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Properties

Label 2-90-45.4-c1-0-0
Degree 22
Conductor 9090
Sign −0.401−0.915i-0.401 - 0.915i
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  + (−0.866 + 0.5i)2-s + (−0.158 + 1.72i)3-s + (0.499 − 0.866i)4-s + (−1.30 + 1.81i)5-s + (−0.724 − 1.57i)6-s + (−0.389 + 0.224i)7-s + 0.999i·8-s + (−2.94 − 0.548i)9-s + (0.224 − 2.22i)10-s + (1.72 + 2.98i)11-s + (1.41 + 0.999i)12-s + (2.12 + 1.22i)13-s + (0.224 − 0.389i)14-s + (−2.92 − 2.54i)15-s + (−0.5 − 0.866i)16-s − 5.89i·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.0917 + 0.995i)3-s + (0.249 − 0.433i)4-s + (−0.584 + 0.811i)5-s + (−0.295 − 0.642i)6-s + (−0.147 + 0.0849i)7-s + 0.353i·8-s + (−0.983 − 0.182i)9-s + (0.0710 − 0.703i)10-s + (0.520 + 0.900i)11-s + (0.408 + 0.288i)12-s + (0.588 + 0.339i)13-s + (0.0600 − 0.104i)14-s + (−0.754 − 0.656i)15-s + (−0.125 − 0.216i)16-s − 1.43i·17-s + ⋯

Functional equation

Λ(s)=(90s/2 ΓC(s) L(s)=((−0.401−0.915i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(90s/2 ΓC(s+1/2) L(s)=((−0.401−0.915i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9090    =    2⋅32⋅52 \cdot 3^{2} \cdot 5
Sign: −0.401−0.915i-0.401 - 0.915i
Analytic conductor: 0.7186530.718653
Root analytic conductor: 0.8477340.847734
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ90(49,⋅)\chi_{90} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 90, ( :1/2), −0.401−0.915i)(2,\ 90,\ (\ :1/2),\ -0.401 - 0.915i)

Particular Values

L(1)L(1) ≈\approx 0.359965+0.551162i0.359965 + 0.551162i
L(12)L(\frac12) ≈\approx 0.359965+0.551162i0.359965 + 0.551162i
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+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
3 1+(0.158−1.72i)T 1 + (0.158 - 1.72i)T
5 1+(1.30−1.81i)T 1 + (1.30 - 1.81i)T
good7 1+(0.389−0.224i)T+(3.5−6.06i)T2 1 + (0.389 - 0.224i)T + (3.5 - 6.06i)T^{2}
11 1+(−1.72−2.98i)T+(−5.5+9.52i)T2 1 + (-1.72 - 2.98i)T + (-5.5 + 9.52i)T^{2}
13 1+(−2.12−1.22i)T+(6.5+11.2i)T2 1 + (-2.12 - 1.22i)T + (6.5 + 11.2i)T^{2}
17 1+5.89iT−17T2 1 + 5.89iT - 17T^{2}
19 1−5.44T+19T2 1 - 5.44T + 19T^{2}
23 1+(−5.97−3.44i)T+(11.5+19.9i)T2 1 + (-5.97 - 3.44i)T + (11.5 + 19.9i)T^{2}
29 1+(3+5.19i)T+(−14.5+25.1i)T2 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2}
31 1+(0.775−1.34i)T+(−15.5−26.8i)T2 1 + (0.775 - 1.34i)T + (-15.5 - 26.8i)T^{2}
37 1−8iT−37T2 1 - 8iT - 37T^{2}
41 1+(−0.5+0.866i)T+(−20.5−35.5i)T2 1 + (-0.5 + 0.866i)T + (-20.5 - 35.5i)T^{2}
43 1+(2.20−1.27i)T+(21.5−37.2i)T2 1 + (2.20 - 1.27i)T + (21.5 - 37.2i)T^{2}
47 1+(3.85−2.22i)T+(23.5−40.7i)T2 1 + (3.85 - 2.22i)T + (23.5 - 40.7i)T^{2}
53 1+3.55iT−53T2 1 + 3.55iT - 53T^{2}
59 1+(−6.62+11.4i)T+(−29.5−51.0i)T2 1 + (-6.62 + 11.4i)T + (-29.5 - 51.0i)T^{2}
61 1+(2.22+3.85i)T+(−30.5+52.8i)T2 1 + (2.22 + 3.85i)T + (-30.5 + 52.8i)T^{2}
67 1+(3.94+2.27i)T+(33.5+58.0i)T2 1 + (3.94 + 2.27i)T + (33.5 + 58.0i)T^{2}
71 1+2.44T+71T2 1 + 2.44T + 71T^{2}
73 1+14.7iT−73T2 1 + 14.7iT - 73T^{2}
79 1+(−3.67−6.36i)T+(−39.5+68.4i)T2 1 + (-3.67 - 6.36i)T + (-39.5 + 68.4i)T^{2}
83 1+(−3.46+2i)T+(41.5−71.8i)T2 1 + (-3.46 + 2i)T + (41.5 - 71.8i)T^{2}
89 1+3.10T+89T2 1 + 3.10T + 89T^{2}
97 1+(−11.2+6.5i)T+(48.5−84.0i)T2 1 + (-11.2 + 6.5i)T + (48.5 - 84.0i)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

−14.80252802390489462235161184135, −13.80910749006305444922646164171, −11.73164429480254117775794839483, −11.22838965258597206306847151996, −9.862181415399079699869606501466, −9.236603134252167103375156632271, −7.68032047276718662452747175486, −6.56722115902533300852606747793, −4.91569702126479824903541430576, −3.25308499998496659481242528402, 1.13154269407539701542357772075, 3.49130015533201372340507188095, 5.69972505269856589164084678426, 7.15732078721980594169475192652, 8.339848874134542507746416653377, 8.975974143258665538829426203126, 10.80949910862099754801095379361, 11.65291547592846516934788169979, 12.68020831287898065998586212391, 13.34870993727116602127202696599

Graph of the ZZ-function along the critical line