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Properties

Label 2-20-4.3-c10-0-15
Degree $2$
Conductor $20$
Sign $-0.748 - 0.663i$
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  + (−29.9 − 11.3i)2-s − 137. i·3-s + (765. + 679. i)4-s − 1.39e3·5-s + (−1.55e3 + 4.10e3i)6-s − 2.49e4i·7-s + (−1.51e4 − 2.90e4i)8-s + 4.02e4·9-s + (4.18e4 + 1.58e4i)10-s + 1.13e4i·11-s + (9.32e4 − 1.05e5i)12-s − 4.07e5·13-s + (−2.82e5 + 7.45e5i)14-s + 1.91e5i·15-s + (1.24e5 + 1.04e6i)16-s − 1.49e6·17-s + ⋯
L(s)  = 1  + (−0.934 − 0.354i)2-s − 0.564i·3-s + (0.748 + 0.663i)4-s − 0.447·5-s + (−0.200 + 0.527i)6-s − 1.48i·7-s + (−0.463 − 0.885i)8-s + 0.681·9-s + (0.418 + 0.158i)10-s + 0.0702i·11-s + (0.374 − 0.422i)12-s − 1.09·13-s + (−0.526 + 1.38i)14-s + 0.252i·15-s + (0.119 + 0.992i)16-s − 1.05·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 - 0.663i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.748 - 0.663i$
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: \((2,\ 20,\ (\ :5),\ -0.748 - 0.663i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.0825284 + 0.217366i\)
\(L(\frac12)\) \(\approx\) \(0.0825284 + 0.217366i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (29.9 + 11.3i)T \)
5 \( 1 + 1.39e3T \)
good3 \( 1 + 137. iT - 5.90e4T^{2} \)
7 \( 1 + 2.49e4iT - 2.82e8T^{2} \)
11 \( 1 - 1.13e4iT - 2.59e10T^{2} \)
13 \( 1 + 4.07e5T + 1.37e11T^{2} \)
17 \( 1 + 1.49e6T + 2.01e12T^{2} \)
19 \( 1 - 2.09e6iT - 6.13e12T^{2} \)
23 \( 1 - 1.00e7iT - 4.14e13T^{2} \)
29 \( 1 + 2.68e7T + 4.20e14T^{2} \)
31 \( 1 + 9.47e6iT - 8.19e14T^{2} \)
37 \( 1 + 6.03e7T + 4.80e15T^{2} \)
41 \( 1 + 3.12e7T + 1.34e16T^{2} \)
43 \( 1 + 2.55e8iT - 2.16e16T^{2} \)
47 \( 1 - 3.78e8iT - 5.25e16T^{2} \)
53 \( 1 - 7.29e6T + 1.74e17T^{2} \)
59 \( 1 - 2.59e8iT - 5.11e17T^{2} \)
61 \( 1 + 3.30e8T + 7.13e17T^{2} \)
67 \( 1 + 7.94e7iT - 1.82e18T^{2} \)
71 \( 1 + 1.16e9iT - 3.25e18T^{2} \)
73 \( 1 - 3.86e9T + 4.29e18T^{2} \)
79 \( 1 + 3.82e9iT - 9.46e18T^{2} \)
83 \( 1 - 9.01e8iT - 1.55e19T^{2} \)
89 \( 1 + 4.02e9T + 3.11e19T^{2} \)
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

−15.43592288166592830618065880282, −13.51919654395701701259888188089, −12.28023374398217498775119370083, −10.87351767204992808509678779174, −9.667625785525239722576238747847, −7.68341586125637000497368556455, −7.05751754138778767938092304954, −3.91178120260535290691991600828, −1.65395149580676539622245214496, −0.13082286609648885400281449266, 2.34663762544384721590489086277, 4.99903049963655759432875098242, 6.85766857880933754390594492597, 8.577160658546069813469496798429, 9.610523182022447042532248115475, 11.08800386290198460906191515361, 12.44436771318571606336286017092, 14.92311764992781030341105256113, 15.46564615511118598030645324049, 16.56887002775672329815485930068

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