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Properties

Label 2-600-1.1-c3-0-22
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 − 20·7-s + 9·9-s − 56·11-s + 86·13-s + 106·17-s + 4·19-s − 60·21-s − 136·23-s + 27·27-s − 206·29-s − 152·31-s − 168·33-s − 282·37-s + 258·39-s − 246·41-s − 412·43-s − 40·47-s + 57·49-s + 318·51-s + 126·53-s + 12·57-s + 56·59-s − 2·61-s − 180·63-s + 388·67-s − 408·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.07·7-s + 1/3·9-s − 1.53·11-s + 1.83·13-s + 1.51·17-s + 0.0482·19-s − 0.623·21-s − 1.23·23-s + 0.192·27-s − 1.31·29-s − 0.880·31-s − 0.886·33-s − 1.25·37-s + 1.05·39-s − 0.937·41-s − 1.46·43-s − 0.124·47-s + 0.166·49-s + 0.873·51-s + 0.326·53-s + 0.0278·57-s + 0.123·59-s − 0.00419·61-s − 0.359·63-s + 0.707·67-s − 0.711·69-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+20T+p3T2 1 + 20 T + p^{3} T^{2}
11 1+56T+p3T2 1 + 56 T + p^{3} T^{2}
13 1−86T+p3T2 1 - 86 T + p^{3} T^{2}
17 1−106T+p3T2 1 - 106 T + p^{3} T^{2}
19 1−4T+p3T2 1 - 4 T + p^{3} T^{2}
23 1+136T+p3T2 1 + 136 T + p^{3} T^{2}
29 1+206T+p3T2 1 + 206 T + p^{3} T^{2}
31 1+152T+p3T2 1 + 152 T + p^{3} T^{2}
37 1+282T+p3T2 1 + 282 T + p^{3} T^{2}
41 1+6pT+p3T2 1 + 6 p T + p^{3} T^{2}
43 1+412T+p3T2 1 + 412 T + p^{3} T^{2}
47 1+40T+p3T2 1 + 40 T + p^{3} T^{2}
53 1−126T+p3T2 1 - 126 T + p^{3} T^{2}
59 1−56T+p3T2 1 - 56 T + p^{3} T^{2}
61 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
67 1−388T+p3T2 1 - 388 T + p^{3} T^{2}
71 1+672T+p3T2 1 + 672 T + p^{3} T^{2}
73 1+1170T+p3T2 1 + 1170 T + p^{3} T^{2}
79 1−408T+p3T2 1 - 408 T + p^{3} T^{2}
83 1+668T+p3T2 1 + 668 T + p^{3} T^{2}
89 1−66T+p3T2 1 - 66 T + p^{3} T^{2}
97 1−926T+p3T2 1 - 926 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

−10.00949230384556564167374542793, −8.864320722600307491350582166437, −8.117285377954784462420338163054, −7.28748501609370505265861103468, −6.09605233773722902826135059366, −5.36731941521076290106281303232, −3.67323308812194972118082389671, −3.21716071887988136811979169068, −1.71878145410417427398019709314, 0, 1.71878145410417427398019709314, 3.21716071887988136811979169068, 3.67323308812194972118082389671, 5.36731941521076290106281303232, 6.09605233773722902826135059366, 7.28748501609370505265861103468, 8.117285377954784462420338163054, 8.864320722600307491350582166437, 10.00949230384556564167374542793

Graph of the ZZ-function along the critical line