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Properties

Label 2-1584-1.1-c3-0-47
Degree 22
Conductor 15841584
Sign −1-1
Analytic cond. 93.459093.4590
Root an. cond. 9.667429.66742
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 17.1·5-s + 23.1·7-s − 11·11-s − 59.6·13-s + 80.8·17-s + 11.7·19-s − 56.0·23-s + 168.·25-s + 85.4·29-s + 99.8·31-s − 396.·35-s + 402.·37-s − 27.0·41-s + 74·43-s − 408.·47-s + 192.·49-s − 463.·53-s + 188.·55-s − 498.·59-s − 635.·61-s + 1.02e3·65-s + 701.·67-s + 27.7·71-s + 619.·73-s − 254.·77-s + 208.·79-s − 1.30e3·83-s + ⋯
L(s)  = 1  − 1.53·5-s + 1.24·7-s − 0.301·11-s − 1.27·13-s + 1.15·17-s + 0.141·19-s − 0.508·23-s + 1.34·25-s + 0.547·29-s + 0.578·31-s − 1.91·35-s + 1.79·37-s − 0.103·41-s + 0.262·43-s − 1.26·47-s + 0.560·49-s − 1.20·53-s + 0.462·55-s − 1.10·59-s − 1.33·61-s + 1.95·65-s + 1.27·67-s + 0.0463·71-s + 0.993·73-s − 0.376·77-s + 0.296·79-s − 1.72·83-s + ⋯

Functional equation

Λ(s)=(1584s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1584s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15841584    =    24⋅32⋅112^{4} \cdot 3^{2} \cdot 11
Sign: −1-1
Analytic conductor: 93.459093.4590
Root analytic conductor: 9.667429.66742
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1584, ( :3/2), −1)(2,\ 1584,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
11 1+11T 1 + 11T
good5 1+17.1T+125T2 1 + 17.1T + 125T^{2}
7 1−23.1T+343T2 1 - 23.1T + 343T^{2}
13 1+59.6T+2.19e3T2 1 + 59.6T + 2.19e3T^{2}
17 1−80.8T+4.91e3T2 1 - 80.8T + 4.91e3T^{2}
19 1−11.7T+6.85e3T2 1 - 11.7T + 6.85e3T^{2}
23 1+56.0T+1.21e4T2 1 + 56.0T + 1.21e4T^{2}
29 1−85.4T+2.43e4T2 1 - 85.4T + 2.43e4T^{2}
31 1−99.8T+2.97e4T2 1 - 99.8T + 2.97e4T^{2}
37 1−402.T+5.06e4T2 1 - 402.T + 5.06e4T^{2}
41 1+27.0T+6.89e4T2 1 + 27.0T + 6.89e4T^{2}
43 1−74T+7.95e4T2 1 - 74T + 7.95e4T^{2}
47 1+408.T+1.03e5T2 1 + 408.T + 1.03e5T^{2}
53 1+463.T+1.48e5T2 1 + 463.T + 1.48e5T^{2}
59 1+498.T+2.05e5T2 1 + 498.T + 2.05e5T^{2}
61 1+635.T+2.26e5T2 1 + 635.T + 2.26e5T^{2}
67 1−701.T+3.00e5T2 1 - 701.T + 3.00e5T^{2}
71 1−27.7T+3.57e5T2 1 - 27.7T + 3.57e5T^{2}
73 1−619.T+3.89e5T2 1 - 619.T + 3.89e5T^{2}
79 1−208.T+4.93e5T2 1 - 208.T + 4.93e5T^{2}
83 1+1.30e3T+5.71e5T2 1 + 1.30e3T + 5.71e5T^{2}
89 1−1.39e3T+7.04e5T2 1 - 1.39e3T + 7.04e5T^{2}
97 1−1.05e3T+9.12e5T2 1 - 1.05e3T + 9.12e5T^{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

−8.278595329809874528341132007612, −7.78244830374909943685011737400, −7.53580027999244594433788440635, −6.26277538124417949784904758899, −4.94827193607931544864424037013, −4.65395880685323897777534853217, −3.57516889948019528588592057699, −2.55890113389633905797180988300, −1.17760953987608182624534403609, 0, 1.17760953987608182624534403609, 2.55890113389633905797180988300, 3.57516889948019528588592057699, 4.65395880685323897777534853217, 4.94827193607931544864424037013, 6.26277538124417949784904758899, 7.53580027999244594433788440635, 7.78244830374909943685011737400, 8.278595329809874528341132007612

Graph of the ZZ-function along the critical line