↑

Properties

Label 2-90-45.2-c1-0-3
Degree $2$
Conductor $90$
Sign $0.803 - 0.595i$
Analytic cond. $0.718653$
Root an. cond. $0.847734$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 + 0.965i)2-s + (1.45 − 0.933i)3-s + (−0.866 + 0.499i)4-s + (−1.29 + 1.82i)5-s + (1.27 + 1.16i)6-s + (1.94 − 0.521i)7-s + (−0.707 − 0.707i)8-s + (1.25 − 2.72i)9-s + (−2.09 − 0.779i)10-s + (−1.70 − 0.984i)11-s + (−0.796 + 1.53i)12-s + (−3.92 − 1.05i)13-s + (1.00 + 1.74i)14-s + (−0.187 + 3.86i)15-s + (0.500 − 0.866i)16-s + (−2.35 + 2.35i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.842 − 0.539i)3-s + (−0.433 + 0.249i)4-s + (−0.579 + 0.815i)5-s + (0.522 + 0.476i)6-s + (0.736 − 0.197i)7-s + (−0.249 − 0.249i)8-s + (0.418 − 0.908i)9-s + (−0.662 − 0.246i)10-s + (−0.514 − 0.296i)11-s + (−0.229 + 0.444i)12-s + (−1.08 − 0.291i)13-s + (0.269 + 0.466i)14-s + (−0.0484 + 0.998i)15-s + (0.125 − 0.216i)16-s + (−0.572 + 0.572i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.803 - 0.595i$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :1/2),\ 0.803 - 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14374 + 0.377975i\)
\(L(\frac12)\) \(\approx\) \(1.14374 + 0.377975i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (-1.45 + 0.933i)T \)
5 \( 1 + (1.29 - 1.82i)T \)
good7 \( 1 + (-1.94 + 0.521i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.70 + 0.984i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.92 + 1.05i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (2.35 - 2.35i)T - 17iT^{2} \)
19 \( 1 + 3.70iT - 19T^{2} \)
23 \( 1 + (1.62 - 6.05i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-3.74 + 6.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.48 - 6.04i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.26 - 4.26i)T + 37iT^{2} \)
41 \( 1 + (-6.13 + 3.54i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.43 - 9.09i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.00 - 7.49i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (7.03 + 7.03i)T + 53iT^{2} \)
59 \( 1 + (1.34 + 2.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.37 - 7.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.19 + 8.18i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 5.68iT - 71T^{2} \)
73 \( 1 + (-1.14 + 1.14i)T - 73iT^{2} \)
79 \( 1 + (-10.0 - 5.80i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.64 - 0.440i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 2.04T + 89T^{2} \)
97 \( 1 + (9.71 - 2.60i)T + (84.0 - 48.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.30647357169344817195482801218, −13.51868030299500603936399549402, −12.31654178707843737963216896321, −11.09521796026550901867230792122, −9.636281961279550132279867415156, −8.102640160480083131188038168101, −7.63012315094024931438546242373, −6.42997738599395217856712038548, −4.50347187072117973144532228450, −2.84177481793653947149061835986, 2.36571584037524598992534039297, 4.25458958365631750342899274197, 5.02709937150538163781308489180, 7.64721682709318273820158167845, 8.593927754594814781826961274143, 9.610452743447340643031176876979, 10.74973554638285262186854348916, 11.99506816446919096323644198660, 12.82466364931759484864639121987, 14.09049377651745275838757244315

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