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Properties

Label 2-600-1.1-c5-0-13
Degree 22
Conductor 600600
Sign 11
Analytic cond. 96.230296.2302
Root an. cond. 9.809709.80970
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·3-s + 160·7-s + 81·9-s − 596·11-s + 122·13-s + 1.07e3·17-s + 796·19-s − 1.44e3·21-s + 1.08e3·23-s − 729·27-s + 46·29-s − 4.95e3·31-s + 5.36e3·33-s + 6.11e3·37-s − 1.09e3·39-s − 6·41-s + 2.41e4·43-s − 1.34e4·47-s + 8.79e3·49-s − 9.70e3·51-s − 2.05e4·53-s − 7.16e3·57-s − 4.67e4·59-s − 9.60e3·61-s + 1.29e4·63-s + 1.74e4·67-s − 9.79e3·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.23·7-s + 1/3·9-s − 1.48·11-s + 0.200·13-s + 0.904·17-s + 0.505·19-s − 0.712·21-s + 0.428·23-s − 0.192·27-s + 0.0101·29-s − 0.925·31-s + 0.857·33-s + 0.734·37-s − 0.115·39-s − 0.000557·41-s + 1.98·43-s − 0.890·47-s + 0.523·49-s − 0.522·51-s − 1.00·53-s − 0.292·57-s − 1.74·59-s − 0.330·61-s + 0.411·63-s + 0.473·67-s − 0.247·69-s + ⋯

Functional equation

Λ(s)=(600s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(600s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 600600    =    23⋅3⋅522^{3} \cdot 3 \cdot 5^{2}
Sign: 11
Analytic conductor: 96.230296.2302
Root analytic conductor: 9.809709.80970
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 600, ( :5/2), 1)(2,\ 600,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 1.8897571001.889757100
L(12)L(\frac12) ≈\approx 1.8897571001.889757100
L(72)L(\frac{7}{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+p2T 1 + p^{2} T
5 1 1
good7 1−160T+p5T2 1 - 160 T + p^{5} T^{2}
11 1+596T+p5T2 1 + 596 T + p^{5} T^{2}
13 1−122T+p5T2 1 - 122 T + p^{5} T^{2}
17 1−1078T+p5T2 1 - 1078 T + p^{5} T^{2}
19 1−796T+p5T2 1 - 796 T + p^{5} T^{2}
23 1−1088T+p5T2 1 - 1088 T + p^{5} T^{2}
29 1−46T+p5T2 1 - 46 T + p^{5} T^{2}
31 1+4952T+p5T2 1 + 4952 T + p^{5} T^{2}
37 1−6114T+p5T2 1 - 6114 T + p^{5} T^{2}
41 1+6T+p5T2 1 + 6 T + p^{5} T^{2}
43 1−24116T+p5T2 1 - 24116 T + p^{5} T^{2}
47 1+13480T+p5T2 1 + 13480 T + p^{5} T^{2}
53 1+20598T+p5T2 1 + 20598 T + p^{5} T^{2}
59 1+46756T+p5T2 1 + 46756 T + p^{5} T^{2}
61 1+9602T+p5T2 1 + 9602 T + p^{5} T^{2}
67 1−17404T+p5T2 1 - 17404 T + p^{5} T^{2}
71 1−26568T+p5T2 1 - 26568 T + p^{5} T^{2}
73 1+75450T+p5T2 1 + 75450 T + p^{5} T^{2}
79 1−50472T+p5T2 1 - 50472 T + p^{5} T^{2}
83 1+33236T+p5T2 1 + 33236 T + p^{5} T^{2}
89 1−133194T+p5T2 1 - 133194 T + p^{5} T^{2}
97 1−42878T+p5T2 1 - 42878 T + p^{5} 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

−10.08440328236203547300207458633, −9.009726542165932015827347259018, −7.75200736924549397896838519065, −7.61263854758241228592848050033, −6.04627138683383733585836325597, −5.25204493797301607811032300958, −4.59051548108154405697701411926, −3.13908949074049935725557939115, −1.84005812748791400887809234109, −0.70145077181122262909665879081, 0.70145077181122262909665879081, 1.84005812748791400887809234109, 3.13908949074049935725557939115, 4.59051548108154405697701411926, 5.25204493797301607811032300958, 6.04627138683383733585836325597, 7.61263854758241228592848050033, 7.75200736924549397896838519065, 9.009726542165932015827347259018, 10.08440328236203547300207458633

Graph of the ZZ-function along the critical line