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Properties

Label 2-20-4.3-c10-0-2
Degree $2$
Conductor $20$
Sign $-0.902 - 0.430i$
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  + (7.05 − 31.2i)2-s + 442. i·3-s + (−924. − 440. i)4-s + 1.39e3·5-s + (1.38e4 + 3.12e3i)6-s − 2.45e3i·7-s + (−2.02e4 + 2.57e4i)8-s − 1.36e5·9-s + (9.86e3 − 4.36e4i)10-s − 6.73e4i·11-s + (1.94e5 − 4.08e5i)12-s − 7.23e5·13-s + (−7.66e4 − 1.73e4i)14-s + 6.18e5i·15-s + (6.60e5 + 8.14e5i)16-s − 8.17e5·17-s + ⋯
L(s)  = 1  + (0.220 − 0.975i)2-s + 1.82i·3-s + (−0.902 − 0.430i)4-s + 0.447·5-s + (1.77 + 0.401i)6-s − 0.146i·7-s + (−0.618 + 0.785i)8-s − 2.31·9-s + (0.0986 − 0.436i)10-s − 0.418i·11-s + (0.783 − 1.64i)12-s − 1.94·13-s + (−0.142 − 0.0322i)14-s + 0.814i·15-s + (0.629 + 0.776i)16-s − 0.575·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.430i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.902 - 0.430i$
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: \((2,\ 20,\ (\ :5),\ -0.902 - 0.430i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.111281 + 0.492254i\)
\(L(\frac12)\) \(\approx\) \(0.111281 + 0.492254i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.05 + 31.2i)T \)
5 \( 1 - 1.39e3T \)
good3 \( 1 - 442. iT - 5.90e4T^{2} \)
7 \( 1 + 2.45e3iT - 2.82e8T^{2} \)
11 \( 1 + 6.73e4iT - 2.59e10T^{2} \)
13 \( 1 + 7.23e5T + 1.37e11T^{2} \)
17 \( 1 + 8.17e5T + 2.01e12T^{2} \)
19 \( 1 - 2.29e6iT - 6.13e12T^{2} \)
23 \( 1 + 6.87e6iT - 4.14e13T^{2} \)
29 \( 1 + 9.72e6T + 4.20e14T^{2} \)
31 \( 1 - 3.08e7iT - 8.19e14T^{2} \)
37 \( 1 + 1.03e8T + 4.80e15T^{2} \)
41 \( 1 - 1.28e8T + 1.34e16T^{2} \)
43 \( 1 - 7.29e7iT - 2.16e16T^{2} \)
47 \( 1 - 1.33e8iT - 5.25e16T^{2} \)
53 \( 1 - 2.17e8T + 1.74e17T^{2} \)
59 \( 1 - 9.48e8iT - 5.11e17T^{2} \)
61 \( 1 + 9.30e7T + 7.13e17T^{2} \)
67 \( 1 - 1.87e9iT - 1.82e18T^{2} \)
71 \( 1 + 5.98e8iT - 3.25e18T^{2} \)
73 \( 1 + 3.66e8T + 4.29e18T^{2} \)
79 \( 1 + 2.46e9iT - 9.46e18T^{2} \)
83 \( 1 - 4.63e9iT - 1.55e19T^{2} \)
89 \( 1 - 2.24e9T + 3.11e19T^{2} \)
97 \( 1 + 6.10e9T + 7.37e19T^{2} \)
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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

−16.50927225599381853691755964425, −14.90425852842152556003615806533, −14.13932917326441049505981906141, −12.22051387562478777223266895017, −10.71475084358820296066179002279, −9.967706761539320633942258746375, −8.879444735269090519058499393192, −5.39405892686584339713469758220, −4.26577243044723788051334578581, −2.70247047959358543624944534130, 0.18858045780583006625171716176, 2.29414133591609325082284998674, 5.33976978376258277284770665609, 6.83528058620078676445440897910, 7.61651574756511640397888037777, 9.252022721939161476227167032213, 11.99883300562051691517517784167, 12.97695308197816504254149117707, 13.91879959800183642034619198742, 15.10087810124363453857922769421

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