↑

Properties

Label 2-90-5.3-c2-0-2
Degree 22
Conductor 9090
Sign 0.767+0.640i0.767 + 0.640i
Analytic cond. 2.452322.45232
Root an. cond. 1.565981.56598
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s − 2i·4-s + (−3 − 4i)5-s + (8 − 8i)7-s + (2 + 2i)8-s + (7 + i)10-s + 4·11-s + (−3 − 3i)13-s + 16i·14-s − 4·16-s + (19 − 19i)17-s + 8i·19-s + (−8 + 6i)20-s + (−4 + 4i)22-s + (−20 − 20i)23-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s − 0.5i·4-s + (−0.600 − 0.800i)5-s + (1.14 − 1.14i)7-s + (0.250 + 0.250i)8-s + (0.700 + 0.100i)10-s + 0.363·11-s + (−0.230 − 0.230i)13-s + 1.14i·14-s − 0.250·16-s + (1.11 − 1.11i)17-s + 0.421i·19-s + (−0.400 + 0.300i)20-s + (−0.181 + 0.181i)22-s + (−0.869 − 0.869i)23-s + ⋯

Functional equation

Λ(s)=(90s/2 ΓC(s) L(s)=((0.767+0.640i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 + 0.640i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(90s/2 ΓC(s+1) L(s)=((0.767+0.640i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9090    =    2⋅32⋅52 \cdot 3^{2} \cdot 5
Sign: 0.767+0.640i0.767 + 0.640i
Analytic conductor: 2.452322.45232
Root analytic conductor: 1.565981.56598
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ90(73,⋅)\chi_{90} (73, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 90, ( :1), 0.767+0.640i)(2,\ 90,\ (\ :1),\ 0.767 + 0.640i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.929943−0.337071i0.929943 - 0.337071i
L(12)L(\frac12) ≈\approx 0.929943−0.337071i0.929943 - 0.337071i
L(2)L(2) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(1−i)T 1 + (1 - i)T
3 1 1
5 1+(3+4i)T 1 + (3 + 4i)T
good7 1+(−8+8i)T−49iT2 1 + (-8 + 8i)T - 49iT^{2}
11 1−4T+121T2 1 - 4T + 121T^{2}
13 1+(3+3i)T+169iT2 1 + (3 + 3i)T + 169iT^{2}
17 1+(−19+19i)T−289iT2 1 + (-19 + 19i)T - 289iT^{2}
19 1−8iT−361T2 1 - 8iT - 361T^{2}
23 1+(20+20i)T+529iT2 1 + (20 + 20i)T + 529iT^{2}
29 1−38iT−841T2 1 - 38iT - 841T^{2}
31 1+44T+961T2 1 + 44T + 961T^{2}
37 1+(3−3i)T−1.36e3iT2 1 + (3 - 3i)T - 1.36e3iT^{2}
41 1−70T+1.68e3T2 1 - 70T + 1.68e3T^{2}
43 1+(−36−36i)T+1.84e3iT2 1 + (-36 - 36i)T + 1.84e3iT^{2}
47 1−2.20e3iT2 1 - 2.20e3iT^{2}
53 1+(−17−17i)T+2.80e3iT2 1 + (-17 - 17i)T + 2.80e3iT^{2}
59 1−92iT−3.48e3T2 1 - 92iT - 3.48e3T^{2}
61 1−72T+3.72e3T2 1 - 72T + 3.72e3T^{2}
67 1+(−44+44i)T−4.48e3iT2 1 + (-44 + 44i)T - 4.48e3iT^{2}
71 1+88T+5.04e3T2 1 + 88T + 5.04e3T^{2}
73 1+(−55−55i)T+5.32e3iT2 1 + (-55 - 55i)T + 5.32e3iT^{2}
79 1+12iT−6.24e3T2 1 + 12iT - 6.24e3T^{2}
83 1+(24+24i)T+6.88e3iT2 1 + (24 + 24i)T + 6.88e3iT^{2}
89 1+26iT−7.92e3T2 1 + 26iT - 7.92e3T^{2}
97 1+(57−57i)T−9.40e3iT2 1 + (57 - 57i)T - 9.40e3iT^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−14.15886357832638319280918651839, −12.62924959685238902986063294303, −11.49901903319005434993482542731, −10.44556285674008257262123730504, −9.138106399083541669950966414132, −7.923884175503687284138153359726, −7.32213427269648630663092833128, −5.35773864414447254213450436352, −4.17151599413298949152826555678, −1.04637085856452170468959364729, 2.13577502529342843796375824345, 3.89414961913487455801271052203, 5.77026034946687879387660380905, 7.52441484965829760354957143396, 8.362558544002381284914929252314, 9.639434373450386469354397017371, 10.96430840121788125065986836627, 11.67597479320255105685364595619, 12.47636677792100653882590862659, 14.23602651858445477500073167637

Graph of the ZZ-function along the critical line