Properties

Label 2-20-4.3-c10-0-0
Degree $2$
Conductor $20$
Sign $0.669 + 0.743i$
Analytic cond. $12.7071$
Root an. cond. $3.56470$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−13.0 + 29.2i)2-s + 448. i·3-s + (−685. − 760. i)4-s − 1.39e3·5-s + (−1.31e4 − 5.84e3i)6-s − 1.93e4i·7-s + (3.11e4 − 1.01e4i)8-s − 1.42e5·9-s + (1.81e4 − 4.08e4i)10-s + 2.07e5i·11-s + (3.41e5 − 3.07e5i)12-s + 6.51e4·13-s + (5.65e5 + 2.51e5i)14-s − 6.27e5i·15-s + (−1.09e5 + 1.04e6i)16-s − 8.05e5·17-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + 1.84i·3-s + (−0.669 − 0.743i)4-s − 0.447·5-s + (−1.68 − 0.751i)6-s − 1.15i·7-s + (0.951 − 0.309i)8-s − 2.40·9-s + (0.181 − 0.408i)10-s + 1.28i·11-s + (1.37 − 1.23i)12-s + 0.175·13-s + (1.05 + 0.467i)14-s − 0.825i·15-s + (−0.104 + 0.994i)16-s − 0.567·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.743i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.669 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.669 + 0.743i$
Analytic conductor: \(12.7071\)
Root analytic conductor: \(3.56470\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5),\ 0.669 + 0.743i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.0189898 - 0.00845473i\)
\(L(\frac12)\) \(\approx\) \(0.0189898 - 0.00845473i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (13.0 - 29.2i)T \)
5 \( 1 + 1.39e3T \)
good3 \( 1 - 448. iT - 5.90e4T^{2} \)
7 \( 1 + 1.93e4iT - 2.82e8T^{2} \)
11 \( 1 - 2.07e5iT - 2.59e10T^{2} \)
13 \( 1 - 6.51e4T + 1.37e11T^{2} \)
17 \( 1 + 8.05e5T + 2.01e12T^{2} \)
19 \( 1 + 4.08e6iT - 6.13e12T^{2} \)
23 \( 1 - 4.40e6iT - 4.14e13T^{2} \)
29 \( 1 + 1.06e7T + 4.20e14T^{2} \)
31 \( 1 - 2.90e6iT - 8.19e14T^{2} \)
37 \( 1 + 1.45e7T + 4.80e15T^{2} \)
41 \( 1 + 6.38e7T + 1.34e16T^{2} \)
43 \( 1 + 2.40e8iT - 2.16e16T^{2} \)
47 \( 1 + 1.51e8iT - 5.25e16T^{2} \)
53 \( 1 + 3.71e8T + 1.74e17T^{2} \)
59 \( 1 - 4.69e8iT - 5.11e17T^{2} \)
61 \( 1 + 7.97e8T + 7.13e17T^{2} \)
67 \( 1 + 8.39e8iT - 1.82e18T^{2} \)
71 \( 1 - 4.87e8iT - 3.25e18T^{2} \)
73 \( 1 + 3.05e9T + 4.29e18T^{2} \)
79 \( 1 - 4.58e9iT - 9.46e18T^{2} \)
83 \( 1 + 2.95e8iT - 1.55e19T^{2} \)
89 \( 1 + 3.76e9T + 3.11e19T^{2} \)
97 \( 1 + 1.37e10T + 7.37e19T^{2} \)
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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

−16.84810624676600089604346580008, −15.61638784863825840563200537022, −15.11950530605686762710706087615, −13.71740374432449866365668181785, −11.03551500338788280648649044494, −10.03481059399219208611067964833, −8.944125781714116022391344826971, −7.16687777061367211394987889980, −4.94965499225494605669986021786, −4.02316015453618254405685050975, 0.01044341629908163003304503329, 1.54788947822112263941658400160, 2.96298089212258536703315888481, 6.03284008391549787241601638871, 7.936685535672601435658896877204, 8.735575800626071798387480855543, 11.22195224747618731683732798556, 12.12373072044653292178192930060, 13.00040161753751471368097727884, 14.26041285822297613115214468845

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