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Properties

Label 2-600-1.1-c3-0-27
Degree 22
Conductor 600600
Sign −1-1
Analytic cond. 35.401135.4011
Root an. cond. 5.949885.94988
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  + 3·3-s + 10·7-s + 9·9-s − 14·11-s − 82·13-s + 18·17-s − 136·19-s + 30·21-s − 140·23-s + 27·27-s + 112·29-s + 72·31-s − 42·33-s + 26·37-s − 246·39-s − 446·41-s + 396·43-s − 144·47-s − 243·49-s + 54·51-s + 158·53-s − 408·57-s − 342·59-s + 314·61-s + 90·63-s − 152·67-s − 420·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.539·7-s + 1/3·9-s − 0.383·11-s − 1.74·13-s + 0.256·17-s − 1.64·19-s + 0.311·21-s − 1.26·23-s + 0.192·27-s + 0.717·29-s + 0.417·31-s − 0.221·33-s + 0.115·37-s − 1.01·39-s − 1.69·41-s + 1.40·43-s − 0.446·47-s − 0.708·49-s + 0.148·51-s + 0.409·53-s − 0.948·57-s − 0.754·59-s + 0.659·61-s + 0.179·63-s − 0.277·67-s − 0.732·69-s + ⋯

Functional equation

Λ(s)=(600s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(600s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 600600    =    23⋅3⋅522^{3} \cdot 3 \cdot 5^{2}
Sign: −1-1
Analytic conductor: 35.401135.4011
Root analytic conductor: 5.949885.94988
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 600, ( :3/2), −1)(2,\ 600,\ (\ :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−pT 1 - p T
5 1 1
good7 1−10T+p3T2 1 - 10 T + p^{3} T^{2}
11 1+14T+p3T2 1 + 14 T + p^{3} T^{2}
13 1+82T+p3T2 1 + 82 T + p^{3} T^{2}
17 1−18T+p3T2 1 - 18 T + p^{3} T^{2}
19 1+136T+p3T2 1 + 136 T + p^{3} T^{2}
23 1+140T+p3T2 1 + 140 T + p^{3} T^{2}
29 1−112T+p3T2 1 - 112 T + p^{3} T^{2}
31 1−72T+p3T2 1 - 72 T + p^{3} T^{2}
37 1−26T+p3T2 1 - 26 T + p^{3} T^{2}
41 1+446T+p3T2 1 + 446 T + p^{3} T^{2}
43 1−396T+p3T2 1 - 396 T + p^{3} T^{2}
47 1+144T+p3T2 1 + 144 T + p^{3} T^{2}
53 1−158T+p3T2 1 - 158 T + p^{3} T^{2}
59 1+342T+p3T2 1 + 342 T + p^{3} T^{2}
61 1−314T+p3T2 1 - 314 T + p^{3} T^{2}
67 1+152T+p3T2 1 + 152 T + p^{3} T^{2}
71 1+932T+p3T2 1 + 932 T + p^{3} T^{2}
73 1+548T+p3T2 1 + 548 T + p^{3} T^{2}
79 1+512T+p3T2 1 + 512 T + p^{3} T^{2}
83 1−284T+p3T2 1 - 284 T + p^{3} T^{2}
89 1+810T+p3T2 1 + 810 T + p^{3} T^{2}
97 1−1304T+p3T2 1 - 1304 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

−9.996212525013104910012155093589, −8.827560756351837793419150428842, −8.065542766505303470110054412559, −7.34921525731902082441982806981, −6.25133811720376164162224463118, −4.97316070405265128123452208969, −4.22429663654816161189254778080, −2.76333385498009938861823142694, −1.88396817716477764064835813873, 0, 1.88396817716477764064835813873, 2.76333385498009938861823142694, 4.22429663654816161189254778080, 4.97316070405265128123452208969, 6.25133811720376164162224463118, 7.34921525731902082441982806981, 8.065542766505303470110054412559, 8.827560756351837793419150428842, 9.996212525013104910012155093589

Graph of the ZZ-function along the critical line