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Properties

Label 2-600-1.1-c3-0-11
Degree 22
Conductor 600600
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 5·7-s + 9·9-s + 14·11-s + 13-s + 46·17-s + 19·19-s + 15·21-s − 46·23-s + 27·27-s + 14·29-s + 133·31-s + 42·33-s + 258·37-s + 3·39-s + 84·41-s − 167·43-s + 410·47-s − 318·49-s + 138·51-s + 456·53-s + 57·57-s − 194·59-s − 17·61-s + 45·63-s + 653·67-s − 138·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.269·7-s + 1/3·9-s + 0.383·11-s + 0.0213·13-s + 0.656·17-s + 0.229·19-s + 0.155·21-s − 0.417·23-s + 0.192·27-s + 0.0896·29-s + 0.770·31-s + 0.221·33-s + 1.14·37-s + 0.0123·39-s + 0.319·41-s − 0.592·43-s + 1.27·47-s − 0.927·49-s + 0.378·51-s + 1.18·53-s + 0.132·57-s − 0.428·59-s − 0.0356·61-s + 0.0899·63-s + 1.19·67-s − 0.240·69-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: 11
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: 00
Selberg data: (2, 600, ( :3/2), 1)(2,\ 600,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.7404824752.740482475
L(12)L(\frac12) ≈\approx 2.7404824752.740482475
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 1−5T+p3T2 1 - 5 T + p^{3} T^{2}
11 1−14T+p3T2 1 - 14 T + p^{3} T^{2}
13 1−T+p3T2 1 - T + p^{3} T^{2}
17 1−46T+p3T2 1 - 46 T + p^{3} T^{2}
19 1−pT+p3T2 1 - p T + p^{3} T^{2}
23 1+2pT+p3T2 1 + 2 p T + p^{3} T^{2}
29 1−14T+p3T2 1 - 14 T + p^{3} T^{2}
31 1−133T+p3T2 1 - 133 T + p^{3} T^{2}
37 1−258T+p3T2 1 - 258 T + p^{3} T^{2}
41 1−84T+p3T2 1 - 84 T + p^{3} T^{2}
43 1+167T+p3T2 1 + 167 T + p^{3} T^{2}
47 1−410T+p3T2 1 - 410 T + p^{3} T^{2}
53 1−456T+p3T2 1 - 456 T + p^{3} T^{2}
59 1+194T+p3T2 1 + 194 T + p^{3} T^{2}
61 1+17T+p3T2 1 + 17 T + p^{3} T^{2}
67 1−653T+p3T2 1 - 653 T + p^{3} T^{2}
71 1−828T+p3T2 1 - 828 T + p^{3} T^{2}
73 1−570T+p3T2 1 - 570 T + p^{3} T^{2}
79 1+552T+p3T2 1 + 552 T + p^{3} T^{2}
83 1−142T+p3T2 1 - 142 T + p^{3} T^{2}
89 1+1104T+p3T2 1 + 1104 T + p^{3} T^{2}
97 1−841T+p3T2 1 - 841 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

−10.08739957229836474837263440771, −9.432632991089721204452018249294, −8.429155463864919476718953001235, −7.75940072420533081688199362022, −6.75825932127965148456750396998, −5.69519306085952846566476914670, −4.53211551748147768776432429537, −3.52904289823064767762286064968, −2.35146608250949449895435806483, −1.01640433081996302184915470479, 1.01640433081996302184915470479, 2.35146608250949449895435806483, 3.52904289823064767762286064968, 4.53211551748147768776432429537, 5.69519306085952846566476914670, 6.75825932127965148456750396998, 7.75940072420533081688199362022, 8.429155463864919476718953001235, 9.432632991089721204452018249294, 10.08739957229836474837263440771

Graph of the ZZ-function along the critical line