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Properties

Label 2-3840-1.1-c1-0-1
Degree 22
Conductor 38403840
Sign 11
Analytic cond. 30.662530.6625
Root an. cond. 5.537375.53737
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  − 3-s + 5-s − 4.68·7-s + 9-s − 2.29·11-s − 4.97·13-s − 15-s − 2.97·17-s + 2.68·19-s + 4.68·21-s − 2.68·23-s + 25-s − 27-s + 2·29-s − 6.97·31-s + 2.29·33-s − 4.68·35-s − 4.39·37-s + 4.97·39-s + 11.3·41-s − 9.37·43-s + 45-s + 7.27·47-s + 14.9·49-s + 2.97·51-s + 2·53-s − 2.29·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 1.77·7-s + 0.333·9-s − 0.691·11-s − 1.38·13-s − 0.258·15-s − 0.722·17-s + 0.616·19-s + 1.02·21-s − 0.560·23-s + 0.200·25-s − 0.192·27-s + 0.371·29-s − 1.25·31-s + 0.399·33-s − 0.792·35-s − 0.722·37-s + 0.797·39-s + 1.77·41-s − 1.42·43-s + 0.149·45-s + 1.06·47-s + 2.13·49-s + 0.417·51-s + 0.274·53-s − 0.309·55-s + ⋯

Functional equation

Λ(s)=(3840s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3840s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38403840    =    28⋅3⋅52^{8} \cdot 3 \cdot 5
Sign: 11
Analytic conductor: 30.662530.6625
Root analytic conductor: 5.537375.53737
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3840, ( :1/2), 1)(2,\ 3840,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.62226937560.6222693756
L(12)L(\frac12) ≈\approx 0.62226937560.6222693756
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+T 1 + T
5 1−T 1 - T
good7 1+4.68T+7T2 1 + 4.68T + 7T^{2}
11 1+2.29T+11T2 1 + 2.29T + 11T^{2}
13 1+4.97T+13T2 1 + 4.97T + 13T^{2}
17 1+2.97T+17T2 1 + 2.97T + 17T^{2}
19 1−2.68T+19T2 1 - 2.68T + 19T^{2}
23 1+2.68T+23T2 1 + 2.68T + 23T^{2}
29 1−2T+29T2 1 - 2T + 29T^{2}
31 1+6.97T+31T2 1 + 6.97T + 31T^{2}
37 1+4.39T+37T2 1 + 4.39T + 37T^{2}
41 1−11.3T+41T2 1 - 11.3T + 41T^{2}
43 1+9.37T+43T2 1 + 9.37T + 43T^{2}
47 1−7.27T+47T2 1 - 7.27T + 47T^{2}
53 1−2T+53T2 1 - 2T + 53T^{2}
59 1+1.70T+59T2 1 + 1.70T + 59T^{2}
61 1−4.58T+61T2 1 - 4.58T + 61T^{2}
67 1−4T+67T2 1 - 4T + 67T^{2}
71 1+0.585T+71T2 1 + 0.585T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1−1.02T+79T2 1 - 1.02T + 79T^{2}
83 1+13.3T+83T2 1 + 13.3T + 83T^{2}
89 1+3.37T+89T2 1 + 3.37T + 89T^{2}
97 1+3.95T+97T2 1 + 3.95T + 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

−8.630390342237459692406257615233, −7.38972203634515397626425043578, −7.06277198156207755689974452437, −6.20592562068817042967307285068, −5.60713997679657661864174451610, −4.87100626249852007035624277326, −3.83977398129389436157749585977, −2.89107152275067715985896925480, −2.14639701933083279396015740907, −0.43721770791480943591712467496, 0.43721770791480943591712467496, 2.14639701933083279396015740907, 2.89107152275067715985896925480, 3.83977398129389436157749585977, 4.87100626249852007035624277326, 5.60713997679657661864174451610, 6.20592562068817042967307285068, 7.06277198156207755689974452437, 7.38972203634515397626425043578, 8.630390342237459692406257615233

Graph of the ZZ-function along the critical line