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Properties

Label 2-90-45.4-c1-0-2
Degree 22
Conductor 9090
Sign 0.993−0.114i0.993 - 0.114i
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 + (−0.866 + 1.5i)3-s + (0.499 − 0.866i)4-s + (2.23 + 0.133i)5-s + 1.73i·6-s + (−0.866 + 0.5i)7-s − 0.999i·8-s + (−1.5 − 2.59i)9-s + (1.99 − i)10-s + (1 + 1.73i)11-s + (0.866 + 1.49i)12-s + (−5.19 − 3i)13-s + (−0.499 + 0.866i)14-s + (−2.13 + 3.23i)15-s + (−0.5 − 0.866i)16-s − 2i·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.499 + 0.866i)3-s + (0.249 − 0.433i)4-s + (0.998 + 0.0599i)5-s + 0.707i·6-s + (−0.327 + 0.188i)7-s − 0.353i·8-s + (−0.5 − 0.866i)9-s + (0.632 − 0.316i)10-s + (0.301 + 0.522i)11-s + (0.250 + 0.433i)12-s + (−1.44 − 0.832i)13-s + (−0.133 + 0.231i)14-s + (−0.550 + 0.834i)15-s + (−0.125 − 0.216i)16-s − 0.485i·17-s + ⋯

Functional equation

Λ(s)=(90s/2 ΓC(s) L(s)=((0.993−0.114i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(90s/2 ΓC(s+1/2) L(s)=((0.993−0.114i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9090    =    2⋅32⋅52 \cdot 3^{2} \cdot 5
Sign: 0.993−0.114i0.993 - 0.114i
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), 0.993−0.114i)(2,\ 90,\ (\ :1/2),\ 0.993 - 0.114i)

Particular Values

L(1)L(1) ≈\approx 1.22505+0.0702613i1.22505 + 0.0702613i
L(12)L(\frac12) ≈\approx 1.22505+0.0702613i1.22505 + 0.0702613i
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+(0.866−1.5i)T 1 + (0.866 - 1.5i)T
5 1+(−2.23−0.133i)T 1 + (-2.23 - 0.133i)T
good7 1+(0.866−0.5i)T+(3.5−6.06i)T2 1 + (0.866 - 0.5i)T + (3.5 - 6.06i)T^{2}
11 1+(−1−1.73i)T+(−5.5+9.52i)T2 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2}
13 1+(5.19+3i)T+(6.5+11.2i)T2 1 + (5.19 + 3i)T + (6.5 + 11.2i)T^{2}
17 1+2iT−17T2 1 + 2iT - 17T^{2}
19 1+6T+19T2 1 + 6T + 19T^{2}
23 1+(−0.866−0.5i)T+(11.5+19.9i)T2 1 + (-0.866 - 0.5i)T + (11.5 + 19.9i)T^{2}
29 1+(−4.5−7.79i)T+(−14.5+25.1i)T2 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2}
31 1+(−1+1.73i)T+(−15.5−26.8i)T2 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2}
37 1−2iT−37T2 1 - 2iT - 37T^{2}
41 1+(−5.5+9.52i)T+(−20.5−35.5i)T2 1 + (-5.5 + 9.52i)T + (-20.5 - 35.5i)T^{2}
43 1+(−3.46+2i)T+(21.5−37.2i)T2 1 + (-3.46 + 2i)T + (21.5 - 37.2i)T^{2}
47 1+(6.06−3.5i)T+(23.5−40.7i)T2 1 + (6.06 - 3.5i)T + (23.5 - 40.7i)T^{2}
53 1−53T2 1 - 53T^{2}
59 1+(2−3.46i)T+(−29.5−51.0i)T2 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2}
61 1+(−3.5−6.06i)T+(−30.5+52.8i)T2 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2}
67 1+(−9.52−5.5i)T+(33.5+58.0i)T2 1 + (-9.52 - 5.5i)T + (33.5 + 58.0i)T^{2}
71 1+6T+71T2 1 + 6T + 71T^{2}
73 1−4iT−73T2 1 - 4iT - 73T^{2}
79 1+(6+10.3i)T+(−39.5+68.4i)T2 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2}
83 1+(−9.52+5.5i)T+(41.5−71.8i)T2 1 + (-9.52 + 5.5i)T + (41.5 - 71.8i)T^{2}
89 1+T+89T2 1 + T + 89T^{2}
97 1+(6.92−4i)T+(48.5−84.0i)T2 1 + (6.92 - 4i)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

−14.35544749301079013769529954766, −12.85295718903619488271351804739, −12.19062534326430898163797428721, −10.73168638131145457989544693944, −10.02758857383745713976956920938, −9.107794971338959898382363204612, −6.82043002097545461274690869636, −5.60810404130365919907561901899, −4.60762477961140238604786618984, −2.73906414150220695599472359919, 2.31534766162471871481446780922, 4.72530765235595003728604336779, 6.14906564400756608464136136607, 6.75993903544224902772219265772, 8.254375647433727327560887201038, 9.779747887727566840538168300942, 11.13066402154609963959697028987, 12.32381791215817728156578994821, 13.06284365441183633661331090682, 13.98216398320997143553975093400

Graph of the ZZ-function along the critical line