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Properties

Label 2-520-65.64-c1-0-13
Degree 22
Conductor 520520
Sign 0.761+0.647i0.761 + 0.647i
Analytic cond. 4.152224.15222
Root an. cond. 2.037692.03769
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  − 0.773i·3-s + (1.98 − 1.02i)5-s + 1.12·7-s + 2.40·9-s + 2.52i·11-s + (−1.37 − 3.33i)13-s + (−0.789 − 1.53i)15-s − 0.870i·17-s + 6.27i·19-s − 0.870i·21-s − 2.81i·23-s + (2.91 − 4.06i)25-s − 4.17i·27-s − 0.176·29-s − 0.388i·31-s + ⋯
L(s)  = 1  − 0.446i·3-s + (0.889 − 0.456i)5-s + 0.425·7-s + 0.800·9-s + 0.760i·11-s + (−0.381 − 0.924i)13-s + (−0.203 − 0.397i)15-s − 0.211i·17-s + 1.43i·19-s − 0.189i·21-s − 0.586i·23-s + (0.583 − 0.812i)25-s − 0.803i·27-s − 0.0328·29-s − 0.0698i·31-s + ⋯

Functional equation

Λ(s)=(520s/2 ΓC(s) L(s)=((0.761+0.647i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(520s/2 ΓC(s+1/2) L(s)=((0.761+0.647i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 520520    =    23⋅5⋅132^{3} \cdot 5 \cdot 13
Sign: 0.761+0.647i0.761 + 0.647i
Analytic conductor: 4.152224.15222
Root analytic conductor: 2.037692.03769
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ520(129,⋅)\chi_{520} (129, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 520, ( :1/2), 0.761+0.647i)(2,\ 520,\ (\ :1/2),\ 0.761 + 0.647i)

Particular Values

L(1)L(1) ≈\approx 1.69277−0.622608i1.69277 - 0.622608i
L(12)L(\frac12) ≈\approx 1.69277−0.622608i1.69277 - 0.622608i
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
5 1+(−1.98+1.02i)T 1 + (-1.98 + 1.02i)T
13 1+(1.37+3.33i)T 1 + (1.37 + 3.33i)T
good3 1+0.773iT−3T2 1 + 0.773iT - 3T^{2}
7 1−1.12T+7T2 1 - 1.12T + 7T^{2}
11 1−2.52iT−11T2 1 - 2.52iT - 11T^{2}
17 1+0.870iT−17T2 1 + 0.870iT - 17T^{2}
19 1−6.27iT−19T2 1 - 6.27iT - 19T^{2}
23 1+2.81iT−23T2 1 + 2.81iT - 23T^{2}
29 1+0.176T+29T2 1 + 0.176T + 29T^{2}
31 1+0.388iT−31T2 1 + 0.388iT - 31T^{2}
37 1−3.12T+37T2 1 - 3.12T + 37T^{2}
41 1+7.25iT−41T2 1 + 7.25iT - 41T^{2}
43 1+4.52iT−43T2 1 + 4.52iT - 43T^{2}
47 1+1.12T+47T2 1 + 1.12T + 47T^{2}
53 1−9.83iT−53T2 1 - 9.83iT - 53T^{2}
59 1−12.1iT−59T2 1 - 12.1iT - 59T^{2}
61 1−10.3T+61T2 1 - 10.3T + 61T^{2}
67 1+6.70T+67T2 1 + 6.70T + 67T^{2}
71 1+4.37iT−71T2 1 + 4.37iT - 71T^{2}
73 1+4.70T+73T2 1 + 4.70T + 73T^{2}
79 1+8.31T+79T2 1 + 8.31T + 79T^{2}
83 1+9.15T+83T2 1 + 9.15T + 83T^{2}
89 1−6.15iT−89T2 1 - 6.15iT - 89T^{2}
97 1−1.55T+97T2 1 - 1.55T + 97T^{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

−10.37715043579732770046743051276, −10.11432955624298570368964408280, −9.034724274563823483094871465044, −7.976587831811759546212910733595, −7.23699199403839414579213658976, −6.11020233211826157792150330055, −5.19531851779143253139319062839, −4.17772850600034976520188607975, −2.39996787610079345999889883147, −1.32088740219816192054630510980, 1.65363070470037908090416834079, 3.01282771885362301650031284521, 4.38091122251793873821007132733, 5.26520391729427714890506664225, 6.45668881895852296288889660693, 7.17155456288815571989411243659, 8.441214950151437721211002035718, 9.484137513237650571285687312137, 9.895944431486315547510046979384, 11.09203994583435442109110931748

Graph of the ZZ-function along the critical line