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Properties

Label 2-1008-63.47-c1-0-6
Degree 22
Conductor 10081008
Sign 0.0614−0.998i0.0614 - 0.998i
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  + (−1.33 − 1.10i)3-s − 0.0676·5-s + (2.64 − 0.142i)7-s + (0.554 + 2.94i)9-s + 3.92i·11-s + (−3.32 + 1.92i)13-s + (0.0901 + 0.0747i)15-s + (−0.775 − 1.34i)17-s + (−5.06 − 2.92i)19-s + (−3.67 − 2.73i)21-s + 5.52i·23-s − 4.99·25-s + (2.52 − 4.54i)27-s + (1.20 + 0.697i)29-s + (−1.09 − 0.632i)31-s + ⋯
L(s)  = 1  + (−0.769 − 0.638i)3-s − 0.0302·5-s + (0.998 − 0.0538i)7-s + (0.184 + 0.982i)9-s + 1.18i·11-s + (−0.922 + 0.532i)13-s + (0.0232 + 0.0193i)15-s + (−0.188 − 0.325i)17-s + (−1.16 − 0.670i)19-s + (−0.802 − 0.596i)21-s + 1.15i·23-s − 0.999·25-s + (0.485 − 0.874i)27-s + (0.224 + 0.129i)29-s + (−0.196 − 0.113i)31-s + ⋯

Functional equation

Λ(s)=(1008s/2 ΓC(s) L(s)=((0.0614−0.998i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0614 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1008s/2 ΓC(s+1/2) L(s)=((0.0614−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0614 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10081008    =    24⋅32⋅72^{4} \cdot 3^{2} \cdot 7
Sign: 0.0614−0.998i0.0614 - 0.998i
Analytic conductor: 8.048928.04892
Root analytic conductor: 2.837062.83706
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1008(929,⋅)\chi_{1008} (929, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1008, ( :1/2), 0.0614−0.998i)(2,\ 1008,\ (\ :1/2),\ 0.0614 - 0.998i)

Particular Values

L(1)L(1) ≈\approx 0.79088556770.7908855677
L(12)L(\frac12) ≈\approx 0.79088556770.7908855677
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 1
3 1+(1.33+1.10i)T 1 + (1.33 + 1.10i)T
7 1+(−2.64+0.142i)T 1 + (-2.64 + 0.142i)T
good5 1+0.0676T+5T2 1 + 0.0676T + 5T^{2}
11 1−3.92iT−11T2 1 - 3.92iT - 11T^{2}
13 1+(3.32−1.92i)T+(6.5−11.2i)T2 1 + (3.32 - 1.92i)T + (6.5 - 11.2i)T^{2}
17 1+(0.775+1.34i)T+(−8.5+14.7i)T2 1 + (0.775 + 1.34i)T + (-8.5 + 14.7i)T^{2}
19 1+(5.06+2.92i)T+(9.5+16.4i)T2 1 + (5.06 + 2.92i)T + (9.5 + 16.4i)T^{2}
23 1−5.52iT−23T2 1 - 5.52iT - 23T^{2}
29 1+(−1.20−0.697i)T+(14.5+25.1i)T2 1 + (-1.20 - 0.697i)T + (14.5 + 25.1i)T^{2}
31 1+(1.09+0.632i)T+(15.5+26.8i)T2 1 + (1.09 + 0.632i)T + (15.5 + 26.8i)T^{2}
37 1+(4.35−7.54i)T+(−18.5−32.0i)T2 1 + (4.35 - 7.54i)T + (-18.5 - 32.0i)T^{2}
41 1+(−5.17−8.96i)T+(−20.5+35.5i)T2 1 + (-5.17 - 8.96i)T + (-20.5 + 35.5i)T^{2}
43 1+(0.735−1.27i)T+(−21.5−37.2i)T2 1 + (0.735 - 1.27i)T + (-21.5 - 37.2i)T^{2}
47 1+(−1.77−3.06i)T+(−23.5+40.7i)T2 1 + (-1.77 - 3.06i)T + (-23.5 + 40.7i)T^{2}
53 1+(−6.28+3.63i)T+(26.5−45.8i)T2 1 + (-6.28 + 3.63i)T + (26.5 - 45.8i)T^{2}
59 1+(4.70−8.14i)T+(−29.5−51.0i)T2 1 + (4.70 - 8.14i)T + (-29.5 - 51.0i)T^{2}
61 1+(−0.0705+0.0407i)T+(30.5−52.8i)T2 1 + (-0.0705 + 0.0407i)T + (30.5 - 52.8i)T^{2}
67 1+(7.67−13.2i)T+(−33.5−58.0i)T2 1 + (7.67 - 13.2i)T + (-33.5 - 58.0i)T^{2}
71 1+4.30iT−71T2 1 + 4.30iT - 71T^{2}
73 1+(−6.12+3.53i)T+(36.5−63.2i)T2 1 + (-6.12 + 3.53i)T + (36.5 - 63.2i)T^{2}
79 1+(3.42+5.92i)T+(−39.5+68.4i)T2 1 + (3.42 + 5.92i)T + (-39.5 + 68.4i)T^{2}
83 1+(−3.93+6.81i)T+(−41.5−71.8i)T2 1 + (-3.93 + 6.81i)T + (-41.5 - 71.8i)T^{2}
89 1+(5.84−10.1i)T+(−44.5−77.0i)T2 1 + (5.84 - 10.1i)T + (-44.5 - 77.0i)T^{2}
97 1+(−0.363−0.209i)T+(48.5+84.0i)T2 1 + (-0.363 - 0.209i)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

−10.24194658091037190022032312139, −9.443899279440264791916002384051, −8.305558859744558111004344548689, −7.43183454627138959873682529935, −6.99156605344460878072391365743, −5.89336936896134596964502783824, −4.81777433474584738018151555459, −4.42398981646540851942794403915, −2.39704831178248623151733206339, −1.54940154682292819554105158155, 0.39351911477433075571870285987, 2.17612897720076527735205360514, 3.70468273803819480515008981185, 4.50739250855118497925897005975, 5.50198900261242795083476725287, 6.05166215988889410896953622494, 7.22129852684413699663274403208, 8.270552847157342723264931798364, 8.854611915544535277920262958475, 9.985926671074871150222395868595

Graph of the ZZ-function along the critical line