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Properties

Label 2-90-45.2-c1-0-4
Degree 22
Conductor 9090
Sign 0.615+0.787i0.615 + 0.787i
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 + (1.62 − 0.599i)3-s + (−0.866 + 0.499i)4-s + (2.20 + 0.358i)5-s + (−1 − 1.41i)6-s + (−4.40 + 1.18i)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 + (−1.10 + 1.33i)12-s + (−3.34 − 0.896i)13-s + (2.28 + 3.94i)14-s + (3.80 − 0.741i)15-s + (0.500 − 0.866i)16-s + (0.317 − 0.317i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.938 − 0.346i)3-s + (−0.433 + 0.249i)4-s + (0.987 + 0.160i)5-s + (−0.408 − 0.577i)6-s + (−1.66 + 0.446i)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 + (−0.319 + 0.384i)12-s + (−0.928 − 0.248i)13-s + (0.609 + 1.05i)14-s + (0.981 − 0.191i)15-s + (0.125 − 0.216i)16-s + (0.0770 − 0.0770i)17-s + ⋯

Functional equation

Λ(s)=(90s/2 ΓC(s) L(s)=((0.615+0.787i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(90s/2 ΓC(s+1/2) L(s)=((0.615+0.787i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9090    =    2⋅32⋅52 \cdot 3^{2} \cdot 5
Sign: 0.615+0.787i0.615 + 0.787i
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), 0.615+0.787i)(2,\ 90,\ (\ :1/2),\ 0.615 + 0.787i)

Particular Values

L(1)L(1) ≈\approx 1.00182−0.488464i1.00182 - 0.488464i
L(12)L(\frac12) ≈\approx 1.00182−0.488464i1.00182 - 0.488464i
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+(−1.62+0.599i)T 1 + (-1.62 + 0.599i)T
5 1+(−2.20−0.358i)T 1 + (-2.20 - 0.358i)T
good7 1+(4.40−1.18i)T+(6.06−3.5i)T2 1 + (4.40 - 1.18i)T + (6.06 - 3.5i)T^{2}
11 1+(0.550+0.317i)T+(5.5+9.52i)T2 1 + (0.550 + 0.317i)T + (5.5 + 9.52i)T^{2}
13 1+(3.34+0.896i)T+(11.2+6.5i)T2 1 + (3.34 + 0.896i)T + (11.2 + 6.5i)T^{2}
17 1+(−0.317+0.317i)T−17iT2 1 + (-0.317 + 0.317i)T - 17iT^{2}
19 1−6.44iT−19T2 1 - 6.44iT - 19T^{2}
23 1+(0.258−0.965i)T+(−19.9−11.5i)T2 1 + (0.258 - 0.965i)T + (-19.9 - 11.5i)T^{2}
29 1+(0.158−0.275i)T+(−14.5−25.1i)T2 1 + (0.158 - 0.275i)T + (-14.5 - 25.1i)T^{2}
31 1+(0.224+0.389i)T+(−15.5+26.8i)T2 1 + (0.224 + 0.389i)T + (-15.5 + 26.8i)T^{2}
37 1+(3+3i)T+37iT2 1 + (3 + 3i)T + 37iT^{2}
41 1+(−6.39+3.69i)T+(20.5−35.5i)T2 1 + (-6.39 + 3.69i)T + (20.5 - 35.5i)T^{2}
43 1+(0.896+3.34i)T+(−37.2+21.5i)T2 1 + (0.896 + 3.34i)T + (-37.2 + 21.5i)T^{2}
47 1+(2.32+8.69i)T+(−40.7+23.5i)T2 1 + (2.32 + 8.69i)T + (-40.7 + 23.5i)T^{2}
53 1+(−3.78−3.78i)T+53iT2 1 + (-3.78 - 3.78i)T + 53iT^{2}
59 1+(4.48+7.77i)T+(−29.5+51.0i)T2 1 + (4.48 + 7.77i)T + (-29.5 + 51.0i)T^{2}
61 1+(−0.275+0.476i)T+(−30.5−52.8i)T2 1 + (-0.275 + 0.476i)T + (-30.5 - 52.8i)T^{2}
67 1+(1.71−6.38i)T+(−58.0−33.5i)T2 1 + (1.71 - 6.38i)T + (-58.0 - 33.5i)T^{2}
71 1−6.29iT−71T2 1 - 6.29iT - 71T^{2}
73 1+(6.89−6.89i)T−73iT2 1 + (6.89 - 6.89i)T - 73iT^{2}
79 1+(2.12+1.22i)T+(39.5+68.4i)T2 1 + (2.12 + 1.22i)T + (39.5 + 68.4i)T^{2}
83 1+(−5.26+1.41i)T+(71.8−41.5i)T2 1 + (-5.26 + 1.41i)T + (71.8 - 41.5i)T^{2}
89 1+8.02T+89T2 1 + 8.02T + 89T^{2}
97 1+(2.59−0.695i)T+(84.0−48.5i)T2 1 + (2.59 - 0.695i)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

−13.75390838752650612222274233985, −12.82975141676855669380480755959, −12.28544385574707569462784434033, −10.18899968214190585881307450664, −9.738465556796882741992324694516, −8.761182982633527232832837116383, −7.19338605129095804100231117999, −5.82559861955877040318123449483, −3.45443420619949451364967755639, −2.31256053256784956475585516413, 2.85018954803007261721190497747, 4.69661118540076006162861913531, 6.37784630758992714555097048811, 7.37512029690529663562849923539, 9.032593130855053680423076343793, 9.608040547364422430860932352312, 10.40793916549395773361012592721, 12.77800970167359361823309551339, 13.37567498932463299362309710398, 14.25340444945879559499823752920

Graph of the ZZ-function along the critical line