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Properties

Label 2-1575-1.1-c3-0-133
Degree 22
Conductor 15751575
Sign −1-1
Analytic cond. 92.928092.9280
Root an. cond. 9.639919.63991
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  + 4.42·2-s + 11.5·4-s − 7·7-s + 15.7·8-s + 23.3·11-s − 3.56·13-s − 30.9·14-s − 22.7·16-s − 57.5·17-s − 51.1·19-s + 103.·22-s − 65.6·23-s − 15.7·26-s − 80.9·28-s − 41.6·29-s − 167.·31-s − 226.·32-s − 254.·34-s + 224.·37-s − 226.·38-s − 196.·41-s − 58.9·43-s + 270.·44-s − 290.·46-s − 41.9·47-s + 49·49-s − 41.2·52-s + ⋯
L(s)  = 1  + 1.56·2-s + 1.44·4-s − 0.377·7-s + 0.697·8-s + 0.640·11-s − 0.0761·13-s − 0.591·14-s − 0.354·16-s − 0.820·17-s − 0.617·19-s + 1.00·22-s − 0.595·23-s − 0.119·26-s − 0.546·28-s − 0.266·29-s − 0.970·31-s − 1.25·32-s − 1.28·34-s + 0.997·37-s − 0.965·38-s − 0.749·41-s − 0.209·43-s + 0.926·44-s − 0.930·46-s − 0.130·47-s + 0.142·49-s − 0.110·52-s + ⋯

Functional equation

Λ(s)=(1575s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1575s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15751575    =    32⋅52⋅73^{2} \cdot 5^{2} \cdot 7
Sign: −1-1
Analytic conductor: 92.928092.9280
Root analytic conductor: 9.639919.63991
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1575, ( :3/2), −1)(2,\ 1575,\ (\ :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)
bad3 1 1
5 1 1
7 1+7T 1 + 7T
good2 1−4.42T+8T2 1 - 4.42T + 8T^{2}
11 1−23.3T+1.33e3T2 1 - 23.3T + 1.33e3T^{2}
13 1+3.56T+2.19e3T2 1 + 3.56T + 2.19e3T^{2}
17 1+57.5T+4.91e3T2 1 + 57.5T + 4.91e3T^{2}
19 1+51.1T+6.85e3T2 1 + 51.1T + 6.85e3T^{2}
23 1+65.6T+1.21e4T2 1 + 65.6T + 1.21e4T^{2}
29 1+41.6T+2.43e4T2 1 + 41.6T + 2.43e4T^{2}
31 1+167.T+2.97e4T2 1 + 167.T + 2.97e4T^{2}
37 1−224.T+5.06e4T2 1 - 224.T + 5.06e4T^{2}
41 1+196.T+6.89e4T2 1 + 196.T + 6.89e4T^{2}
43 1+58.9T+7.95e4T2 1 + 58.9T + 7.95e4T^{2}
47 1+41.9T+1.03e5T2 1 + 41.9T + 1.03e5T^{2}
53 1−33.3T+1.48e5T2 1 - 33.3T + 1.48e5T^{2}
59 1−229.T+2.05e5T2 1 - 229.T + 2.05e5T^{2}
61 1+700.T+2.26e5T2 1 + 700.T + 2.26e5T^{2}
67 1−453.T+3.00e5T2 1 - 453.T + 3.00e5T^{2}
71 1+930.T+3.57e5T2 1 + 930.T + 3.57e5T^{2}
73 1+370.T+3.89e5T2 1 + 370.T + 3.89e5T^{2}
79 1+54.6T+4.93e5T2 1 + 54.6T + 4.93e5T^{2}
83 1−430.T+5.71e5T2 1 - 430.T + 5.71e5T^{2}
89 1−737.T+7.04e5T2 1 - 737.T + 7.04e5T^{2}
97 1+150.T+9.12e5T2 1 + 150.T + 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

−8.735356245539814843809548937832, −7.55973658675769768918584504454, −6.63696174865574883611972844360, −6.17110395030828208201658990292, −5.26852909626236097040674487360, −4.33825213797983399165174840191, −3.77225699248095646594731869118, −2.76650522359898395153222623748, −1.78746356226343367574356846662, 0, 1.78746356226343367574356846662, 2.76650522359898395153222623748, 3.77225699248095646594731869118, 4.33825213797983399165174840191, 5.26852909626236097040674487360, 6.17110395030828208201658990292, 6.63696174865574883611972844360, 7.55973658675769768918584504454, 8.735356245539814843809548937832

Graph of the ZZ-function along the critical line