Properties

Label 2-90-9.7-c1-0-0
Degree 22
Conductor 9090
Sign 0.9390.342i0.939 - 0.342i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (1.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + 1.73i·6-s + (0.5 − 0.866i)7-s + 0.999·8-s + (1.5 − 2.59i)9-s − 0.999·10-s + (−3 + 5.19i)11-s + (−1.49 − 0.866i)12-s + (−1 − 1.73i)13-s + (0.499 + 0.866i)14-s + (1.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.866 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s + 0.707i·6-s + (0.188 − 0.327i)7-s + 0.353·8-s + (0.5 − 0.866i)9-s − 0.316·10-s + (−0.904 + 1.56i)11-s + (−0.433 − 0.250i)12-s + (−0.277 − 0.480i)13-s + (0.133 + 0.231i)14-s + (0.387 + 0.223i)15-s + (−0.125 + 0.216i)16-s + ⋯

Functional equation

Λ(s)=(90s/2ΓC(s)L(s)=((0.9390.342i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(90s/2ΓC(s+1/2)L(s)=((0.9390.342i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9090    =    23252 \cdot 3^{2} \cdot 5
Sign: 0.9390.342i0.939 - 0.342i
Analytic conductor: 0.7186530.718653
Root analytic conductor: 0.8477340.847734
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ90(61,)\chi_{90} (61, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 90, ( :1/2), 0.9390.342i)(2,\ 90,\ (\ :1/2),\ 0.939 - 0.342i)

Particular Values

L(1)L(1) \approx 1.02091+0.180015i1.02091 + 0.180015i
L(12)L(\frac12) \approx 1.02091+0.180015i1.02091 + 0.180015i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
3 1+(1.5+0.866i)T 1 + (-1.5 + 0.866i)T
5 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
good7 1+(0.5+0.866i)T+(3.56.06i)T2 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2}
11 1+(35.19i)T+(5.59.52i)T2 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2}
13 1+(1+1.73i)T+(6.5+11.2i)T2 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2}
17 1+17T2 1 + 17T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1+(4.5+7.79i)T+(11.5+19.9i)T2 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2}
29 1+(1.52.59i)T+(14.525.1i)T2 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2}
31 1+(23.46i)T+(15.5+26.8i)T2 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2}
37 18T+37T2 1 - 8T + 37T^{2}
41 1+(1.52.59i)T+(20.5+35.5i)T2 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2}
43 1+(46.92i)T+(21.537.2i)T2 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2}
47 1+(1.5+2.59i)T+(23.540.7i)T2 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2}
53 16T+53T2 1 - 6T + 53T^{2}
59 1+(3+5.19i)T+(29.5+51.0i)T2 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2}
61 1+(6.5+11.2i)T+(30.552.8i)T2 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2}
67 1+(6.511.2i)T+(33.5+58.0i)T2 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2}
71 1+6T+71T2 1 + 6T + 71T^{2}
73 1+4T+73T2 1 + 4T + 73T^{2}
79 1+(5+8.66i)T+(39.568.4i)T2 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2}
83 1+(4.5+7.79i)T+(41.571.8i)T2 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2}
89 19T+89T2 1 - 9T + 89T^{2}
97 1+(11.73i)T+(48.584.0i)T2 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2}
show more
show less
   L(s)=p j=12(1αj,pps)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

−14.56079602071985426039705926516, −13.20930217081189068297682058650, −12.45137356857302849413930973901, −10.50162404784282696573836333329, −9.747830227416623325614311522859, −8.321306931986745331309819886925, −7.48422726612557706065549337647, −6.45954662637191414006009843123, −4.55038068692036260792144174367, −2.33919010819884478167126981392, 2.37537030969457984752294204023, 3.93544610495337274078412802121, 5.57075319935115414868322336336, 7.83575954475383675984806026531, 8.641736633502856368430856265107, 9.612859932227793954121340908248, 10.68583135433577144212687711208, 11.76290228817964208844447116448, 13.27077342351155635351936184337, 13.74390613949249611289264562549

Graph of the ZZ-function along the critical line