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Properties

Label 2-20-4.3-c10-0-6
Degree $2$
Conductor $20$
Sign $-0.149 - 0.988i$
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  + (24.2 + 20.8i)2-s − 208. i·3-s + (153. + 1.01e3i)4-s − 1.39e3·5-s + (4.35e3 − 5.06e3i)6-s + 1.75e4i·7-s + (−1.74e4 + 2.77e4i)8-s + 1.54e4·9-s + (−3.39e4 − 2.91e4i)10-s + 2.65e5i·11-s + (2.11e5 − 3.20e4i)12-s + 6.47e5·13-s + (−3.66e5 + 4.25e5i)14-s + 2.91e5i·15-s + (−1.00e6 + 3.10e5i)16-s − 2.51e6·17-s + ⋯
L(s)  = 1  + (0.758 + 0.652i)2-s − 0.859i·3-s + (0.149 + 0.988i)4-s − 0.447·5-s + (0.560 − 0.651i)6-s + 1.04i·7-s + (−0.531 + 0.847i)8-s + 0.261·9-s + (−0.339 − 0.291i)10-s + 1.65i·11-s + (0.849 − 0.128i)12-s + 1.74·13-s + (−0.681 + 0.792i)14-s + 0.384i·15-s + (−0.955 + 0.296i)16-s − 1.77·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.149 - 0.988i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.149 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.149 - 0.988i$
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: \((2,\ 20,\ (\ :5),\ -0.149 - 0.988i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.54203 + 1.79314i\)
\(L(\frac12)\) \(\approx\) \(1.54203 + 1.79314i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-24.2 - 20.8i)T \)
5 \( 1 + 1.39e3T \)
good3 \( 1 + 208. iT - 5.90e4T^{2} \)
7 \( 1 - 1.75e4iT - 2.82e8T^{2} \)
11 \( 1 - 2.65e5iT - 2.59e10T^{2} \)
13 \( 1 - 6.47e5T + 1.37e11T^{2} \)
17 \( 1 + 2.51e6T + 2.01e12T^{2} \)
19 \( 1 + 1.70e5iT - 6.13e12T^{2} \)
23 \( 1 - 5.21e6iT - 4.14e13T^{2} \)
29 \( 1 - 6.80e6T + 4.20e14T^{2} \)
31 \( 1 + 2.47e7iT - 8.19e14T^{2} \)
37 \( 1 - 9.23e6T + 4.80e15T^{2} \)
41 \( 1 - 1.44e8T + 1.34e16T^{2} \)
43 \( 1 + 2.79e7iT - 2.16e16T^{2} \)
47 \( 1 + 1.10e8iT - 5.25e16T^{2} \)
53 \( 1 - 1.09e8T + 1.74e17T^{2} \)
59 \( 1 - 6.65e8iT - 5.11e17T^{2} \)
61 \( 1 + 7.40e8T + 7.13e17T^{2} \)
67 \( 1 - 4.17e7iT - 1.82e18T^{2} \)
71 \( 1 - 6.30e8iT - 3.25e18T^{2} \)
73 \( 1 - 9.52e8T + 4.29e18T^{2} \)
79 \( 1 - 2.24e9iT - 9.46e18T^{2} \)
83 \( 1 + 6.38e9iT - 1.55e19T^{2} \)
89 \( 1 + 1.29e9T + 3.11e19T^{2} \)
97 \( 1 - 1.43e10T + 7.37e19T^{2} \)
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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

−15.73010922923198395667063849711, −15.26465708339660445099113985094, −13.44984178298846854703950178887, −12.62396877678165622186528733134, −11.49911765342890319821380370612, −8.822173120974523236400010008010, −7.39287951547270239294221484900, −6.20295015021262126620423069998, −4.30584480642030849814173817287, −2.11405611591938873897870463638, 0.841148629795394217740821762575, 3.48925697348988138901248096957, 4.40201365137162240724790435859, 6.38293140896036442294596092677, 8.830112835698597050962678371559, 10.74633420452196806667387858965, 11.03858828484476784511085486988, 13.13843762057746323702239107485, 14.03039414103940965776592667427, 15.66270295362222884268763373200

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