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Properties

Label 2-20-4.3-c4-0-2
Degree $2$
Conductor $20$
Sign $0.258 - 0.965i$
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  + (2.43 + 3.17i)2-s + 3.20i·3-s + (−4.13 + 15.4i)4-s + 11.1·5-s + (−10.1 + 7.80i)6-s − 30.6i·7-s + (−59.1 + 24.4i)8-s + 70.7·9-s + (27.2 + 35.4i)10-s − 168. i·11-s + (−49.5 − 13.2i)12-s + 3.15·13-s + (97.1 − 74.5i)14-s + 35.8i·15-s + (−221. − 127. i)16-s − 229.·17-s + ⋯
L(s)  = 1  + (0.608 + 0.793i)2-s + 0.356i·3-s + (−0.258 + 0.965i)4-s + 0.447·5-s + (−0.282 + 0.216i)6-s − 0.624i·7-s + (−0.923 + 0.382i)8-s + 0.873·9-s + (0.272 + 0.354i)10-s − 1.38i·11-s + (−0.344 − 0.0921i)12-s + 0.0186·13-s + (0.495 − 0.380i)14-s + 0.159i·15-s + (−0.866 − 0.499i)16-s − 0.793·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.258 - 0.965i$
Analytic conductor: \(2.06739\)
Root analytic conductor: \(1.43784\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :2),\ 0.258 - 0.965i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.33583 + 1.02511i\)
\(L(\frac12)\) \(\approx\) \(1.33583 + 1.02511i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.43 - 3.17i)T \)
5 \( 1 - 11.1T \)
good3 \( 1 - 3.20iT - 81T^{2} \)
7 \( 1 + 30.6iT - 2.40e3T^{2} \)
11 \( 1 + 168. iT - 1.46e4T^{2} \)
13 \( 1 - 3.15T + 2.85e4T^{2} \)
17 \( 1 + 229.T + 8.35e4T^{2} \)
19 \( 1 - 151. iT - 1.30e5T^{2} \)
23 \( 1 - 916. iT - 2.79e5T^{2} \)
29 \( 1 + 1.00e3T + 7.07e5T^{2} \)
31 \( 1 + 1.72e3iT - 9.23e5T^{2} \)
37 \( 1 + 841.T + 1.87e6T^{2} \)
41 \( 1 - 2.52e3T + 2.82e6T^{2} \)
43 \( 1 + 1.65e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.66e3iT - 4.87e6T^{2} \)
53 \( 1 + 3.14e3T + 7.89e6T^{2} \)
59 \( 1 - 3.26e3iT - 1.21e7T^{2} \)
61 \( 1 + 767.T + 1.38e7T^{2} \)
67 \( 1 - 1.85e3iT - 2.01e7T^{2} \)
71 \( 1 - 2.77e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.25e3T + 2.83e7T^{2} \)
79 \( 1 + 906. iT - 3.89e7T^{2} \)
83 \( 1 - 3.83e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.82e3T + 6.27e7T^{2} \)
97 \( 1 - 1.58e4T + 8.85e7T^{2} \)
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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

−17.42845833837956121632572836414, −16.39640850287349614765762224649, −15.36774508123418342351583938902, −13.87924488822210625605550710863, −13.08730210297430793132338777404, −11.14978042903116615102205827623, −9.325136789413783728127649526665, −7.52117797679546948201621203769, −5.82080121133201692574235608822, −3.91401636656093178141542123574, 2.07210826989337508159763204260, 4.71969000723616621845960033482, 6.69358384960516761804748623318, 9.249105998536536462317778802839, 10.56535111633708951167984621059, 12.31277330932764530546147544709, 12.99676901462732018616288329969, 14.52296118416410817867776830213, 15.65737701951826482847856346102, 17.80864292204767684780805736436

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