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Properties

Label 2-20-20.3-c5-0-5
Degree $2$
Conductor $20$
Sign $-0.0729 + 0.997i$
Analytic cond. $3.20767$
Root an. cond. $1.79099$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.11 + 2.41i)2-s + (−16.4 + 16.4i)3-s + (20.3 − 24.6i)4-s + (21.8 − 51.4i)5-s + (44.5 − 124. i)6-s + (−78.4 − 78.4i)7-s + (−44.4 + 175. i)8-s − 300. i·9-s + (12.5 + 315. i)10-s − 208. i·11-s + (71.6 + 742. i)12-s + (−401. − 401. i)13-s + (590. + 212. i)14-s + (488. + 1.20e3i)15-s + (−195. − 1.00e3i)16-s + (−515. + 515. i)17-s + ⋯
L(s)  = 1  + (−0.904 + 0.426i)2-s + (−1.05 + 1.05i)3-s + (0.635 − 0.771i)4-s + (0.390 − 0.920i)5-s + (0.505 − 1.40i)6-s + (−0.605 − 0.605i)7-s + (−0.245 + 0.969i)8-s − 1.23i·9-s + (0.0397 + 0.999i)10-s − 0.518i·11-s + (0.143 + 1.48i)12-s + (−0.658 − 0.658i)13-s + (0.805 + 0.289i)14-s + (0.560 + 1.38i)15-s + (−0.191 − 0.981i)16-s + (−0.432 + 0.432i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0729 + 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0729 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.0729 + 0.997i$
Analytic conductor: \(3.20767\)
Root analytic conductor: \(1.79099\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5/2),\ -0.0729 + 0.997i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.188565 - 0.202862i\)
\(L(\frac12)\) \(\approx\) \(0.188565 - 0.202862i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.11 - 2.41i)T \)
5 \( 1 + (-21.8 + 51.4i)T \)
good3 \( 1 + (16.4 - 16.4i)T - 243iT^{2} \)
7 \( 1 + (78.4 + 78.4i)T + 1.68e4iT^{2} \)
11 \( 1 + 208. iT - 1.61e5T^{2} \)
13 \( 1 + (401. + 401. i)T + 3.71e5iT^{2} \)
17 \( 1 + (515. - 515. i)T - 1.41e6iT^{2} \)
19 \( 1 + 2.46e3T + 2.47e6T^{2} \)
23 \( 1 + (-2.92e3 + 2.92e3i)T - 6.43e6iT^{2} \)
29 \( 1 - 6.39e3iT - 2.05e7T^{2} \)
31 \( 1 + 556. iT - 2.86e7T^{2} \)
37 \( 1 + (1.96e3 - 1.96e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 6.13e3T + 1.15e8T^{2} \)
43 \( 1 + (7.37e3 - 7.37e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (825. + 825. i)T + 2.29e8iT^{2} \)
53 \( 1 + (1.44e4 + 1.44e4i)T + 4.18e8iT^{2} \)
59 \( 1 + 771.T + 7.14e8T^{2} \)
61 \( 1 - 1.09e4T + 8.44e8T^{2} \)
67 \( 1 + (-3.35e3 - 3.35e3i)T + 1.35e9iT^{2} \)
71 \( 1 - 1.02e4iT - 1.80e9T^{2} \)
73 \( 1 + (2.36e4 + 2.36e4i)T + 2.07e9iT^{2} \)
79 \( 1 - 3.34e4T + 3.07e9T^{2} \)
83 \( 1 + (-4.38e4 + 4.38e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 1.06e4iT - 5.58e9T^{2} \)
97 \( 1 + (-6.84e4 + 6.84e4i)T - 8.58e9iT^{2} \)
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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

−16.80311104224875682796721198892, −16.28621183760639556681604627127, −14.94133564266568467642875467075, −12.77047958776924167902045816887, −10.90303719965400253028462158810, −10.08697294331283295847334740053, −8.721637091942237530055548393710, −6.38163737125555515532658653398, −4.91545814871272176790115656239, −0.29372918972902658030754424574, 2.20857377604083393700133851118, 6.32146986354790149934580401089, 7.24122308867172569598796807080, 9.473756306053443959807195755325, 10.98315727945866499178610053516, 12.03270211950002848453093500784, 13.14391781244495734969287109524, 15.32043629507197382163660241296, 17.05701188406782593132320688080, 17.66692425913291599130741055761

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