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Properties

Label 2-20-20.19-c8-0-12
Degree $2$
Conductor $20$
Sign $0.744 + 0.667i$
Analytic cond. $8.14757$
Root an. cond. $2.85439$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.05 + 15.4i)2-s − 75.2·3-s + (−223. + 125. i)4-s + (−200. − 591. i)5-s + (−305. − 1.16e3i)6-s + 4.41e3·7-s + (−2.85e3 − 2.94e3i)8-s − 900.·9-s + (8.34e3 − 5.51e3i)10-s − 330. i·11-s + (1.67e4 − 9.45e3i)12-s − 2.67e4i·13-s + (1.79e4 + 6.82e4i)14-s + (1.51e4 + 4.45e4i)15-s + (3.39e4 − 5.60e4i)16-s − 6.86e4i·17-s + ⋯
L(s)  = 1  + (0.253 + 0.967i)2-s − 0.928·3-s + (−0.871 + 0.490i)4-s + (−0.321 − 0.946i)5-s + (−0.235 − 0.898i)6-s + 1.83·7-s + (−0.695 − 0.718i)8-s − 0.137·9-s + (0.834 − 0.551i)10-s − 0.0225i·11-s + (0.809 − 0.455i)12-s − 0.935i·13-s + (0.466 + 1.77i)14-s + (0.298 + 0.879i)15-s + (0.518 − 0.855i)16-s − 0.822i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.744 + 0.667i$
Analytic conductor: \(8.14757\)
Root analytic conductor: \(2.85439\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :4),\ 0.744 + 0.667i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.914126 - 0.349594i\)
\(L(\frac12)\) \(\approx\) \(0.914126 - 0.349594i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.05 - 15.4i)T \)
5 \( 1 + (200. + 591. i)T \)
good3 \( 1 + 75.2T + 6.56e3T^{2} \)
7 \( 1 - 4.41e3T + 5.76e6T^{2} \)
11 \( 1 + 330. iT - 2.14e8T^{2} \)
13 \( 1 + 2.67e4iT - 8.15e8T^{2} \)
17 \( 1 + 6.86e4iT - 6.97e9T^{2} \)
19 \( 1 + 1.94e5iT - 1.69e10T^{2} \)
23 \( 1 + 2.48e5T + 7.83e10T^{2} \)
29 \( 1 + 4.96e5T + 5.00e11T^{2} \)
31 \( 1 - 7.34e5iT - 8.52e11T^{2} \)
37 \( 1 + 1.92e6iT - 3.51e12T^{2} \)
41 \( 1 + 8.79e5T + 7.98e12T^{2} \)
43 \( 1 - 1.87e6T + 1.16e13T^{2} \)
47 \( 1 + 5.21e5T + 2.38e13T^{2} \)
53 \( 1 - 8.82e5iT - 6.22e13T^{2} \)
59 \( 1 + 3.24e6iT - 1.46e14T^{2} \)
61 \( 1 + 9.76e5T + 1.91e14T^{2} \)
67 \( 1 + 1.58e7T + 4.06e14T^{2} \)
71 \( 1 + 1.19e7iT - 6.45e14T^{2} \)
73 \( 1 + 4.92e7iT - 8.06e14T^{2} \)
79 \( 1 - 5.67e7iT - 1.51e15T^{2} \)
83 \( 1 - 3.12e7T + 2.25e15T^{2} \)
89 \( 1 + 4.74e7T + 3.93e15T^{2} \)
97 \( 1 - 2.28e7iT - 7.83e15T^{2} \)
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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

−16.38071640088599895783070453427, −15.20848869559844878272358948113, −13.84649705132528665249559930280, −12.29220404354037290471932243434, −11.19515168091701539874681507358, −8.785443318800850461303762585961, −7.62308050105687181714181014699, −5.46999178546507836201351213123, −4.71818753386075408358856181721, −0.54278656734402814551813936477, 1.78429991840905008679730200822, 4.20208364928684606515239101096, 5.83275162671434003666583140652, 8.153889560826823563273686293897, 10.39723277305346801499255973517, 11.37486474537350773930178792878, 11.95083247868572614268734845802, 14.12027230409649369471538122846, 14.81457203678200235560459484423, 17.04007715354046804006790632573

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