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Properties

Label 2-5712-1.1-c1-0-24
Degree 22
Conductor 57125712
Sign 11
Analytic cond. 45.610545.6105
Root an. cond. 6.753556.75355
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 − 3.73·5-s + 7-s + 9-s + 5·11-s + 3.19·13-s + 3.73·15-s + 17-s + 7.19·19-s − 21-s + 3·23-s + 8.92·25-s − 27-s + 4·29-s + 4.73·31-s − 5·33-s − 3.73·35-s + 4.73·37-s − 3.19·39-s − 11.7·41-s − 9.39·43-s − 3.73·45-s − 4.73·47-s + 49-s − 51-s − 5.66·53-s − 18.6·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.66·5-s + 0.377·7-s + 0.333·9-s + 1.50·11-s + 0.886·13-s + 0.963·15-s + 0.242·17-s + 1.65·19-s − 0.218·21-s + 0.625·23-s + 1.78·25-s − 0.192·27-s + 0.742·29-s + 0.849·31-s − 0.870·33-s − 0.630·35-s + 0.777·37-s − 0.511·39-s − 1.83·41-s − 1.43·43-s − 0.556·45-s − 0.690·47-s + 0.142·49-s − 0.140·51-s − 0.777·53-s − 2.51·55-s + ⋯

Functional equation

Λ(s)=(5712s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5712s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57125712    =    24⋅3⋅7⋅172^{4} \cdot 3 \cdot 7 \cdot 17
Sign: 11
Analytic conductor: 45.610545.6105
Root analytic conductor: 6.753556.75355
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5712, ( :1/2), 1)(2,\ 5712,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.5317740771.531774077
L(12)L(\frac12) ≈\approx 1.5317740771.531774077
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
7 1−T 1 - T
17 1−T 1 - T
good5 1+3.73T+5T2 1 + 3.73T + 5T^{2}
11 1−5T+11T2 1 - 5T + 11T^{2}
13 1−3.19T+13T2 1 - 3.19T + 13T^{2}
19 1−7.19T+19T2 1 - 7.19T + 19T^{2}
23 1−3T+23T2 1 - 3T + 23T^{2}
29 1−4T+29T2 1 - 4T + 29T^{2}
31 1−4.73T+31T2 1 - 4.73T + 31T^{2}
37 1−4.73T+37T2 1 - 4.73T + 37T^{2}
41 1+11.7T+41T2 1 + 11.7T + 41T^{2}
43 1+9.39T+43T2 1 + 9.39T + 43T^{2}
47 1+4.73T+47T2 1 + 4.73T + 47T^{2}
53 1+5.66T+53T2 1 + 5.66T + 53T^{2}
59 1+2.19T+59T2 1 + 2.19T + 59T^{2}
61 1+6.73T+61T2 1 + 6.73T + 61T^{2}
67 1−11.4T+67T2 1 - 11.4T + 67T^{2}
71 1−3.46T+71T2 1 - 3.46T + 71T^{2}
73 1+3.46T+73T2 1 + 3.46T + 73T^{2}
79 1−0.196T+79T2 1 - 0.196T + 79T^{2}
83 1−12.9T+83T2 1 - 12.9T + 83T^{2}
89 1+12.3T+89T2 1 + 12.3T + 89T^{2}
97 1−13.8T+97T2 1 - 13.8T + 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

−8.166454627089986504467038836104, −7.36294260169292270040878392827, −6.76022319256883022210803061326, −6.14207935237932834531708920944, −4.97041840851185352614315505011, −4.57745338116478962608129149047, −3.52901142143481329884445434404, −3.31857890846164543260823749352, −1.43800941721517619041558483966, −0.77289563369327962261296290212, 0.77289563369327962261296290212, 1.43800941721517619041558483966, 3.31857890846164543260823749352, 3.52901142143481329884445434404, 4.57745338116478962608129149047, 4.97041840851185352614315505011, 6.14207935237932834531708920944, 6.76022319256883022210803061326, 7.36294260169292270040878392827, 8.166454627089986504467038836104

Graph of the ZZ-function along the critical line