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Properties

Label 2-1800-1.1-c3-0-45
Degree 22
Conductor 18001800
Sign −1-1
Analytic cond. 106.203106.203
Root an. cond. 10.305510.3055
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 24·7-s + 44·11-s − 22·13-s + 50·17-s + 44·19-s − 56·23-s − 198·29-s − 160·31-s + 162·37-s + 198·41-s − 52·43-s + 528·47-s + 233·49-s − 242·53-s + 668·59-s + 550·61-s − 188·67-s − 728·71-s − 154·73-s − 1.05e3·77-s − 656·79-s + 236·83-s − 714·89-s + 528·91-s + 478·97-s − 1.56e3·101-s + 968·103-s + ⋯
L(s)  = 1  − 1.29·7-s + 1.20·11-s − 0.469·13-s + 0.713·17-s + 0.531·19-s − 0.507·23-s − 1.26·29-s − 0.926·31-s + 0.719·37-s + 0.754·41-s − 0.184·43-s + 1.63·47-s + 0.679·49-s − 0.627·53-s + 1.47·59-s + 1.15·61-s − 0.342·67-s − 1.21·71-s − 0.246·73-s − 1.56·77-s − 0.934·79-s + 0.312·83-s − 0.850·89-s + 0.608·91-s + 0.500·97-s − 1.54·101-s + 0.926·103-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: −1-1
Analytic conductor: 106.203106.203
Root analytic conductor: 10.305510.3055
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1800, ( :3/2), −1)(2,\ 1800,\ (\ :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
5 1 1
good7 1+24T+p3T2 1 + 24 T + p^{3} T^{2}
11 1−4pT+p3T2 1 - 4 p T + p^{3} T^{2}
13 1+22T+p3T2 1 + 22 T + p^{3} T^{2}
17 1−50T+p3T2 1 - 50 T + p^{3} T^{2}
19 1−44T+p3T2 1 - 44 T + p^{3} T^{2}
23 1+56T+p3T2 1 + 56 T + p^{3} T^{2}
29 1+198T+p3T2 1 + 198 T + p^{3} T^{2}
31 1+160T+p3T2 1 + 160 T + p^{3} T^{2}
37 1−162T+p3T2 1 - 162 T + p^{3} T^{2}
41 1−198T+p3T2 1 - 198 T + p^{3} T^{2}
43 1+52T+p3T2 1 + 52 T + p^{3} T^{2}
47 1−528T+p3T2 1 - 528 T + p^{3} T^{2}
53 1+242T+p3T2 1 + 242 T + p^{3} T^{2}
59 1−668T+p3T2 1 - 668 T + p^{3} T^{2}
61 1−550T+p3T2 1 - 550 T + p^{3} T^{2}
67 1+188T+p3T2 1 + 188 T + p^{3} T^{2}
71 1+728T+p3T2 1 + 728 T + p^{3} T^{2}
73 1+154T+p3T2 1 + 154 T + p^{3} T^{2}
79 1+656T+p3T2 1 + 656 T + p^{3} T^{2}
83 1−236T+p3T2 1 - 236 T + p^{3} T^{2}
89 1+714T+p3T2 1 + 714 T + p^{3} T^{2}
97 1−478T+p3T2 1 - 478 T + p^{3} T^{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

−8.687147679266628718588888118938, −7.52897034336328746954900834937, −6.99768584104102333715941849518, −6.07246550045469698457847110707, −5.48923955664485230872755505642, −4.11217773970782304976909373890, −3.54470403181847861116617730865, −2.50824677694422533718193916738, −1.20808733640550797019789221265, 0, 1.20808733640550797019789221265, 2.50824677694422533718193916738, 3.54470403181847861116617730865, 4.11217773970782304976909373890, 5.48923955664485230872755505642, 6.07246550045469698457847110707, 6.99768584104102333715941849518, 7.52897034336328746954900834937, 8.687147679266628718588888118938

Graph of the ZZ-function along the critical line