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Properties

Label 2-920-1.1-c1-0-3
Degree 22
Conductor 920920
Sign 11
Analytic cond. 7.346237.34623
Root an. cond. 2.710392.71039
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.56·3-s + 5-s − 3.12·7-s − 0.561·9-s − 4·11-s + 3.56·13-s − 1.56·15-s + 5.12·17-s + 4·19-s + 4.87·21-s + 23-s + 25-s + 5.56·27-s − 4.43·29-s + 5.56·31-s + 6.24·33-s − 3.12·35-s + 1.12·37-s − 5.56·39-s − 3.56·41-s − 0.876·43-s − 0.561·45-s + 8.68·47-s + 2.75·49-s − 8·51-s + 12.2·53-s − 4·55-s + ⋯
L(s)  = 1  − 0.901·3-s + 0.447·5-s − 1.18·7-s − 0.187·9-s − 1.20·11-s + 0.987·13-s − 0.403·15-s + 1.24·17-s + 0.917·19-s + 1.06·21-s + 0.208·23-s + 0.200·25-s + 1.07·27-s − 0.824·29-s + 0.998·31-s + 1.08·33-s − 0.527·35-s + 0.184·37-s − 0.890·39-s − 0.556·41-s − 0.133·43-s − 0.0837·45-s + 1.26·47-s + 0.393·49-s − 1.12·51-s + 1.68·53-s − 0.539·55-s + ⋯

Functional equation

Λ(s)=(920s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(920s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 920920    =    23⋅5⋅232^{3} \cdot 5 \cdot 23
Sign: 11
Analytic conductor: 7.346237.34623
Root analytic conductor: 2.710392.71039
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 920, ( :1/2), 1)(2,\ 920,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.96800079200.9680007920
L(12)L(\frac12) ≈\approx 0.96800079200.9680007920
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−T 1 - T
23 1−T 1 - T
good3 1+1.56T+3T2 1 + 1.56T + 3T^{2}
7 1+3.12T+7T2 1 + 3.12T + 7T^{2}
11 1+4T+11T2 1 + 4T + 11T^{2}
13 1−3.56T+13T2 1 - 3.56T + 13T^{2}
17 1−5.12T+17T2 1 - 5.12T + 17T^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
29 1+4.43T+29T2 1 + 4.43T + 29T^{2}
31 1−5.56T+31T2 1 - 5.56T + 31T^{2}
37 1−1.12T+37T2 1 - 1.12T + 37T^{2}
41 1+3.56T+41T2 1 + 3.56T + 41T^{2}
43 1+0.876T+43T2 1 + 0.876T + 43T^{2}
47 1−8.68T+47T2 1 - 8.68T + 47T^{2}
53 1−12.2T+53T2 1 - 12.2T + 53T^{2}
59 1−10.2T+59T2 1 - 10.2T + 59T^{2}
61 1−2.87T+61T2 1 - 2.87T + 61T^{2}
67 1+10.2T+67T2 1 + 10.2T + 67T^{2}
71 1+8.68T+71T2 1 + 8.68T + 71T^{2}
73 1−12.4T+73T2 1 - 12.4T + 73T^{2}
79 1−6.24T+79T2 1 - 6.24T + 79T^{2}
83 1−12T+83T2 1 - 12T + 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1−0.246T+97T2 1 - 0.246T + 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

−10.22479266136955488063613525487, −9.445456308088901965682713131926, −8.434560251187830677688666690251, −7.41832271260702522578550728966, −6.42466880405790236018053499478, −5.69534474055076709306143378642, −5.21122514948191233506734493260, −3.60696458038525539065121510260, −2.70859664445996752798783022122, −0.808908830703287407846678614885, 0.808908830703287407846678614885, 2.70859664445996752798783022122, 3.60696458038525539065121510260, 5.21122514948191233506734493260, 5.69534474055076709306143378642, 6.42466880405790236018053499478, 7.41832271260702522578550728966, 8.434560251187830677688666690251, 9.445456308088901965682713131926, 10.22479266136955488063613525487

Graph of the ZZ-function along the critical line