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Properties

Label 2-5760-1.1-c1-0-47
Degree 22
Conductor 57605760
Sign −1-1
Analytic cond. 45.993845.9938
Root an. cond. 6.781876.78187
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 3.23·7-s − 2·11-s + 4.47·13-s − 4.47·17-s + 4.47·19-s + 4.76·23-s + 25-s − 2·29-s + 6.47·31-s + 3.23·35-s − 6.94·37-s − 12.4·41-s + 7.70·43-s + 7.23·47-s + 3.47·49-s + 0.472·53-s + 2·55-s − 8.47·59-s + 6·61-s − 4.47·65-s − 7.70·67-s + 2.47·71-s + 4.47·73-s + 6.47·77-s + 12.9·79-s − 3.70·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.22·7-s − 0.603·11-s + 1.24·13-s − 1.08·17-s + 1.02·19-s + 0.993·23-s + 0.200·25-s − 0.371·29-s + 1.16·31-s + 0.546·35-s − 1.14·37-s − 1.94·41-s + 1.17·43-s + 1.05·47-s + 0.496·49-s + 0.0648·53-s + 0.269·55-s − 1.10·59-s + 0.768·61-s − 0.554·65-s − 0.941·67-s + 0.293·71-s + 0.523·73-s + 0.737·77-s + 1.45·79-s − 0.407·83-s + ⋯

Functional equation

Λ(s)=(5760s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(5760s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57605760    =    27⋅32⋅52^{7} \cdot 3^{2} \cdot 5
Sign: −1-1
Analytic conductor: 45.993845.9938
Root analytic conductor: 6.781876.78187
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 5760, ( :1/2), −1)(2,\ 5760,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1
5 1+T 1 + T
good7 1+3.23T+7T2 1 + 3.23T + 7T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
13 1−4.47T+13T2 1 - 4.47T + 13T^{2}
17 1+4.47T+17T2 1 + 4.47T + 17T^{2}
19 1−4.47T+19T2 1 - 4.47T + 19T^{2}
23 1−4.76T+23T2 1 - 4.76T + 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1−6.47T+31T2 1 - 6.47T + 31T^{2}
37 1+6.94T+37T2 1 + 6.94T + 37T^{2}
41 1+12.4T+41T2 1 + 12.4T + 41T^{2}
43 1−7.70T+43T2 1 - 7.70T + 43T^{2}
47 1−7.23T+47T2 1 - 7.23T + 47T^{2}
53 1−0.472T+53T2 1 - 0.472T + 53T^{2}
59 1+8.47T+59T2 1 + 8.47T + 59T^{2}
61 1−6T+61T2 1 - 6T + 61T^{2}
67 1+7.70T+67T2 1 + 7.70T + 67T^{2}
71 1−2.47T+71T2 1 - 2.47T + 71T^{2}
73 1−4.47T+73T2 1 - 4.47T + 73T^{2}
79 1−12.9T+79T2 1 - 12.9T + 79T^{2}
83 1+3.70T+83T2 1 + 3.70T + 83T^{2}
89 1−14.9T+89T2 1 - 14.9T + 89T^{2}
97 1+16.4T+97T2 1 + 16.4T + 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

−7.76296485347228859662553314243, −6.88353000676477203223288685244, −6.52868974567642153735063781421, −5.62893881024335508772399412649, −4.89077693448926163411348733024, −3.88404107900890110028005616200, −3.29523348158617410245571352354, −2.55744828075660061295288842500, −1.17557750453053293088730808952, 0, 1.17557750453053293088730808952, 2.55744828075660061295288842500, 3.29523348158617410245571352354, 3.88404107900890110028005616200, 4.89077693448926163411348733024, 5.62893881024335508772399412649, 6.52868974567642153735063781421, 6.88353000676477203223288685244, 7.76296485347228859662553314243

Graph of the ZZ-function along the critical line