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Properties

Label 2-760-5.4-c1-0-2
Degree 22
Conductor 760760
Sign −0.950+0.309i-0.950 + 0.309i
Analytic cond. 6.068636.06863
Root an. cond. 2.463452.46345
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  + 2.84i·3-s + (0.691 + 2.12i)5-s − 0.145i·7-s − 5.10·9-s − 5.71·11-s − 5.24i·13-s + (−6.05 + 1.96i)15-s + 7.15i·17-s − 19-s + 0.412·21-s + 0.622i·23-s + (−4.04 + 2.94i)25-s − 6.00i·27-s + 5.46·29-s − 3.77·31-s + ⋯
L(s)  = 1  + 1.64i·3-s + (0.309 + 0.950i)5-s − 0.0548i·7-s − 1.70·9-s − 1.72·11-s − 1.45i·13-s + (−1.56 + 0.508i)15-s + 1.73i·17-s − 0.229·19-s + 0.0901·21-s + 0.129i·23-s + (−0.808 + 0.588i)25-s − 1.15i·27-s + 1.01·29-s − 0.678·31-s + ⋯

Functional equation

Λ(s)=(760s/2 ΓC(s) L(s)=((−0.950+0.309i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(760s/2 ΓC(s+1/2) L(s)=((−0.950+0.309i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 760760    =    23⋅5⋅192^{3} \cdot 5 \cdot 19
Sign: −0.950+0.309i-0.950 + 0.309i
Analytic conductor: 6.068636.06863
Root analytic conductor: 2.463452.46345
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ760(609,⋅)\chi_{760} (609, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 760, ( :1/2), −0.950+0.309i)(2,\ 760,\ (\ :1/2),\ -0.950 + 0.309i)

Particular Values

L(1)L(1) ≈\approx 0.151603−0.956297i0.151603 - 0.956297i
L(12)L(\frac12) ≈\approx 0.151603−0.956297i0.151603 - 0.956297i
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+(−0.691−2.12i)T 1 + (-0.691 - 2.12i)T
19 1+T 1 + T
good3 1−2.84iT−3T2 1 - 2.84iT - 3T^{2}
7 1+0.145iT−7T2 1 + 0.145iT - 7T^{2}
11 1+5.71T+11T2 1 + 5.71T + 11T^{2}
13 1+5.24iT−13T2 1 + 5.24iT - 13T^{2}
17 1−7.15iT−17T2 1 - 7.15iT - 17T^{2}
23 1−0.622iT−23T2 1 - 0.622iT - 23T^{2}
29 1−5.46T+29T2 1 - 5.46T + 29T^{2}
31 1+3.77T+31T2 1 + 3.77T + 31T^{2}
37 1+5.03iT−37T2 1 + 5.03iT - 37T^{2}
41 1−5.77T+41T2 1 - 5.77T + 41T^{2}
43 1−3.32iT−43T2 1 - 3.32iT - 43T^{2}
47 1−5.85iT−47T2 1 - 5.85iT - 47T^{2}
53 1−6.97iT−53T2 1 - 6.97iT - 53T^{2}
59 1+9.09T+59T2 1 + 9.09T + 59T^{2}
61 1+8.16T+61T2 1 + 8.16T + 61T^{2}
67 1−13.6iT−67T2 1 - 13.6iT - 67T^{2}
71 1−2.41T+71T2 1 - 2.41T + 71T^{2}
73 1+7.44iT−73T2 1 + 7.44iT - 73T^{2}
79 1−9.69T+79T2 1 - 9.69T + 79T^{2}
83 1−2.17iT−83T2 1 - 2.17iT - 83T^{2}
89 1+3.90T+89T2 1 + 3.90T + 89T^{2}
97 1−4.98iT−97T2 1 - 4.98iT - 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

−10.67461728148270695704379857903, −10.30045173411237971477761480055, −9.372002458760344938961405143459, −8.259201495191166037945507385241, −7.59105270761020454155853895154, −5.98209622805055950594434696624, −5.53012861508510805039556109674, −4.39883544928620694971361244307, −3.34341541818690546905218718374, −2.59539537682821163918249438841, 0.46195769092392563376062838461, 1.87250313635537423160202761721, 2.70649220817531741865704564532, 4.66919489724046233047418091916, 5.44338573742752083447910001005, 6.50066745918766078389819877394, 7.30091413216339621862414989664, 8.029701919971746771911160665621, 8.835433214466052244505533613055, 9.664074768467531297959307941657

Graph of the ZZ-function along the critical line