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Properties

Label 2-90-45.38-c1-0-3
Degree 22
Conductor 9090
Sign 0.970−0.241i0.970 - 0.241i
Analytic cond. 0.7186530.718653
Root an. cond. 0.8477340.847734
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 − 0.258i)2-s + (0.933 + 1.45i)3-s + (0.866 − 0.499i)4-s + (−2.22 + 0.210i)5-s + (1.27 + 1.16i)6-s + (−0.521 − 1.94i)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 + (1.53 + 0.796i)12-s + (1.05 − 3.92i)13-s + (−1.00 − 1.74i)14-s + (−2.38 − 3.05i)15-s + (0.500 − 0.866i)16-s + (2.35 + 2.35i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.539 + 0.842i)3-s + (0.433 − 0.249i)4-s + (−0.995 + 0.0942i)5-s + (0.522 + 0.476i)6-s + (−0.197 − 0.736i)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 + (0.444 + 0.229i)12-s + (0.291 − 1.08i)13-s + (−0.269 − 0.466i)14-s + (−0.616 − 0.787i)15-s + (0.125 − 0.216i)16-s + (0.572 + 0.572i)17-s + ⋯

Functional equation

Λ(s)=(90s/2 ΓC(s) L(s)=((0.970−0.241i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(90s/2 ΓC(s+1/2) L(s)=((0.970−0.241i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9090    =    2⋅32⋅52 \cdot 3^{2} \cdot 5
Sign: 0.970−0.241i0.970 - 0.241i
Analytic conductor: 0.7186530.718653
Root analytic conductor: 0.8477340.847734
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ90(83,⋅)\chi_{90} (83, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 90, ( :1/2), 0.970−0.241i)(2,\ 90,\ (\ :1/2),\ 0.970 - 0.241i)

Particular Values

L(1)L(1) ≈\approx 1.36376+0.166895i1.36376 + 0.166895i
L(12)L(\frac12) ≈\approx 1.36376+0.166895i1.36376 + 0.166895i
L(32)L(\frac{3}{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+(−0.965+0.258i)T 1 + (-0.965 + 0.258i)T
3 1+(−0.933−1.45i)T 1 + (-0.933 - 1.45i)T
5 1+(2.22−0.210i)T 1 + (2.22 - 0.210i)T
good7 1+(0.521+1.94i)T+(−6.06+3.5i)T2 1 + (0.521 + 1.94i)T + (-6.06 + 3.5i)T^{2}
11 1+(1.70+0.984i)T+(5.5+9.52i)T2 1 + (1.70 + 0.984i)T + (5.5 + 9.52i)T^{2}
13 1+(−1.05+3.92i)T+(−11.2−6.5i)T2 1 + (-1.05 + 3.92i)T + (-11.2 - 6.5i)T^{2}
17 1+(−2.35−2.35i)T+17iT2 1 + (-2.35 - 2.35i)T + 17iT^{2}
19 1−3.70iT−19T2 1 - 3.70iT - 19T^{2}
23 1+(6.05+1.62i)T+(19.9+11.5i)T2 1 + (6.05 + 1.62i)T + (19.9 + 11.5i)T^{2}
29 1+(3.74−6.49i)T+(−14.5−25.1i)T2 1 + (3.74 - 6.49i)T + (-14.5 - 25.1i)T^{2}
31 1+(−3.48−6.04i)T+(−15.5+26.8i)T2 1 + (-3.48 - 6.04i)T + (-15.5 + 26.8i)T^{2}
37 1+(−4.26+4.26i)T−37iT2 1 + (-4.26 + 4.26i)T - 37iT^{2}
41 1+(−6.13+3.54i)T+(20.5−35.5i)T2 1 + (-6.13 + 3.54i)T + (20.5 - 35.5i)T^{2}
43 1+(9.09−2.43i)T+(37.2−21.5i)T2 1 + (9.09 - 2.43i)T + (37.2 - 21.5i)T^{2}
47 1+(−7.49+2.00i)T+(40.7−23.5i)T2 1 + (-7.49 + 2.00i)T + (40.7 - 23.5i)T^{2}
53 1+(−7.03+7.03i)T−53iT2 1 + (-7.03 + 7.03i)T - 53iT^{2}
59 1+(−1.34−2.33i)T+(−29.5+51.0i)T2 1 + (-1.34 - 2.33i)T + (-29.5 + 51.0i)T^{2}
61 1+(4.37−7.57i)T+(−30.5−52.8i)T2 1 + (4.37 - 7.57i)T + (-30.5 - 52.8i)T^{2}
67 1+(8.18+2.19i)T+(58.0+33.5i)T2 1 + (8.18 + 2.19i)T + (58.0 + 33.5i)T^{2}
71 1+5.68iT−71T2 1 + 5.68iT - 71T^{2}
73 1+(−1.14−1.14i)T+73iT2 1 + (-1.14 - 1.14i)T + 73iT^{2}
79 1+(10.0+5.80i)T+(39.5+68.4i)T2 1 + (10.0 + 5.80i)T + (39.5 + 68.4i)T^{2}
83 1+(0.440+1.64i)T+(−71.8+41.5i)T2 1 + (0.440 + 1.64i)T + (-71.8 + 41.5i)T^{2}
89 1+2.04T+89T2 1 + 2.04T + 89T^{2}
97 1+(−2.60−9.71i)T+(−84.0+48.5i)T2 1 + (-2.60 - 9.71i)T + (-84.0 + 48.5i)T^{2}
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   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

−14.31298212003126384897969386363, −13.19856269370687314248402722393, −12.11088207473277873107977593277, −10.66859833482839975679205238925, −10.31432511492175483454994881310, −8.427541330882447641857029951900, −7.52274720129954352861200684514, −5.60027414188458140727662036094, −4.09583677000796742339092850936, −3.24414664008117669448514210312, 2.59758594820897388503820025075, 4.19993852546629065683394378293, 5.99345562269768015513279761027, 7.28033951315223620773979660609, 8.162315258809296139594649527987, 9.430361765285760861297004396465, 11.55984880446351189933468688980, 11.96834174409633152322071004812, 13.09559632320528467789870257467, 13.96027856464549760364042502919

Graph of the ZZ-function along the critical line