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Properties

Label 2-5712-1.1-c1-0-1
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 − 2.16·5-s − 7-s + 9-s − 3·11-s − 0.162·13-s + 2.16·15-s − 17-s + 1.83·19-s + 21-s − 7.32·23-s − 0.324·25-s − 27-s + 10.3·29-s − 1.16·31-s + 3·33-s + 2.16·35-s − 9.16·37-s + 0.162·39-s − 3.83·41-s − 9.32·43-s − 2.16·45-s − 5.16·47-s + 49-s + 51-s + 9.48·53-s + 6.48·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.966·5-s − 0.377·7-s + 0.333·9-s − 0.904·11-s − 0.0450·13-s + 0.558·15-s − 0.242·17-s + 0.421·19-s + 0.218·21-s − 1.52·23-s − 0.0649·25-s − 0.192·27-s + 1.91·29-s − 0.208·31-s + 0.522·33-s + 0.365·35-s − 1.50·37-s + 0.0259·39-s − 0.599·41-s − 1.42·43-s − 0.322·45-s − 0.752·47-s + 0.142·49-s + 0.140·51-s + 1.30·53-s + 0.874·55-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 0.50929029090.5092902909
L(12)L(\frac12) ≈\approx 0.50929029090.5092902909
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+2.16T+5T2 1 + 2.16T + 5T^{2}
11 1+3T+11T2 1 + 3T + 11T^{2}
13 1+0.162T+13T2 1 + 0.162T + 13T^{2}
19 1−1.83T+19T2 1 - 1.83T + 19T^{2}
23 1+7.32T+23T2 1 + 7.32T + 23T^{2}
29 1−10.3T+29T2 1 - 10.3T + 29T^{2}
31 1+1.16T+31T2 1 + 1.16T + 31T^{2}
37 1+9.16T+37T2 1 + 9.16T + 37T^{2}
41 1+3.83T+41T2 1 + 3.83T + 41T^{2}
43 1+9.32T+43T2 1 + 9.32T + 43T^{2}
47 1+5.16T+47T2 1 + 5.16T + 47T^{2}
53 1−9.48T+53T2 1 - 9.48T + 53T^{2}
59 1+6.83T+59T2 1 + 6.83T + 59T^{2}
61 1+3.16T+61T2 1 + 3.16T + 61T^{2}
67 1−10T+67T2 1 - 10T + 67T^{2}
71 1+14.6T+71T2 1 + 14.6T + 71T^{2}
73 1−12.3T+73T2 1 - 12.3T + 73T^{2}
79 1+7.16T+79T2 1 + 7.16T + 79T^{2}
83 1+6T+83T2 1 + 6T + 83T^{2}
89 1+10.3T+89T2 1 + 10.3T + 89T^{2}
97 1−16.6T+97T2 1 - 16.6T + 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

−8.220094241865914411058141860379, −7.34744559564930105584374954016, −6.77212343936618090169346311650, −5.98871350518501796244378121884, −5.18938886017655228095240878419, −4.52788835494880625515537032491, −3.70110526043721509999793455388, −2.95321312026719671999130632615, −1.79962660521271483312640355686, −0.37762336384834728616715225670, 0.37762336384834728616715225670, 1.79962660521271483312640355686, 2.95321312026719671999130632615, 3.70110526043721509999793455388, 4.52788835494880625515537032491, 5.18938886017655228095240878419, 5.98871350518501796244378121884, 6.77212343936618090169346311650, 7.34744559564930105584374954016, 8.220094241865914411058141860379

Graph of the ZZ-function along the critical line