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Properties

Label 2-20-5.4-c3-0-0
Degree $2$
Conductor $20$
Sign $0.626 - 0.779i$
Analytic cond. $1.18003$
Root an. cond. $1.08629$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.71i·3-s + (7 − 8.71i)5-s − 8.71i·7-s − 49.0·9-s + 20·11-s − 52.3i·13-s + (76.0 + 61.0i)15-s + 69.7i·17-s − 84·19-s + 76.0·21-s + 61.0i·23-s + (−27.0 − 122. i)25-s − 191. i·27-s + 6·29-s − 224·31-s + ⋯
L(s)  = 1  + 1.67i·3-s + (0.626 − 0.779i)5-s − 0.470i·7-s − 1.81·9-s + 0.548·11-s − 1.11i·13-s + (1.30 + 1.05i)15-s + 0.995i·17-s − 1.01·19-s + 0.789·21-s + 0.553i·23-s + (−0.216 − 0.976i)25-s − 1.36i·27-s + 0.0384·29-s − 1.29·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.626 - 0.779i$
Analytic conductor: \(1.18003\)
Root analytic conductor: \(1.08629\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :3/2),\ 0.626 - 0.779i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.01897 + 0.488615i\)
\(L(\frac12)\) \(\approx\) \(1.01897 + 0.488615i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-7 + 8.71i)T \)
good3 \( 1 - 8.71iT - 27T^{2} \)
7 \( 1 + 8.71iT - 343T^{2} \)
11 \( 1 - 20T + 1.33e3T^{2} \)
13 \( 1 + 52.3iT - 2.19e3T^{2} \)
17 \( 1 - 69.7iT - 4.91e3T^{2} \)
19 \( 1 + 84T + 6.85e3T^{2} \)
23 \( 1 - 61.0iT - 1.21e4T^{2} \)
29 \( 1 - 6T + 2.43e4T^{2} \)
31 \( 1 + 224T + 2.97e4T^{2} \)
37 \( 1 - 122. iT - 5.06e4T^{2} \)
41 \( 1 - 266T + 6.89e4T^{2} \)
43 \( 1 + 305. iT - 7.95e4T^{2} \)
47 \( 1 - 374. iT - 1.03e5T^{2} \)
53 \( 1 + 366. iT - 1.48e5T^{2} \)
59 \( 1 + 28T + 2.05e5T^{2} \)
61 \( 1 - 182T + 2.26e5T^{2} \)
67 \( 1 - 427. iT - 3.00e5T^{2} \)
71 \( 1 - 408T + 3.57e5T^{2} \)
73 \( 1 - 1.08e3iT - 3.89e5T^{2} \)
79 \( 1 - 48T + 4.93e5T^{2} \)
83 \( 1 + 200. iT - 5.71e5T^{2} \)
89 \( 1 + 1.52e3T + 7.04e5T^{2} \)
97 \( 1 - 557. iT - 9.12e5T^{2} \)
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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

−17.43398165083592770586898744393, −16.78745847343915461491075569813, −15.57366033980970387120580076788, −14.45371790187605513640779832026, −12.84060968202092836931152176590, −10.84802298015777363479779749574, −9.846903727540078403039718758298, −8.608894150981030211255287408288, −5.60960235659322356340998779670, −4.04559734726076053379783345269, 2.17574836811444390244578782448, 6.19234148341793526883379278252, 7.20124157754998590148797684238, 9.068424520127858991695121819413, 11.25278276316939258229292415292, 12.46815259895530776601873678406, 13.75028849132029244324851523436, 14.61699373435934343023098856324, 16.80385419446652342359555965183, 18.08083386241452452491186441512

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