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Properties

Label 2-20-20.3-c3-0-2
Degree $2$
Conductor $20$
Sign $0.920 + 0.389i$
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  + (−2.78 + 0.510i)2-s + (4.02 − 4.02i)3-s + (7.47 − 2.83i)4-s + (10.9 + 2.32i)5-s + (−9.15 + 13.2i)6-s + (−14.4 − 14.4i)7-s + (−19.3 + 11.7i)8-s − 5.46i·9-s + (−31.6 − 0.899i)10-s + 47.0i·11-s + (18.6 − 41.5i)12-s + (−8.79 − 8.79i)13-s + (47.5 + 32.8i)14-s + (53.4 − 34.6i)15-s + (47.8 − 42.4i)16-s + (−26.4 + 26.4i)17-s + ⋯
L(s)  = 1  + (−0.983 + 0.180i)2-s + (0.775 − 0.775i)3-s + (0.934 − 0.354i)4-s + (0.978 + 0.208i)5-s + (−0.622 + 0.902i)6-s + (−0.779 − 0.779i)7-s + (−0.855 + 0.517i)8-s − 0.202i·9-s + (−0.999 − 0.0284i)10-s + 1.28i·11-s + (0.449 − 1.00i)12-s + (−0.187 − 0.187i)13-s + (0.907 + 0.626i)14-s + (0.919 − 0.596i)15-s + (0.747 − 0.663i)16-s + (−0.377 + 0.377i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.389i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.920 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.920 + 0.389i$
Analytic conductor: \(1.18003\)
Root analytic conductor: \(1.08629\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :3/2),\ 0.920 + 0.389i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.922629 - 0.187307i\)
\(L(\frac12)\) \(\approx\) \(0.922629 - 0.187307i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.78 - 0.510i)T \)
5 \( 1 + (-10.9 - 2.32i)T \)
good3 \( 1 + (-4.02 + 4.02i)T - 27iT^{2} \)
7 \( 1 + (14.4 + 14.4i)T + 343iT^{2} \)
11 \( 1 - 47.0iT - 1.33e3T^{2} \)
13 \( 1 + (8.79 + 8.79i)T + 2.19e3iT^{2} \)
17 \( 1 + (26.4 - 26.4i)T - 4.91e3iT^{2} \)
19 \( 1 + 49.8T + 6.85e3T^{2} \)
23 \( 1 + (41.2 - 41.2i)T - 1.21e4iT^{2} \)
29 \( 1 + 247. iT - 2.43e4T^{2} \)
31 \( 1 + 62.3iT - 2.97e4T^{2} \)
37 \( 1 + (73.2 - 73.2i)T - 5.06e4iT^{2} \)
41 \( 1 - 118.T + 6.89e4T^{2} \)
43 \( 1 + (-245. + 245. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-125. - 125. i)T + 1.03e5iT^{2} \)
53 \( 1 + (326. + 326. i)T + 1.48e5iT^{2} \)
59 \( 1 + 365.T + 2.05e5T^{2} \)
61 \( 1 + 268.T + 2.26e5T^{2} \)
67 \( 1 + (112. + 112. i)T + 3.00e5iT^{2} \)
71 \( 1 + 559. iT - 3.57e5T^{2} \)
73 \( 1 + (-215. - 215. i)T + 3.89e5iT^{2} \)
79 \( 1 - 1.17e3T + 4.93e5T^{2} \)
83 \( 1 + (592. - 592. i)T - 5.71e5iT^{2} \)
89 \( 1 - 552. iT - 7.04e5T^{2} \)
97 \( 1 + (-460. + 460. i)T - 9.12e5iT^{2} \)
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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

−17.77985279507936935910659355076, −16.95429441126145176392909871858, −15.23722280067712443264168892071, −13.87626873853453457571264852068, −12.70178199799571476384060143630, −10.44073548831785995159392234835, −9.419400006574715774991886483152, −7.68443594585726623506568097750, −6.52607985161147397187458486045, −2.15758103342252538819102954414, 2.89403133439151454233574161653, 6.21998371934022356453435387675, 8.761339985971014058958984736693, 9.315663151450989476924707477077, 10.64407147283354037402567126233, 12.55786558546095883013634867832, 14.25875957073140071857047132065, 15.73151580688921947150806196523, 16.57250002342352185544134088693, 18.06528834771601513016425541526

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