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Properties

Label 2-20-20.7-c5-0-4
Degree $2$
Conductor $20$
Sign $-0.349 - 0.936i$
Analytic cond. $3.20767$
Root an. cond. $1.79099$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.294 + 5.64i)2-s + (20.2 + 20.2i)3-s + (−31.8 + 3.32i)4-s + (16.5 − 53.4i)5-s + (−108. + 120. i)6-s + (−9.02 + 9.02i)7-s + (−28.1 − 178. i)8-s + 574. i·9-s + (306. + 77.6i)10-s − 200. i·11-s + (−710. − 576. i)12-s + (362. − 362. i)13-s + (−53.6 − 48.3i)14-s + (1.41e3 − 745. i)15-s + (1.00e3 − 211. i)16-s + (465. + 465. i)17-s + ⋯
L(s)  = 1  + (0.0520 + 0.998i)2-s + (1.29 + 1.29i)3-s + (−0.994 + 0.104i)4-s + (0.295 − 0.955i)5-s + (−1.22 + 1.36i)6-s + (−0.0695 + 0.0695i)7-s + (−0.155 − 0.987i)8-s + 2.36i·9-s + (0.969 + 0.245i)10-s − 0.500i·11-s + (−1.42 − 1.15i)12-s + (0.595 − 0.595i)13-s + (−0.0731 − 0.0658i)14-s + (1.62 − 0.855i)15-s + (0.978 − 0.206i)16-s + (0.391 + 0.391i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 - 0.936i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.349 - 0.936i$
Analytic conductor: \(3.20767\)
Root analytic conductor: \(1.79099\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5/2),\ -0.349 - 0.936i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.07900 + 1.55499i\)
\(L(\frac12)\) \(\approx\) \(1.07900 + 1.55499i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.294 - 5.64i)T \)
5 \( 1 + (-16.5 + 53.4i)T \)
good3 \( 1 + (-20.2 - 20.2i)T + 243iT^{2} \)
7 \( 1 + (9.02 - 9.02i)T - 1.68e4iT^{2} \)
11 \( 1 + 200. iT - 1.61e5T^{2} \)
13 \( 1 + (-362. + 362. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-465. - 465. i)T + 1.41e6iT^{2} \)
19 \( 1 + 582.T + 2.47e6T^{2} \)
23 \( 1 + (-285. - 285. i)T + 6.43e6iT^{2} \)
29 \( 1 - 1.27e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.26e3iT - 2.86e7T^{2} \)
37 \( 1 + (6.27e3 + 6.27e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 1.16e4T + 1.15e8T^{2} \)
43 \( 1 + (1.19e4 + 1.19e4i)T + 1.47e8iT^{2} \)
47 \( 1 + (-4.13e3 + 4.13e3i)T - 2.29e8iT^{2} \)
53 \( 1 + (1.37e4 - 1.37e4i)T - 4.18e8iT^{2} \)
59 \( 1 - 3.81e4T + 7.14e8T^{2} \)
61 \( 1 - 3.00e4T + 8.44e8T^{2} \)
67 \( 1 + (-3.07e3 + 3.07e3i)T - 1.35e9iT^{2} \)
71 \( 1 - 4.94e4iT - 1.80e9T^{2} \)
73 \( 1 + (-2.54e4 + 2.54e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 9.90e4T + 3.07e9T^{2} \)
83 \( 1 + (-3.98e4 - 3.98e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 6.51e4iT - 5.58e9T^{2} \)
97 \( 1 + (4.46e4 + 4.46e4i)T + 8.58e9iT^{2} \)
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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

−17.10359850975128744280523989743, −16.10521726488165235687565482715, −15.28573690362640693866207194866, −14.07611752778313434128231467436, −13.08276881718898005382721688472, −10.17350287124825224301043561705, −8.946958474529818489805168541648, −8.188601659013092789363166860324, −5.37256440118461801826644843127, −3.80654180545507355459495116940, 1.82172705817174291008327689619, 3.27420568838000422613445892132, 6.82250367384850787713001916882, 8.444388403916957993369298968198, 9.884402676685886346071464887348, 11.72110494090821775097951707759, 13.08710932992093638098698752526, 13.97083339095602761305336889960, 14.83608325546256486860204322575, 17.69987703106288698023111982054

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