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Properties

Label 2-600-1.1-c3-0-6
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 + 19·7-s + 9·9-s + 22·11-s − 13-s + 58·17-s − 53·19-s − 57·21-s − 58·23-s − 27·27-s + 22·29-s − 35·31-s − 66·33-s + 270·37-s + 3·39-s − 468·41-s + 431·43-s + 230·47-s + 18·49-s − 174·51-s + 159·57-s + 446·59-s + 127·61-s + 171·63-s + 811·67-s + 174·69-s + 36·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.02·7-s + 1/3·9-s + 0.603·11-s − 0.0213·13-s + 0.827·17-s − 0.639·19-s − 0.592·21-s − 0.525·23-s − 0.192·27-s + 0.140·29-s − 0.202·31-s − 0.348·33-s + 1.19·37-s + 0.0123·39-s − 1.78·41-s + 1.52·43-s + 0.713·47-s + 0.0524·49-s − 0.477·51-s + 0.369·57-s + 0.984·59-s + 0.266·61-s + 0.341·63-s + 1.47·67-s + 0.303·69-s + 0.0601·71-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 1.9139366691.913936669
L(12)L(\frac12) ≈\approx 1.9139366691.913936669
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 1−19T+p3T2 1 - 19 T + p^{3} T^{2}
11 1−2pT+p3T2 1 - 2 p T + p^{3} T^{2}
13 1+T+p3T2 1 + T + p^{3} T^{2}
17 1−58T+p3T2 1 - 58 T + p^{3} T^{2}
19 1+53T+p3T2 1 + 53 T + p^{3} T^{2}
23 1+58T+p3T2 1 + 58 T + p^{3} T^{2}
29 1−22T+p3T2 1 - 22 T + p^{3} T^{2}
31 1+35T+p3T2 1 + 35 T + p^{3} T^{2}
37 1−270T+p3T2 1 - 270 T + p^{3} T^{2}
41 1+468T+p3T2 1 + 468 T + p^{3} T^{2}
43 1−431T+p3T2 1 - 431 T + p^{3} T^{2}
47 1−230T+p3T2 1 - 230 T + p^{3} T^{2}
53 1+p3T2 1 + p^{3} T^{2}
59 1−446T+p3T2 1 - 446 T + p^{3} T^{2}
61 1−127T+p3T2 1 - 127 T + p^{3} T^{2}
67 1−811T+p3T2 1 - 811 T + p^{3} T^{2}
71 1−36T+p3T2 1 - 36 T + p^{3} T^{2}
73 1+522T+p3T2 1 + 522 T + p^{3} T^{2}
79 1−1368T+p3T2 1 - 1368 T + p^{3} T^{2}
83 1−1138T+p3T2 1 - 1138 T + p^{3} T^{2}
89 1−144T+p3T2 1 - 144 T + p^{3} T^{2}
97 1−1079T+p3T2 1 - 1079 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

−10.39088651658994165538008491560, −9.468698687870373270937583522282, −8.407663077807796529324863441937, −7.64239262351513397028794959460, −6.58926865025042990638046631450, −5.65269792343558904515760488527, −4.72975676078603205047560489751, −3.76354237765496788760058019501, −2.08867865651075280038107703353, −0.887623715705447190301704751726, 0.887623715705447190301704751726, 2.08867865651075280038107703353, 3.76354237765496788760058019501, 4.72975676078603205047560489751, 5.65269792343558904515760488527, 6.58926865025042990638046631450, 7.64239262351513397028794959460, 8.407663077807796529324863441937, 9.468698687870373270937583522282, 10.39088651658994165538008491560

Graph of the ZZ-function along the critical line