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Properties

Label 2-20-20.7-c3-0-0
Degree $2$
Conductor $20$
Sign $0.0689 - 0.997i$
Analytic cond. $1.18003$
Root an. cond. $1.08629$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.813 + 2.70i)2-s + (2.61 + 2.61i)3-s + (−6.67 − 4.40i)4-s + (−0.435 + 11.1i)5-s + (−9.22 + 4.96i)6-s + (17.7 − 17.7i)7-s + (17.3 − 14.4i)8-s − 13.2i·9-s + (−29.9 − 10.2i)10-s + 7.37i·11-s + (−5.93 − 29.0i)12-s + (−2.68 + 2.68i)13-s + (33.6 + 62.6i)14-s + (−30.3 + 28.1i)15-s + (25.1 + 58.8i)16-s + (−20.2 − 20.2i)17-s + ⋯
L(s)  = 1  + (−0.287 + 0.957i)2-s + (0.503 + 0.503i)3-s + (−0.834 − 0.551i)4-s + (−0.0389 + 0.999i)5-s + (−0.627 + 0.337i)6-s + (0.959 − 0.959i)7-s + (0.767 − 0.640i)8-s − 0.492i·9-s + (−0.945 − 0.324i)10-s + 0.202i·11-s + (−0.142 − 0.698i)12-s + (−0.0572 + 0.0572i)13-s + (0.643 + 1.19i)14-s + (−0.523 + 0.483i)15-s + (0.392 + 0.919i)16-s + (−0.288 − 0.288i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0689 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0689 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.0689 - 0.997i$
Analytic conductor: \(1.18003\)
Root analytic conductor: \(1.08629\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :3/2),\ 0.0689 - 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.767720 + 0.716469i\)
\(L(\frac12)\) \(\approx\) \(0.767720 + 0.716469i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.813 - 2.70i)T \)
5 \( 1 + (0.435 - 11.1i)T \)
good3 \( 1 + (-2.61 - 2.61i)T + 27iT^{2} \)
7 \( 1 + (-17.7 + 17.7i)T - 343iT^{2} \)
11 \( 1 - 7.37iT - 1.33e3T^{2} \)
13 \( 1 + (2.68 - 2.68i)T - 2.19e3iT^{2} \)
17 \( 1 + (20.2 + 20.2i)T + 4.91e3iT^{2} \)
19 \( 1 + 135.T + 6.85e3T^{2} \)
23 \( 1 + (-71.0 - 71.0i)T + 1.21e4iT^{2} \)
29 \( 1 + 34.2iT - 2.43e4T^{2} \)
31 \( 1 + 187. iT - 2.97e4T^{2} \)
37 \( 1 + (-250. - 250. i)T + 5.06e4iT^{2} \)
41 \( 1 + 211.T + 6.89e4T^{2} \)
43 \( 1 + (46.7 + 46.7i)T + 7.95e4iT^{2} \)
47 \( 1 + (189. - 189. i)T - 1.03e5iT^{2} \)
53 \( 1 + (74.5 - 74.5i)T - 1.48e5iT^{2} \)
59 \( 1 + 101.T + 2.05e5T^{2} \)
61 \( 1 - 232.T + 2.26e5T^{2} \)
67 \( 1 + (-34.7 + 34.7i)T - 3.00e5iT^{2} \)
71 \( 1 - 614. iT - 3.57e5T^{2} \)
73 \( 1 + (37.4 - 37.4i)T - 3.89e5iT^{2} \)
79 \( 1 - 1.00e3T + 4.93e5T^{2} \)
83 \( 1 + (-423. - 423. i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.04e3iT - 7.04e5T^{2} \)
97 \( 1 + (536. + 536. i)T + 9.12e5iT^{2} \)
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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

−17.89191280119431106756791071822, −16.99465369633861029122404903318, −15.21369577111709536489444345271, −14.74449790639662437803013437598, −13.58156956375976491228886300974, −10.99740939227831827610610328226, −9.735239426817691807917541632119, −8.077323041626304831663517728510, −6.68156544128264921793062924906, −4.24966230655131450597625971537, 2.00358922228275972764457728887, 4.86748403966537872443889530926, 8.213996252559873179681797719489, 8.829687128124095342017662194647, 10.90226513816481719029646969690, 12.32776712472532127904171319281, 13.21251725643605531490933266890, 14.68511718395812670323069746590, 16.63249650174215171300059074201, 17.86881611474240504907332638523

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