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Properties

Label 2-1800-1.1-c3-0-23
Degree 22
Conductor 18001800
Sign 11
Analytic cond. 106.203106.203
Root an. cond. 10.305510.3055
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 35.0·7-s − 25.6·11-s − 37.6·13-s − 95.7·17-s + 50.8·19-s + 110.·23-s + 54.5·29-s + 198.·31-s − 266.·37-s − 103.·41-s − 108·43-s + 597.·47-s + 887.·49-s − 305.·53-s + 223.·59-s + 485.·61-s + 876.·67-s − 585.·71-s + 1.13e3·73-s − 899.·77-s + 685.·79-s + 305.·83-s − 887.·89-s − 1.32e3·91-s + 556.·97-s − 1.59e3·101-s − 1.35e3·103-s + ⋯
L(s)  = 1  + 1.89·7-s − 0.702·11-s − 0.803·13-s − 1.36·17-s + 0.614·19-s + 1.00·23-s + 0.349·29-s + 1.14·31-s − 1.18·37-s − 0.395·41-s − 0.383·43-s + 1.85·47-s + 2.58·49-s − 0.792·53-s + 0.493·59-s + 1.01·61-s + 1.59·67-s − 0.978·71-s + 1.82·73-s − 1.33·77-s + 0.975·79-s + 0.404·83-s − 1.05·89-s − 1.52·91-s + 0.582·97-s − 1.56·101-s − 1.29·103-s + ⋯

Functional equation

Λ(s)=(1800s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1800s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18001800    =    23⋅32⋅522^{3} \cdot 3^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 106.203106.203
Root analytic conductor: 10.305510.3055
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1800, ( :3/2), 1)(2,\ 1800,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.5702752072.570275207
L(12)L(\frac12) ≈\approx 2.5702752072.570275207
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
5 1 1
good7 1−35.0T+343T2 1 - 35.0T + 343T^{2}
11 1+25.6T+1.33e3T2 1 + 25.6T + 1.33e3T^{2}
13 1+37.6T+2.19e3T2 1 + 37.6T + 2.19e3T^{2}
17 1+95.7T+4.91e3T2 1 + 95.7T + 4.91e3T^{2}
19 1−50.8T+6.85e3T2 1 - 50.8T + 6.85e3T^{2}
23 1−110.T+1.21e4T2 1 - 110.T + 1.21e4T^{2}
29 1−54.5T+2.43e4T2 1 - 54.5T + 2.43e4T^{2}
31 1−198.T+2.97e4T2 1 - 198.T + 2.97e4T^{2}
37 1+266.T+5.06e4T2 1 + 266.T + 5.06e4T^{2}
41 1+103.T+6.89e4T2 1 + 103.T + 6.89e4T^{2}
43 1+108T+7.95e4T2 1 + 108T + 7.95e4T^{2}
47 1−597.T+1.03e5T2 1 - 597.T + 1.03e5T^{2}
53 1+305.T+1.48e5T2 1 + 305.T + 1.48e5T^{2}
59 1−223.T+2.05e5T2 1 - 223.T + 2.05e5T^{2}
61 1−485.T+2.26e5T2 1 - 485.T + 2.26e5T^{2}
67 1−876.T+3.00e5T2 1 - 876.T + 3.00e5T^{2}
71 1+585.T+3.57e5T2 1 + 585.T + 3.57e5T^{2}
73 1−1.13e3T+3.89e5T2 1 - 1.13e3T + 3.89e5T^{2}
79 1−685.T+4.93e5T2 1 - 685.T + 4.93e5T^{2}
83 1−305.T+5.71e5T2 1 - 305.T + 5.71e5T^{2}
89 1+887.T+7.04e5T2 1 + 887.T + 7.04e5T^{2}
97 1−556.T+9.12e5T2 1 - 556.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

−8.688414636296261576290979469402, −8.209161153596871372877784083059, −7.39748835658298214572321304864, −6.71705555981179115729392254027, −5.27171269892024354760214251897, −5.01244367438998317976151494092, −4.14631113891326389578988216549, −2.69835740062118030375420233112, −1.94006380646689545956139097520, −0.76345755174280897898008155279, 0.76345755174280897898008155279, 1.94006380646689545956139097520, 2.69835740062118030375420233112, 4.14631113891326389578988216549, 5.01244367438998317976151494092, 5.27171269892024354760214251897, 6.71705555981179115729392254027, 7.39748835658298214572321304864, 8.209161153596871372877784083059, 8.688414636296261576290979469402

Graph of the ZZ-function along the critical line