)
73 \( 1 - 2.04e10iT - 3.13e20T^{2} \)
79 \( 1 + 2.27e10T + 7.47e20T^{2} \)
83 \( 1 - 8.74e9iT - 1.28e21T^{2} \)
89 \( 1 + 6.71e10T + 2.77e21T^{2} \)
97 \( 1 - 5.17e10iT - 7.15e21T^{2} \)
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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

−15.32777855627375502846212612242, −13.94172217677130033310510479755, −12.71700583073271481757112952542, −10.97003245447737392291587506457, −10.06813777831638391710330518651, −7.80897742137490446748481116329, −6.81360293435333475676406033385, −4.53104160879952833389209261609, −2.72197099416164675640839306310, −0.36979763856101680322616396707, 1.77891590037431060162781594801, 4.11377967944746239481871467344, 5.65053365843206122528623388215, 7.78701956373817394631629946727, 9.058019588312729522232668245131, 10.64842282136836115066693267686, 12.38690810416928363416996792779, 13.09324898989239816376298050229, 15.20462598627024062830387059004, 15.85724001366610426193725060947

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