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Properties

Label 2-600-1.1-c1-0-8
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 − 5·7-s + 9-s − 6·11-s − 3·13-s − 2·17-s + 19-s − 5·21-s − 2·23-s + 27-s + 6·29-s + 3·31-s − 6·33-s − 6·37-s − 3·39-s + 4·41-s + 11·43-s − 10·47-s + 18·49-s − 2·51-s − 8·53-s + 57-s − 6·59-s + 3·61-s − 5·63-s − 67-s − 2·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.88·7-s + 1/3·9-s − 1.80·11-s − 0.832·13-s − 0.485·17-s + 0.229·19-s − 1.09·21-s − 0.417·23-s + 0.192·27-s + 1.11·29-s + 0.538·31-s − 1.04·33-s − 0.986·37-s − 0.480·39-s + 0.624·41-s + 1.67·43-s − 1.45·47-s + 18/7·49-s − 0.280·51-s − 1.09·53-s + 0.132·57-s − 0.781·59-s + 0.384·61-s − 0.629·63-s − 0.122·67-s − 0.240·69-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+5T+pT2 1 + 5 T + p T^{2}
11 1+6T+pT2 1 + 6 T + p T^{2}
13 1+3T+pT2 1 + 3 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1−T+pT2 1 - T + p T^{2}
23 1+2T+pT2 1 + 2 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−3T+pT2 1 - 3 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1−4T+pT2 1 - 4 T + p T^{2}
43 1−11T+pT2 1 - 11 T + p T^{2}
47 1+10T+pT2 1 + 10 T + p T^{2}
53 1+8T+pT2 1 + 8 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1−3T+pT2 1 - 3 T + p T^{2}
67 1+T+pT2 1 + T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1+16T+pT2 1 + 16 T + p T^{2}
97 1+7T+pT2 1 + 7 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

−10.01640260119628208124792470973, −9.573634131463927078112432516263, −8.483378982015133633479563111090, −7.55110806087691558837157186820, −6.72557170085953961720882600963, −5.70500717305478856615746032890, −4.50519787498977221160424199992, −3.11187956794533611218111257703, −2.55892028284702124429993107097, 0, 2.55892028284702124429993107097, 3.11187956794533611218111257703, 4.50519787498977221160424199992, 5.70500717305478856615746032890, 6.72557170085953961720882600963, 7.55110806087691558837157186820, 8.483378982015133633479563111090, 9.573634131463927078112432516263, 10.01640260119628208124792470973

Graph of the ZZ-function along the critical line