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Properties

Label 2-20-20.3-c5-0-7
Degree $2$
Conductor $20$
Sign $0.964 + 0.262i$
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  + (−2.41 + 5.11i)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 + (175. − 44.4i)8-s − 300. i·9-s + (210. + 235. i)10-s + 208. i·11-s + (−742. − 71.6i)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  + (−0.426 + 0.904i)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 + (0.969 − 0.245i)8-s − 1.23i·9-s + (0.666 + 0.745i)10-s + 0.518i·11-s + (−1.48 − 0.143i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.964 + 0.262i$
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: \((2,\ 20,\ (\ :5/2),\ 0.964 + 0.262i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.53836 - 0.205426i\)
\(L(\frac12)\) \(\approx\) \(1.53836 - 0.205426i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.41 - 5.11i)T \)
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

−17.59936680038104119489555459383, −15.85496691913655932327425813209, −14.57129400475950976259366372489, −13.57175813159085894520996821236, −12.37851420251761324707965592822, −9.582535344514014448573753463198, −8.431808816395812569382062015185, −7.43581889435795569271860337315, −5.36817964089147129245389381632, −1.62451769275097447398450714481, 2.63472741959122147350219962376, 4.21992440200605521349302198039, 7.75638294344185272922390275006, 9.354796206791233619281209875159, 10.26210458899658761288698814593, 11.48600877039998145165377542554, 13.86918378131174706243908674326, 14.31192431869188425506592376768, 16.06876726167049843464102983663, 17.56999760212270245312827289251

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