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Properties

Label 2-20-20.19-c10-0-9
Degree $2$
Conductor $20$
Sign $-0.640 - 0.767i$
Analytic cond. $12.7071$
Root an. cond. $3.56470$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−21.8 + 23.3i)2-s + 416.·3-s + (−71.0 − 1.02e3i)4-s + (−2.53e3 + 1.83e3i)5-s + (−9.09e3 + 9.75e3i)6-s − 9.55e3·7-s + (2.54e4 + 2.06e4i)8-s + 1.14e5·9-s + (1.24e4 − 9.92e4i)10-s + 3.04e5i·11-s + (−2.96e4 − 4.25e5i)12-s + 2.93e4i·13-s + (2.08e5 − 2.23e5i)14-s + (−1.05e6 + 7.63e5i)15-s + (−1.03e6 + 1.45e5i)16-s + 1.45e6i·17-s + ⋯
L(s)  = 1  + (−0.682 + 0.731i)2-s + 1.71·3-s + (−0.0694 − 0.997i)4-s + (−0.810 + 0.585i)5-s + (−1.16 + 1.25i)6-s − 0.568·7-s + (0.776 + 0.629i)8-s + 1.94·9-s + (0.124 − 0.992i)10-s + 1.89i·11-s + (−0.119 − 1.71i)12-s + 0.0791i·13-s + (0.387 − 0.415i)14-s + (−1.38 + 1.00i)15-s + (−0.990 + 0.138i)16-s + 1.02i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 - 0.767i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.640 - 0.767i$
Analytic conductor: \(12.7071\)
Root analytic conductor: \(3.56470\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5),\ -0.640 - 0.767i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.674343 + 1.44135i\)
\(L(\frac12)\) \(\approx\) \(0.674343 + 1.44135i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (21.8 - 23.3i)T \)
5 \( 1 + (2.53e3 - 1.83e3i)T \)
good3 \( 1 - 416.T + 5.90e4T^{2} \)
7 \( 1 + 9.55e3T + 2.82e8T^{2} \)
11 \( 1 - 3.04e5iT - 2.59e10T^{2} \)
13 \( 1 - 2.93e4iT - 1.37e11T^{2} \)
17 \( 1 - 1.45e6iT - 2.01e12T^{2} \)
19 \( 1 - 6.37e5iT - 6.13e12T^{2} \)
23 \( 1 + 3.42e5T + 4.14e13T^{2} \)
29 \( 1 + 2.17e7T + 4.20e14T^{2} \)
31 \( 1 - 2.54e7iT - 8.19e14T^{2} \)
37 \( 1 + 6.66e7iT - 4.80e15T^{2} \)
41 \( 1 - 9.83e7T + 1.34e16T^{2} \)
43 \( 1 - 9.92e7T + 2.16e16T^{2} \)
47 \( 1 - 2.47e8T + 5.25e16T^{2} \)
53 \( 1 + 4.80e8iT - 1.74e17T^{2} \)
59 \( 1 - 4.28e8iT - 5.11e17T^{2} \)
61 \( 1 - 1.43e9T + 7.13e17T^{2} \)
67 \( 1 + 1.31e8T + 1.82e18T^{2} \)
71 \( 1 - 1.02e8iT - 3.25e18T^{2} \)
73 \( 1 + 8.16e8iT - 4.29e18T^{2} \)
79 \( 1 + 1.66e9iT - 9.46e18T^{2} \)
83 \( 1 - 4.49e9T + 1.55e19T^{2} \)
89 \( 1 + 1.63e9T + 3.11e19T^{2} \)
97 \( 1 - 8.94e9iT - 7.37e19T^{2} \)
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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

−15.95420708942311245062774574564, −15.01407716552106745862102696278, −14.46603201081321790803216495913, −12.76384773946842066053982484517, −10.30871822771864304177820887218, −9.214837685099300536985290972095, −7.85518175667995264438985015901, −6.99690720620908192786650436787, −4.00522269261802869478218495265, −2.09219733458712280686474376228, 0.70715787514894123115250868187, 2.81862352141199602605631941029, 3.81168556057219644362109663353, 7.59933144656690430785647143131, 8.613294072006695996626776430257, 9.438673138072242559104832052070, 11.30878828935454284443531779982, 12.93335183053730348627993243264, 13.84051736729807862908768238616, 15.65703826755914155353515333339

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