↑

Properties

Label 2-90-45.38-c1-0-0
Degree $2$
Conductor $90$
Sign $0.601 - 0.799i$
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  + (−0.965 + 0.258i)2-s + (0.599 + 1.62i)3-s + (0.866 − 0.499i)4-s + (0.792 − 2.09i)5-s + (−1 − 1.41i)6-s + (1.18 + 4.40i)7-s + (−0.707 + 0.707i)8-s + (−2.28 + 1.94i)9-s + (−0.224 + 2.22i)10-s + (−0.550 − 0.317i)11-s + (1.33 + 1.10i)12-s + (0.896 − 3.34i)13-s + (−2.28 − 3.94i)14-s + (3.87 + 0.0340i)15-s + (0.500 − 0.866i)16-s + (−0.317 − 0.317i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.346 + 0.938i)3-s + (0.433 − 0.249i)4-s + (0.354 − 0.935i)5-s + (−0.408 − 0.577i)6-s + (0.446 + 1.66i)7-s + (−0.249 + 0.249i)8-s + (−0.760 + 0.649i)9-s + (−0.0710 + 0.703i)10-s + (−0.165 − 0.0958i)11-s + (0.384 + 0.319i)12-s + (0.248 − 0.928i)13-s + (−0.609 − 1.05i)14-s + (0.999 + 0.00879i)15-s + (0.125 − 0.216i)16-s + (−0.0770 − 0.0770i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.601 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.601 - 0.799i$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :1/2),\ 0.601 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.763464 + 0.381005i\)
\(L(\frac12)\) \(\approx\) \(0.763464 + 0.381005i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-0.599 - 1.62i)T \)
5 \( 1 + (-0.792 + 2.09i)T \)
good7 \( 1 + (-1.18 - 4.40i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.550 + 0.317i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.896 + 3.34i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.317 + 0.317i)T + 17iT^{2} \)
19 \( 1 + 6.44iT - 19T^{2} \)
23 \( 1 + (0.965 + 0.258i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.158 + 0.275i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.224 + 0.389i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 + (-6.39 + 3.69i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.34 + 0.896i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (8.69 - 2.32i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.78 - 3.78i)T - 53iT^{2} \)
59 \( 1 + (-4.48 - 7.77i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.275 + 0.476i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.38 - 1.71i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 6.29iT - 71T^{2} \)
73 \( 1 + (6.89 + 6.89i)T + 73iT^{2} \)
79 \( 1 + (-2.12 - 1.22i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.41 - 5.26i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 8.02T + 89T^{2} \)
97 \( 1 + (-0.695 - 2.59i)T + (-84.0 + 48.5i)T^{2} \)
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

−14.65391340243258118597440531703, −13.22839288902140553337330194255, −11.96406076958945852136605620621, −10.86485940193238606585941192427, −9.531633963565565678088429208059, −8.845301417909822178133117310060, −8.118686021830512862527888487670, −5.80845260597834299632988121465, −4.95278368525193385784889916190, −2.55617330786342676836139657284, 1.74013925689599066407174594820, 3.66838064547409474418430929729, 6.36111826115680485824442115002, 7.27409102106321019340703886834, 8.087011789233646609940391795103, 9.686715208188720926488225489638, 10.70737972022764571268550080105, 11.59094119948522173291774775732, 13.01610580681947777752892090188, 14.10626020979474037671008769008

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