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Properties

Label 2-20-4.3-c6-0-5
Degree $2$
Conductor $20$
Sign $0.494 + 0.869i$
Analytic cond. $4.60108$
Root an. cond. $2.14501$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.91 − 4.02i)2-s + 5.41i·3-s + (31.6 + 55.6i)4-s + 55.9·5-s + (21.7 − 37.4i)6-s − 454. i·7-s + (4.80 − 511. i)8-s + 699.·9-s + (−386. − 224. i)10-s − 1.15e3i·11-s + (−301. + 171. i)12-s + 2.20e3·13-s + (−1.82e3 + 3.14e3i)14-s + 302. i·15-s + (−2.09e3 + 3.52e3i)16-s + 1.11e3·17-s + ⋯
L(s)  = 1  + (−0.864 − 0.502i)2-s + 0.200i·3-s + (0.494 + 0.869i)4-s + 0.447·5-s + (0.100 − 0.173i)6-s − 1.32i·7-s + (0.00938 − 0.999i)8-s + 0.959·9-s + (−0.386 − 0.224i)10-s − 0.868i·11-s + (−0.174 + 0.0991i)12-s + 1.00·13-s + (−0.665 + 1.14i)14-s + 0.0896i·15-s + (−0.510 + 0.859i)16-s + 0.226·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.494 + 0.869i$
Analytic conductor: \(4.60108\)
Root analytic conductor: \(2.14501\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :3),\ 0.494 + 0.869i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.984722 - 0.572646i\)
\(L(\frac12)\) \(\approx\) \(0.984722 - 0.572646i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (6.91 + 4.02i)T \)
5 \( 1 - 55.9T \)
good3 \( 1 - 5.41iT - 729T^{2} \)
7 \( 1 + 454. iT - 1.17e5T^{2} \)
11 \( 1 + 1.15e3iT - 1.77e6T^{2} \)
13 \( 1 - 2.20e3T + 4.82e6T^{2} \)
17 \( 1 - 1.11e3T + 2.41e7T^{2} \)
19 \( 1 - 6.70e3iT - 4.70e7T^{2} \)
23 \( 1 + 2.20e4iT - 1.48e8T^{2} \)
29 \( 1 + 5.47e3T + 5.94e8T^{2} \)
31 \( 1 - 2.83e4iT - 8.87e8T^{2} \)
37 \( 1 + 7.30e4T + 2.56e9T^{2} \)
41 \( 1 + 3.12e4T + 4.75e9T^{2} \)
43 \( 1 - 2.28e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.01e5iT - 1.07e10T^{2} \)
53 \( 1 - 1.03e5T + 2.21e10T^{2} \)
59 \( 1 + 1.37e5iT - 4.21e10T^{2} \)
61 \( 1 - 4.52e5T + 5.15e10T^{2} \)
67 \( 1 - 1.43e5iT - 9.04e10T^{2} \)
71 \( 1 + 4.59e5iT - 1.28e11T^{2} \)
73 \( 1 + 3.46e5T + 1.51e11T^{2} \)
79 \( 1 - 8.93e5iT - 2.43e11T^{2} \)
83 \( 1 - 9.27e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.10e6T + 4.96e11T^{2} \)
97 \( 1 + 1.05e6T + 8.32e11T^{2} \)
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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

−16.79094294767159271627940739357, −16.08446523984173208005501756422, −13.95608660064435690926684419994, −12.70907646113149055672640627594, −10.82617019692286079455334162911, −10.11881909667562652895311960957, −8.414781829073464493446085241549, −6.77646380615784766112582266671, −3.78891041213750271398590148897, −1.13730905757155353766513266539, 1.81377496758888941444489507944, 5.55209603253967433880635918741, 7.11557644390096060429476235984, 8.839844473913011170983018070379, 9.953801643830058111283306267456, 11.69161299048797233639498050280, 13.33473410063947562322424581733, 15.16872899111302094595684775464, 15.79580374289155684372191777486, 17.50516303261971937373752012642

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