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Properties

Label 2-90-45.2-c1-0-1
Degree 22
Conductor 9090
Sign 0.947−0.319i0.947 - 0.319i
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  + (−0.258 − 0.965i)2-s + (−0.0795 + 1.73i)3-s + (−0.866 + 0.499i)4-s + (0.661 + 2.13i)5-s + (1.69 − 0.370i)6-s + (3.75 − 1.00i)7-s + (0.707 + 0.707i)8-s + (−2.98 − 0.275i)9-s + (1.89 − 1.19i)10-s + (−3.44 − 1.98i)11-s + (−0.796 − 1.53i)12-s + (0.956 + 0.256i)13-s + (−1.94 − 3.36i)14-s + (−3.74 + 0.974i)15-s + (0.500 − 0.866i)16-s + (−0.120 + 0.120i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.0459 + 0.998i)3-s + (−0.433 + 0.249i)4-s + (0.295 + 0.955i)5-s + (0.690 − 0.151i)6-s + (1.41 − 0.380i)7-s + (0.249 + 0.249i)8-s + (−0.995 − 0.0917i)9-s + (0.598 − 0.376i)10-s + (−1.03 − 0.599i)11-s + (−0.229 − 0.444i)12-s + (0.265 + 0.0710i)13-s + (−0.519 − 0.899i)14-s + (−0.967 + 0.251i)15-s + (0.125 − 0.216i)16-s + (−0.0291 + 0.0291i)17-s + ⋯

Functional equation

Λ(s)=(90s/2 ΓC(s) L(s)=((0.947−0.319i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(90s/2 ΓC(s+1/2) L(s)=((0.947−0.319i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9090    =    2⋅32⋅52 \cdot 3^{2} \cdot 5
Sign: 0.947−0.319i0.947 - 0.319i
Analytic conductor: 0.7186530.718653
Root analytic conductor: 0.8477340.847734
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ90(47,⋅)\chi_{90} (47, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 90, ( :1/2), 0.947−0.319i)(2,\ 90,\ (\ :1/2),\ 0.947 - 0.319i)

Particular Values

L(1)L(1) ≈\approx 0.920564+0.151262i0.920564 + 0.151262i
L(12)L(\frac12) ≈\approx 0.920564+0.151262i0.920564 + 0.151262i
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+(0.258+0.965i)T 1 + (0.258 + 0.965i)T
3 1+(0.0795−1.73i)T 1 + (0.0795 - 1.73i)T
5 1+(−0.661−2.13i)T 1 + (-0.661 - 2.13i)T
good7 1+(−3.75+1.00i)T+(6.06−3.5i)T2 1 + (-3.75 + 1.00i)T + (6.06 - 3.5i)T^{2}
11 1+(3.44+1.98i)T+(5.5+9.52i)T2 1 + (3.44 + 1.98i)T + (5.5 + 9.52i)T^{2}
13 1+(−0.956−0.256i)T+(11.2+6.5i)T2 1 + (-0.956 - 0.256i)T + (11.2 + 6.5i)T^{2}
17 1+(0.120−0.120i)T−17iT2 1 + (0.120 - 0.120i)T - 17iT^{2}
19 1+1.88iT−19T2 1 + 1.88iT - 19T^{2}
23 1+(−1.36+5.08i)T+(−19.9−11.5i)T2 1 + (-1.36 + 5.08i)T + (-19.9 - 11.5i)T^{2}
29 1+(2.15−3.73i)T+(−14.5−25.1i)T2 1 + (2.15 - 3.73i)T + (-14.5 - 25.1i)T^{2}
31 1+(4.70+8.14i)T+(−15.5+26.8i)T2 1 + (4.70 + 8.14i)T + (-15.5 + 26.8i)T^{2}
37 1+(−3.26−3.26i)T+37iT2 1 + (-3.26 - 3.26i)T + 37iT^{2}
41 1+(−7.15+4.13i)T+(20.5−35.5i)T2 1 + (-7.15 + 4.13i)T + (20.5 - 35.5i)T^{2}
43 1+(−0.533−1.99i)T+(−37.2+21.5i)T2 1 + (-0.533 - 1.99i)T + (-37.2 + 21.5i)T^{2}
47 1+(−0.897−3.34i)T+(−40.7+23.5i)T2 1 + (-0.897 - 3.34i)T + (-40.7 + 23.5i)T^{2}
53 1+(3.66+3.66i)T+53iT2 1 + (3.66 + 3.66i)T + 53iT^{2}
59 1+(−2.72−4.72i)T+(−29.5+51.0i)T2 1 + (-2.72 - 4.72i)T + (-29.5 + 51.0i)T^{2}
61 1+(4.35−7.54i)T+(−30.5−52.8i)T2 1 + (4.35 - 7.54i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.10−7.86i)T+(−58.0−33.5i)T2 1 + (2.10 - 7.86i)T + (-58.0 - 33.5i)T^{2}
71 1−6.94iT−71T2 1 - 6.94iT - 71T^{2}
73 1+(8.27−8.27i)T−73iT2 1 + (8.27 - 8.27i)T - 73iT^{2}
79 1+(11.7+6.78i)T+(39.5+68.4i)T2 1 + (11.7 + 6.78i)T + (39.5 + 68.4i)T^{2}
83 1+(−6.75+1.81i)T+(71.8−41.5i)T2 1 + (-6.75 + 1.81i)T + (71.8 - 41.5i)T^{2}
89 1−4.87T+89T2 1 - 4.87T + 89T^{2}
97 1+(−1.44+0.387i)T+(84.0−48.5i)T2 1 + (-1.44 + 0.387i)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

−14.32963689902960724985949060259, −13.23648543805560862225761915740, −11.35702795268597906285999251280, −10.98384114021111656623357494192, −10.19388475575991369234523285945, −8.825428167125322110728291815423, −7.67291254090949274413504052014, −5.62503445204123225441378922322, −4.28407584483394434470350031662, −2.68100260688780801836488358418, 1.73428787292598141776655761801, 4.94451411667770672973877522933, 5.76611260963748113229591339883, 7.54672279069282917293181562776, 8.159730414949622211855171825428, 9.241691220189246874914387106744, 10.95588842666517222264581336446, 12.19657517282070384268762320802, 13.08588362641917830736271997575, 14.02715428475129943885560188053

Graph of the ZZ-function along the critical line