Properties

Label 1-2652-2652.203-r1-0-0
Degree 11
Conductor 26522652
Sign 0.289+0.957i0.289 + 0.957i
Analytic cond. 284.996284.996
Root an. cond. 284.996284.996
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·5-s i·7-s + i·11-s i·19-s + 23-s − 25-s + 29-s + i·31-s + 35-s + i·37-s + i·41-s + 43-s i·47-s − 49-s − 53-s + ⋯
L(s)  = 1  + i·5-s i·7-s + i·11-s i·19-s + 23-s − 25-s + 29-s + i·31-s + 35-s + i·37-s + i·41-s + 43-s i·47-s − 49-s − 53-s + ⋯

Functional equation

Λ(s)=(2652s/2ΓR(s+1)L(s)=((0.289+0.957i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2652s/2ΓR(s+1)L(s)=((0.289+0.957i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 26522652    =    22313172^{2} \cdot 3 \cdot 13 \cdot 17
Sign: 0.289+0.957i0.289 + 0.957i
Analytic conductor: 284.996284.996
Root analytic conductor: 284.996284.996
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2652(203,)\chi_{2652} (203, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 2652, (1: ), 0.289+0.957i)(1,\ 2652,\ (1:\ ),\ 0.289 + 0.957i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.643877838+1.219849362i1.643877838 + 1.219849362i
L(12)L(\frac12) \approx 1.643877838+1.219849362i1.643877838 + 1.219849362i
L(1)L(1) \approx 1.075618898+0.1842456970i1.075618898 + 0.1842456970i
L(1)L(1) \approx 1.075618898+0.1842456970i1.075618898 + 0.1842456970i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
13 1 1
17 1 1
good5 1 1
7 1 1
11 1 1
19 1iT 1 - iT
23 1 1
29 1 1
31 1 1
37 1+iT 1 + iT
41 1 1
43 1 1
47 1 1
53 1 1
59 1 1
61 1 1
67 1 1
71 1iT 1 - iT
73 1 1
79 1 1
83 1 1
89 1+T 1 + T
97 1 1
show more
show less
   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−19.10956064636441266752044591556, −18.39839401862955188135216063870, −17.48981246494809953304747523198, −16.88276877422785777681644106888, −16.03568852323072417840652775504, −15.75098451281606064143654806189, −14.722255341152508721348282572454, −14.03566329615061796427885474951, −13.17834829732965333787803956383, −12.54009602222527183157285568826, −11.96164207785894910024052951095, −11.21429136814709594949187819434, −10.36122982571962732343073083333, −9.28034492782624357656800554962, −8.9417584090894607363851337144, −8.19557378970015794381274494239, −7.50161740104936835524152294297, −6.13157231923907737778152466677, −5.80090288172321568368146165004, −4.97879612694454144513622772018, −4.12869669708688481371458180387, −3.17473240890460697593794793333, −2.28826144508215462356608315416, −1.29488249763888711895309197462, −0.43725738136437255845062952910, 0.76413261241354197522231600514, 1.77795092247748605156493586096, 2.81891249739205050324702558859, 3.41414858487303379050400182490, 4.50768020148246360295445045797, 4.97644780078939820155912710355, 6.393761151498034007044642702302, 6.830356242139791647672047262835, 7.4254792294155592477579334814, 8.23054868929958096977486016744, 9.36123508126607548216047082043, 9.97855572221918606945473014626, 10.72496352159313116172129331176, 11.16976182411773541550636072146, 12.12986818959568642817900725196, 12.94303539004129030711808944684, 13.7427225502222050807743849393, 14.248082290083688128775216467057, 15.12415842868160741006042007330, 15.53129080935302200488606887593, 16.55814624498709684893815778434, 17.36149520085207346143707275365, 17.789749420702633933183960062933, 18.51039017039806790702231562002, 19.48641617190070176595273425397

Graph of the ZZ-function along the critical line