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Properties

Label 2-20-20.19-c4-0-6
Degree $2$
Conductor $20$
Sign $0.371 + 0.928i$
Analytic cond. $2.06739$
Root an. cond. $1.43784$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.35 − 3.76i)2-s + 13.9·3-s + (−12.3 + 10.2i)4-s + (7.64 − 23.8i)5-s + (−18.9 − 52.6i)6-s − 35.6·7-s + (55.1 + 32.5i)8-s + 114.·9-s + (−99.9 + 3.47i)10-s + 169. i·11-s + (−172. + 142. i)12-s + 72.8i·13-s + (48.3 + 134. i)14-s + (107. − 333. i)15-s + (47.7 − 251. i)16-s − 25.2i·17-s + ⋯
L(s)  = 1  + (−0.338 − 0.940i)2-s + 1.55·3-s + (−0.770 + 0.637i)4-s + (0.305 − 0.952i)5-s + (−0.527 − 1.46i)6-s − 0.728·7-s + (0.861 + 0.508i)8-s + 1.41·9-s + (−0.999 + 0.0347i)10-s + 1.40i·11-s + (−1.19 + 0.991i)12-s + 0.430i·13-s + (0.246 + 0.685i)14-s + (0.475 − 1.48i)15-s + (0.186 − 0.982i)16-s − 0.0872i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.371 + 0.928i$
Analytic conductor: \(2.06739\)
Root analytic conductor: \(1.43784\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :2),\ 0.371 + 0.928i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.23612 - 0.836852i\)
\(L(\frac12)\) \(\approx\) \(1.23612 - 0.836852i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 3.76i)T \)
5 \( 1 + (-7.64 + 23.8i)T \)
good3 \( 1 - 13.9T + 81T^{2} \)
7 \( 1 + 35.6T + 2.40e3T^{2} \)
11 \( 1 - 169. iT - 1.46e4T^{2} \)
13 \( 1 - 72.8iT - 2.85e4T^{2} \)
17 \( 1 + 25.2iT - 8.35e4T^{2} \)
19 \( 1 - 156. iT - 1.30e5T^{2} \)
23 \( 1 + 420.T + 2.79e5T^{2} \)
29 \( 1 - 439.T + 7.07e5T^{2} \)
31 \( 1 + 1.00e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.72e3iT - 1.87e6T^{2} \)
41 \( 1 - 2.18e3T + 2.82e6T^{2} \)
43 \( 1 + 180.T + 3.41e6T^{2} \)
47 \( 1 + 270.T + 4.87e6T^{2} \)
53 \( 1 - 924. iT - 7.89e6T^{2} \)
59 \( 1 + 4.12e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.49e3T + 1.38e7T^{2} \)
67 \( 1 + 860.T + 2.01e7T^{2} \)
71 \( 1 - 8.03e3iT - 2.54e7T^{2} \)
73 \( 1 - 5.77e3iT - 2.83e7T^{2} \)
79 \( 1 + 4.40e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.04e4T + 4.74e7T^{2} \)
89 \( 1 - 1.20e4T + 6.27e7T^{2} \)
97 \( 1 - 8.15e3iT - 8.85e7T^{2} \)
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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

−17.61302205932190301293927975197, −16.08372087889311352490684218659, −14.37216906258502452152301432647, −13.19942137901839736850993255520, −12.33397504478160244192328225419, −9.842390381144462123740718833051, −9.216248332609724952081728468795, −7.83455007989266439350711825531, −4.13706915182706263234309562251, −2.13154105056386801887373636923, 3.23206086795810829610067477545, 6.33723497715289373747860416905, 7.896122686740347528096315805390, 9.123534766140329672123022496420, 10.38876913345657435353150395910, 13.40816415266791478807612970366, 14.07540875739039735047332358236, 15.14951901237577000451377293655, 16.20946122343322949821123520024, 17.99219511732158806548869770096

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