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Properties

Label 2-20-20.19-c10-0-15
Degree $2$
Conductor $20$
Sign $0.703 + 0.710i$
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  + (0.302 − 31.9i)2-s + 375.·3-s + (−1.02e3 − 19.3i)4-s + (2.24e3 + 2.17e3i)5-s + (113. − 1.20e4i)6-s + 1.96e4·7-s + (−929. + 3.27e4i)8-s + 8.21e4·9-s + (7.03e4 − 7.10e4i)10-s + 1.63e5i·11-s + (−3.84e5 − 7.28e3i)12-s + 8.65e4i·13-s + (5.94e3 − 6.28e5i)14-s + (8.42e5 + 8.18e5i)15-s + (1.04e6 + 3.96e4i)16-s − 2.09e6i·17-s + ⋯
L(s)  = 1  + (0.00946 − 0.999i)2-s + 1.54·3-s + (−0.999 − 0.0189i)4-s + (0.717 + 0.696i)5-s + (0.0146 − 1.54i)6-s + 1.16·7-s + (−0.0283 + 0.999i)8-s + 1.39·9-s + (0.703 − 0.710i)10-s + 1.01i·11-s + (−1.54 − 0.0292i)12-s + 0.233i·13-s + (0.0110 − 1.16i)14-s + (1.10 + 1.07i)15-s + (0.999 + 0.0378i)16-s − 1.47i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.703 + 0.710i$
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: \((2,\ 20,\ (\ :5),\ 0.703 + 0.710i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(2.98698 - 1.24549i\)
\(L(\frac12)\) \(\approx\) \(2.98698 - 1.24549i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.302 + 31.9i)T \)
5 \( 1 + (-2.24e3 - 2.17e3i)T \)
good3 \( 1 - 375.T + 5.90e4T^{2} \)
7 \( 1 - 1.96e4T + 2.82e8T^{2} \)
11 \( 1 - 1.63e5iT - 2.59e10T^{2} \)
13 \( 1 - 8.65e4iT - 1.37e11T^{2} \)
17 \( 1 + 2.09e6iT - 2.01e12T^{2} \)
19 \( 1 + 3.13e6iT - 6.13e12T^{2} \)
23 \( 1 + 1.05e7T + 4.14e13T^{2} \)
29 \( 1 - 1.13e7T + 4.20e14T^{2} \)
31 \( 1 + 1.21e7iT - 8.19e14T^{2} \)
37 \( 1 - 4.87e7iT - 4.80e15T^{2} \)
41 \( 1 - 4.55e7T + 1.34e16T^{2} \)
43 \( 1 + 7.47e7T + 2.16e16T^{2} \)
47 \( 1 + 9.62e6T + 5.25e16T^{2} \)
53 \( 1 - 1.27e8iT - 1.74e17T^{2} \)
59 \( 1 - 1.18e9iT - 5.11e17T^{2} \)
61 \( 1 - 2.97e8T + 7.13e17T^{2} \)
67 \( 1 + 7.15e8T + 1.82e18T^{2} \)
71 \( 1 + 2.56e9iT - 3.25e18T^{2} \)
73 \( 1 + 2.10e9iT - 4.29e18T^{2} \)
79 \( 1 + 3.44e9iT - 9.46e18T^{2} \)
83 \( 1 + 7.09e9T + 1.55e19T^{2} \)
89 \( 1 + 3.39e9T + 3.11e19T^{2} \)
97 \( 1 - 5.71e9iT - 7.37e19T^{2} \)
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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

−15.08327640741974820686031978622, −14.15746449471751446259370307620, −13.51555738923734545765780957611, −11.66162134453212901660862729475, −10.06728629076461374483724182841, −9.059249706041117979122005276024, −7.59014514433065851453707305426, −4.59765047481343807442299569601, −2.73604364453197285035124886956, −1.83672911307003440530536307451, 1.61296337952159014708752248857, 3.96941460694529783930335939928, 5.78426821975052739172484882106, 8.229995907718952000226964268124, 8.373373742314937645684743908183, 9.997150663501894802183790686173, 12.78101194137243778358110668214, 14.08952094148526997358192080578, 14.42424517792979213383298213542, 15.91669025229337829270168313956

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