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Properties

Label 2-20-5.3-c4-0-0
Degree $2$
Conductor $20$
Sign $0.875 - 0.483i$
Analytic cond. $2.06739$
Root an. cond. $1.43784$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.26 + 5.26i)3-s + (21.7 + 12.2i)5-s + (4.21 − 4.21i)7-s − 25.6i·9-s − 152.·11-s + (−136. − 136. i)13-s + (50.1 + 179. i)15-s + (348. − 348. i)17-s + 527. i·19-s + 44.3·21-s + (−70.5 − 70.5i)23-s + (324. + 534. i)25-s + (561. − 561. i)27-s − 68.9i·29-s − 1.37e3·31-s + ⋯
L(s)  = 1  + (0.584 + 0.584i)3-s + (0.871 + 0.490i)5-s + (0.0859 − 0.0859i)7-s − 0.316i·9-s − 1.26·11-s + (−0.808 − 0.808i)13-s + (0.222 + 0.796i)15-s + (1.20 − 1.20i)17-s + 1.46i·19-s + 0.100·21-s + (−0.133 − 0.133i)23-s + (0.518 + 0.854i)25-s + (0.769 − 0.769i)27-s − 0.0820i·29-s − 1.42·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.875 - 0.483i$
Analytic conductor: \(2.06739\)
Root analytic conductor: \(1.43784\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :2),\ 0.875 - 0.483i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.48726 + 0.383388i\)
\(L(\frac12)\) \(\approx\) \(1.48726 + 0.383388i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-21.7 - 12.2i)T \)
good3 \( 1 + (-5.26 - 5.26i)T + 81iT^{2} \)
7 \( 1 + (-4.21 + 4.21i)T - 2.40e3iT^{2} \)
11 \( 1 + 152.T + 1.46e4T^{2} \)
13 \( 1 + (136. + 136. i)T + 2.85e4iT^{2} \)
17 \( 1 + (-348. + 348. i)T - 8.35e4iT^{2} \)
19 \( 1 - 527. iT - 1.30e5T^{2} \)
23 \( 1 + (70.5 + 70.5i)T + 2.79e5iT^{2} \)
29 \( 1 + 68.9iT - 7.07e5T^{2} \)
31 \( 1 + 1.37e3T + 9.23e5T^{2} \)
37 \( 1 + (1.00e3 - 1.00e3i)T - 1.87e6iT^{2} \)
41 \( 1 - 663.T + 2.82e6T^{2} \)
43 \( 1 + (-1.63e3 - 1.63e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-438. + 438. i)T - 4.87e6iT^{2} \)
53 \( 1 + (-712. - 712. i)T + 7.89e6iT^{2} \)
59 \( 1 + 2.91e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.39e3T + 1.38e7T^{2} \)
67 \( 1 + (1.69e3 - 1.69e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 3.28e3T + 2.54e7T^{2} \)
73 \( 1 + (1.46e3 + 1.46e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 3.38e3iT - 3.89e7T^{2} \)
83 \( 1 + (-8.69e3 - 8.69e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 4.27e3iT - 6.27e7T^{2} \)
97 \( 1 + (-7.21e3 + 7.21e3i)T - 8.85e7iT^{2} \)
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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

−17.80299138168495102709204838838, −16.28351147026054079515060826626, −14.90775390930107239415123839497, −14.06913697064056947916966446842, −12.50771465313888285192842780525, −10.41626048719966863896871139575, −9.603457126009498398754569936812, −7.69255358505708850579176025135, −5.48922813216259480015290794832, −2.97650759875887709875893004600, 2.18162554790584242368735364044, 5.31063784343133071597762670867, 7.44543570724354338097910072290, 8.916423700623931333647762270455, 10.47671239983168355273154478960, 12.55921310098960846433001094683, 13.46448092796117336375624620529, 14.60780000235525228235508819735, 16.32754954420834842567605256616, 17.53965945189041400832316574400

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