Properties

Label 2-1008-63.4-c1-0-36
Degree 22
Conductor 10081008
Sign 0.823+0.566i0.823 + 0.566i
Analytic cond. 8.048928.04892
Root an. cond. 2.837062.83706
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  + (1.5 − 0.866i)3-s + 5-s + (2.5 − 0.866i)7-s + (1.5 − 2.59i)9-s + 3·11-s + (−0.5 + 0.866i)13-s + (1.5 − 0.866i)15-s + (−1.5 + 2.59i)17-s + (2.5 + 4.33i)19-s + (3 − 3.46i)21-s − 23-s − 4·25-s − 5.19i·27-s + (−4.5 − 7.79i)29-s + (2 + 3.46i)31-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + 0.447·5-s + (0.944 − 0.327i)7-s + (0.5 − 0.866i)9-s + 0.904·11-s + (−0.138 + 0.240i)13-s + (0.387 − 0.223i)15-s + (−0.363 + 0.630i)17-s + (0.573 + 0.993i)19-s + (0.654 − 0.755i)21-s − 0.208·23-s − 0.800·25-s − 0.999i·27-s + (−0.835 − 1.44i)29-s + (0.359 + 0.622i)31-s + ⋯

Functional equation

Λ(s)=(1008s/2ΓC(s)L(s)=((0.823+0.566i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1008s/2ΓC(s+1/2)L(s)=((0.823+0.566i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10081008    =    243272^{4} \cdot 3^{2} \cdot 7
Sign: 0.823+0.566i0.823 + 0.566i
Analytic conductor: 8.048928.04892
Root analytic conductor: 2.837062.83706
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1008(193,)\chi_{1008} (193, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1008, ( :1/2), 0.823+0.566i)(2,\ 1008,\ (\ :1/2),\ 0.823 + 0.566i)

Particular Values

L(1)L(1) \approx 2.6618307262.661830726
L(12)L(\frac12) \approx 2.6618307262.661830726
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 1
3 1+(1.5+0.866i)T 1 + (-1.5 + 0.866i)T
7 1+(2.5+0.866i)T 1 + (-2.5 + 0.866i)T
good5 1T+5T2 1 - T + 5T^{2}
11 13T+11T2 1 - 3T + 11T^{2}
13 1+(0.50.866i)T+(6.511.2i)T2 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2}
17 1+(1.52.59i)T+(8.514.7i)T2 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2}
19 1+(2.54.33i)T+(9.5+16.4i)T2 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2}
23 1+T+23T2 1 + T + 23T^{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+(23.46i)T+(15.5+26.8i)T2 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2}
37 1+(2.5+4.33i)T+(18.5+32.0i)T2 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2}
41 1+(3.56.06i)T+(20.535.5i)T2 1 + (3.5 - 6.06i)T + (-20.5 - 35.5i)T^{2}
43 1+(1.52.59i)T+(21.5+37.2i)T2 1 + (-1.5 - 2.59i)T + (-21.5 + 37.2i)T^{2}
47 1+(4+6.92i)T+(23.540.7i)T2 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2}
53 1+(4.57.79i)T+(26.545.8i)T2 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{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+(11.73i)T+(30.552.8i)T2 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2}
67 1+(610.3i)T+(33.5+58.0i)T2 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1+(6.5+11.2i)T+(36.563.2i)T2 1 + (-6.5 + 11.2i)T + (-36.5 - 63.2i)T^{2}
79 1+(4+6.92i)T+(39.568.4i)T2 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2}
83 1+(6.5+11.2i)T+(41.5+71.8i)T2 1 + (6.5 + 11.2i)T + (-41.5 + 71.8i)T^{2}
89 1+(4.57.79i)T+(44.5+77.0i)T2 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2}
97 1+(8.514.7i)T+(48.5+84.0i)T2 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2}
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   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

−9.718064825195812181491392346196, −9.040151919805813430038464962107, −8.118943662606817609951073246647, −7.58504368879306099563759295735, −6.56404412218236368100298542352, −5.74309446452027759966945853436, −4.35161592462145940197429539428, −3.63520668692261658897688127217, −2.13329657963466697794068030030, −1.39928190328830203230809695440, 1.59898483877940763721174448712, 2.60927833244559101138015319216, 3.76790532616847037004619472395, 4.79243623790019665119544464351, 5.49046248812668576999836412958, 6.84199170077041713333442427253, 7.64715858611448764716170142053, 8.572924509691368573195944604944, 9.217845918084275416428514865098, 9.779234120622260103825673264826

Graph of the ZZ-function along the critical line