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Properties

Label 2-20-4.3-c6-0-11
Degree $2$
Conductor $20$
Sign $-0.999 - 0.00842i$
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  + (0.0337 − 7.99i)2-s − 38.3i·3-s + (−63.9 − 0.539i)4-s + 55.9·5-s + (−306. − 1.29i)6-s + 111. i·7-s + (−6.47 + 511. i)8-s − 743.·9-s + (1.88 − 447. i)10-s − 1.81e3i·11-s + (−20.6 + 2.45e3i)12-s − 2.12e3·13-s + (890. + 3.75i)14-s − 2.14e3i·15-s + (4.09e3 + 69.0i)16-s + 7.92e3·17-s + ⋯
L(s)  = 1  + (0.00421 − 0.999i)2-s − 1.42i·3-s + (−0.999 − 0.00842i)4-s + 0.447·5-s + (−1.42 − 0.00598i)6-s + 0.324i·7-s + (−0.0126 + 0.999i)8-s − 1.01·9-s + (0.00188 − 0.447i)10-s − 1.36i·11-s + (−0.0119 + 1.42i)12-s − 0.967·13-s + (0.324 + 0.00136i)14-s − 0.635i·15-s + (0.999 + 0.0168i)16-s + 1.61·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00842i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.999 - 0.00842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.999 - 0.00842i$
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: \((2,\ 20,\ (\ :3),\ -0.999 - 0.00842i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.00549554 + 1.30423i\)
\(L(\frac12)\) \(\approx\) \(0.00549554 + 1.30423i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0337 + 7.99i)T \)
5 \( 1 - 55.9T \)
good3 \( 1 + 38.3iT - 729T^{2} \)
7 \( 1 - 111. iT - 1.17e5T^{2} \)
11 \( 1 + 1.81e3iT - 1.77e6T^{2} \)
13 \( 1 + 2.12e3T + 4.82e6T^{2} \)
17 \( 1 - 7.92e3T + 2.41e7T^{2} \)
19 \( 1 + 9.23e3iT - 4.70e7T^{2} \)
23 \( 1 - 6.15e3iT - 1.48e8T^{2} \)
29 \( 1 + 2.95e4T + 5.94e8T^{2} \)
31 \( 1 - 1.35e4iT - 8.87e8T^{2} \)
37 \( 1 - 8.80e4T + 2.56e9T^{2} \)
41 \( 1 - 7.68e4T + 4.75e9T^{2} \)
43 \( 1 + 6.66e3iT - 6.32e9T^{2} \)
47 \( 1 - 1.47e5iT - 1.07e10T^{2} \)
53 \( 1 + 2.60e4T + 2.21e10T^{2} \)
59 \( 1 - 4.10e4iT - 4.21e10T^{2} \)
61 \( 1 - 5.83e4T + 5.15e10T^{2} \)
67 \( 1 - 7.69e4iT - 9.04e10T^{2} \)
71 \( 1 + 3.00e5iT - 1.28e11T^{2} \)
73 \( 1 - 3.13e5T + 1.51e11T^{2} \)
79 \( 1 + 1.65e5iT - 2.43e11T^{2} \)
83 \( 1 - 6.65e5iT - 3.26e11T^{2} \)
89 \( 1 + 7.58e5T + 4.96e11T^{2} \)
97 \( 1 + 1.48e4T + 8.32e11T^{2} \)
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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

−16.91801857143339409413161774514, −14.39099563901234263339115130106, −13.37434849282917720467723912183, −12.41206405403494294354944249377, −11.23327087544766647739427299544, −9.374456442924699311691731013149, −7.76747743977357427335216089662, −5.69136221150584076905289122547, −2.69460558320962833957871414352, −0.917434886830457395155061266304, 4.13255999547347740044174387716, 5.48066881967778946929900021175, 7.58574758622228223270159038621, 9.614093152059896634281610957382, 10.08541311020419218244429539354, 12.56511186279885836479203966863, 14.48696437720313953247936836997, 14.96378013358569406937351714504, 16.49129757749546214284299831881, 17.02002637955652183238189866254

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