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Properties

Label 2-600-1.1-c3-0-19
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 − 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: −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+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

−9.853776397969120604618428199620, −9.067081188466859593651617878747, −8.037333894389118986660853080570, −6.92828621815004337233921292544, −6.28256720359774686233792727359, −5.18511376787084567594995246369, −4.24248004324726432064316242329, −2.98955944237088359240704654108, −1.46808929779602186516515225409, 0, 1.46808929779602186516515225409, 2.98955944237088359240704654108, 4.24248004324726432064316242329, 5.18511376787084567594995246369, 6.28256720359774686233792727359, 6.92828621815004337233921292544, 8.037333894389118986660853080570, 9.067081188466859593651617878747, 9.853776397969120604618428199620

Graph of the ZZ-function along the critical line