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Properties

Label 2-600-1.1-c1-0-7
Degree 22
Conductor 600600
Sign −1-1
Analytic cond. 4.791024.79102
Root an. cond. 2.188842.18884
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 4·11-s − 6·13-s + 6·17-s − 4·19-s − 27-s − 2·29-s − 8·31-s + 4·33-s + 2·37-s + 6·39-s − 6·41-s − 12·43-s − 8·47-s − 7·49-s − 6·51-s − 6·53-s + 4·57-s + 12·59-s + 14·61-s − 4·67-s + 8·71-s + 6·73-s − 8·79-s + 81-s + 12·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 1.45·17-s − 0.917·19-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.696·33-s + 0.328·37-s + 0.960·39-s − 0.937·41-s − 1.82·43-s − 1.16·47-s − 49-s − 0.840·51-s − 0.824·53-s + 0.529·57-s + 1.56·59-s + 1.79·61-s − 0.488·67-s + 0.949·71-s + 0.702·73-s − 0.900·79-s + 1/9·81-s + 1.31·83-s + ⋯

Functional equation

Λ(s)=(600s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(600s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 4.791024.79102
Root analytic conductor: 2.188842.18884
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 600, ( :1/2), −1)(2,\ 600,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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+T 1 + T
5 1 1
good7 1+pT2 1 + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+12T+pT2 1 + 12 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−10T+pT2 1 - 10 T + p T^{2}
97 1+2T+pT2 1 + 2 T + p 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

−10.09854126146590211624669174065, −9.749205121396688870647577836420, −8.280524075884956092273610432842, −7.55727707153378484774141976097, −6.65963768785974737826480192145, −5.36683559095823513937522588289, −4.95639818252911756604562729940, −3.42683606015106566673715003849, −2.07933314479152275364533906758, 0, 2.07933314479152275364533906758, 3.42683606015106566673715003849, 4.95639818252911756604562729940, 5.36683559095823513937522588289, 6.65963768785974737826480192145, 7.55727707153378484774141976097, 8.280524075884956092273610432842, 9.749205121396688870647577836420, 10.09854126146590211624669174065

Graph of the ZZ-function along the critical line