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Properties

Label 2-650-65.29-c1-0-0
Degree 22
Conductor 650650
Sign −0.658+0.752i-0.658 + 0.752i
Analytic cond. 5.190275.19027
Root an. cond. 2.278212.27821
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 + (−1.99 + 1.15i)3-s + (0.499 − 0.866i)4-s + (1.15 − 1.99i)6-s + (0.603 + 0.348i)7-s + 0.999i·8-s + (1.15 − 1.99i)9-s + (0.348 + 0.603i)11-s + 2.30i·12-s + (−3.12 + 1.80i)13-s − 0.697·14-s + (−0.5 − 0.866i)16-s + (2.51 + 1.45i)17-s + 2.30i·18-s + (−0.197 + 0.341i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−1.15 + 0.664i)3-s + (0.249 − 0.433i)4-s + (0.470 − 0.814i)6-s + (0.228 + 0.131i)7-s + 0.353i·8-s + (0.383 − 0.664i)9-s + (0.105 + 0.182i)11-s + 0.664i·12-s + (−0.866 + 0.499i)13-s − 0.186·14-s + (−0.125 − 0.216i)16-s + (0.610 + 0.352i)17-s + 0.542i·18-s + (−0.0452 + 0.0783i)19-s + ⋯

Functional equation

Λ(s)=(650s/2 ΓC(s) L(s)=((−0.658+0.752i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(650s/2 ΓC(s+1/2) L(s)=((−0.658+0.752i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 650650    =    2⋅52⋅132 \cdot 5^{2} \cdot 13
Sign: −0.658+0.752i-0.658 + 0.752i
Analytic conductor: 5.190275.19027
Root analytic conductor: 2.278212.27821
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ650(549,⋅)\chi_{650} (549, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 650, ( :1/2), −0.658+0.752i)(2,\ 650,\ (\ :1/2),\ -0.658 + 0.752i)

Particular Values

L(1)L(1) ≈\approx 0.0465425−0.102596i0.0465425 - 0.102596i
L(12)L(\frac12) ≈\approx 0.0465425−0.102596i0.0465425 - 0.102596i
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
5 1 1
13 1+(3.12−1.80i)T 1 + (3.12 - 1.80i)T
good3 1+(1.99−1.15i)T+(1.5−2.59i)T2 1 + (1.99 - 1.15i)T + (1.5 - 2.59i)T^{2}
7 1+(−0.603−0.348i)T+(3.5+6.06i)T2 1 + (-0.603 - 0.348i)T + (3.5 + 6.06i)T^{2}
11 1+(−0.348−0.603i)T+(−5.5+9.52i)T2 1 + (-0.348 - 0.603i)T + (-5.5 + 9.52i)T^{2}
17 1+(−2.51−1.45i)T+(8.5+14.7i)T2 1 + (-2.51 - 1.45i)T + (8.5 + 14.7i)T^{2}
19 1+(0.197−0.341i)T+(−9.5−16.4i)T2 1 + (0.197 - 0.341i)T + (-9.5 - 16.4i)T^{2}
23 1+(4.85−2.80i)T+(11.5−19.9i)T2 1 + (4.85 - 2.80i)T + (11.5 - 19.9i)T^{2}
29 1+(−1.65−2.86i)T+(−14.5+25.1i)T2 1 + (-1.65 - 2.86i)T + (-14.5 + 25.1i)T^{2}
31 1+8.60T+31T2 1 + 8.60T + 31T^{2}
37 1+(−3.20+1.84i)T+(18.5−32.0i)T2 1 + (-3.20 + 1.84i)T + (18.5 - 32.0i)T^{2}
41 1+(0.5+0.866i)T+(−20.5+35.5i)T2 1 + (0.5 + 0.866i)T + (-20.5 + 35.5i)T^{2}
43 1+(7.37+4.25i)T+(21.5+37.2i)T2 1 + (7.37 + 4.25i)T + (21.5 + 37.2i)T^{2}
47 1+10.3iT−47T2 1 + 10.3iT - 47T^{2}
53 1+12.2iT−53T2 1 + 12.2iT - 53T^{2}
59 1+(5.10−8.84i)T+(−29.5−51.0i)T2 1 + (5.10 - 8.84i)T + (-29.5 - 51.0i)T^{2}
61 1+(2.10−3.64i)T+(−30.5−52.8i)T2 1 + (2.10 - 3.64i)T + (-30.5 - 52.8i)T^{2}
67 1+(4.85−2.80i)T+(33.5−58.0i)T2 1 + (4.85 - 2.80i)T + (33.5 - 58.0i)T^{2}
71 1+(−5.30+9.18i)T+(−35.5−61.4i)T2 1 + (-5.30 + 9.18i)T + (-35.5 - 61.4i)T^{2}
73 1+8iT−73T2 1 + 8iT - 73T^{2}
79 1+9.30T+79T2 1 + 9.30T + 79T^{2}
83 1−16.8iT−83T2 1 - 16.8iT - 83T^{2}
89 1+(−0.0458−0.0793i)T+(−44.5+77.0i)T2 1 + (-0.0458 - 0.0793i)T + (-44.5 + 77.0i)T^{2}
97 1+(10.1+5.84i)T+(48.5+84.0i)T2 1 + (10.1 + 5.84i)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

−10.94846529698751260310507926371, −10.15395366253292508290144938254, −9.628069624788541997730972762162, −8.559032210168798517841525244070, −7.53296396892150474325904328277, −6.63571245788178473617157325143, −5.60124771807485113457977400203, −5.03394124440056408489309937750, −3.82551392903124168356635352041, −1.89695133153836001977767962087, 0.087509765701600440654719947573, 1.45082208913571055841775943188, 2.93159717839683273713726441280, 4.51031613078481387862183314941, 5.62945784372776124684314319931, 6.44676966278357932119792870065, 7.45431086020353706924004101354, 8.037915678333098284255392065102, 9.323864449564156019801611873778, 10.12116357395872535396397676885

Graph of the ZZ-function along the critical line