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Properties

Label 2-20-20.3-c9-0-21
Degree $2$
Conductor $20$
Sign $0.142 + 0.989i$
Analytic cond. $10.3007$
Root an. cond. $3.20947$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (22.3 − 3.74i)2-s + (68.1 − 68.1i)3-s + (483. − 167. i)4-s + (−497. − 1.30e3i)5-s + (1.26e3 − 1.77e3i)6-s + (−2.74e3 − 2.74e3i)7-s + (1.01e4 − 5.53e3i)8-s + 1.04e4i·9-s + (−1.59e4 − 2.72e4i)10-s − 4.20e4i·11-s + (2.15e4 − 4.43e4i)12-s + (4.82e4 + 4.82e4i)13-s + (−7.15e4 − 5.09e4i)14-s + (−1.22e5 − 5.50e4i)15-s + (2.06e5 − 1.61e5i)16-s + (2.69e4 − 2.69e4i)17-s + ⋯
L(s)  = 1  + (0.986 − 0.165i)2-s + (0.485 − 0.485i)3-s + (0.945 − 0.326i)4-s + (−0.355 − 0.934i)5-s + (0.398 − 0.559i)6-s + (−0.432 − 0.432i)7-s + (0.878 − 0.478i)8-s + 0.528i·9-s + (−0.505 − 0.862i)10-s − 0.865i·11-s + (0.300 − 0.617i)12-s + (0.469 + 0.469i)13-s + (−0.497 − 0.354i)14-s + (−0.626 − 0.280i)15-s + (0.787 − 0.616i)16-s + (0.0783 − 0.0783i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.142 + 0.989i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.142 + 0.989i$
Analytic conductor: \(10.3007\)
Root analytic conductor: \(3.20947\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :9/2),\ 0.142 + 0.989i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.56311 - 2.22116i\)
\(L(\frac12)\) \(\approx\) \(2.56311 - 2.22116i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-22.3 + 3.74i)T \)
5 \( 1 + (497. + 1.30e3i)T \)
good3 \( 1 + (-68.1 + 68.1i)T - 1.96e4iT^{2} \)
7 \( 1 + (2.74e3 + 2.74e3i)T + 4.03e7iT^{2} \)
11 \( 1 + 4.20e4iT - 2.35e9T^{2} \)
13 \( 1 + (-4.82e4 - 4.82e4i)T + 1.06e10iT^{2} \)
17 \( 1 + (-2.69e4 + 2.69e4i)T - 1.18e11iT^{2} \)
19 \( 1 - 3.57e5T + 3.22e11T^{2} \)
23 \( 1 + (-7.98e5 + 7.98e5i)T - 1.80e12iT^{2} \)
29 \( 1 - 5.48e6iT - 1.45e13T^{2} \)
31 \( 1 - 8.10e6iT - 2.64e13T^{2} \)
37 \( 1 + (1.32e7 - 1.32e7i)T - 1.29e14iT^{2} \)
41 \( 1 - 7.49e6T + 3.27e14T^{2} \)
43 \( 1 + (-2.51e7 + 2.51e7i)T - 5.02e14iT^{2} \)
47 \( 1 + (1.82e7 + 1.82e7i)T + 1.11e15iT^{2} \)
53 \( 1 + (-3.95e6 - 3.95e6i)T + 3.29e15iT^{2} \)
59 \( 1 + 1.22e8T + 8.66e15T^{2} \)
61 \( 1 - 3.05e7T + 1.16e16T^{2} \)
67 \( 1 + (-1.72e8 - 1.72e8i)T + 2.72e16iT^{2} \)
71 \( 1 + 2.97e8iT - 4.58e16T^{2} \)
73 \( 1 + (2.23e7 + 2.23e7i)T + 5.88e16iT^{2} \)
79 \( 1 + 3.86e8T + 1.19e17T^{2} \)
83 \( 1 + (-1.87e8 + 1.87e8i)T - 1.86e17iT^{2} \)
89 \( 1 - 2.80e8iT - 3.50e17T^{2} \)
97 \( 1 + (6.97e8 - 6.97e8i)T - 7.60e17iT^{2} \)
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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

−15.94873419777833670952329074895, −14.11855769447055627352190236165, −13.35493329628517585463916812531, −12.26035511457395424462430709500, −10.76788366028426618666739268539, −8.617548605656757537813268574887, −7.02060209373773759938776564563, −5.10106241340407429063723253038, −3.35113935560522005785371464394, −1.28273095433318886653522153543, 2.75000953838695745427830029499, 3.96846766530305685529607460452, 6.08525421505212271828470414628, 7.59153707505891499850005353883, 9.714750650845863728464975616541, 11.29993583277242457664137413807, 12.62062512328281725657174339435, 14.14147182730026820368806820531, 15.25119527380472736476025919283, 15.68934532361048374212857958948

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