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Properties

Label 2-600-1.1-c3-0-7
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 + 24·7-s + 9·9-s − 28·11-s + 74·13-s − 82·17-s + 92·19-s − 72·21-s − 8·23-s − 27·27-s − 138·29-s + 80·31-s + 84·33-s − 30·37-s − 222·39-s + 282·41-s − 4·43-s − 240·47-s + 233·49-s + 246·51-s + 130·53-s − 276·57-s + 596·59-s − 218·61-s + 216·63-s + 436·67-s + 24·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.29·7-s + 1/3·9-s − 0.767·11-s + 1.57·13-s − 1.16·17-s + 1.11·19-s − 0.748·21-s − 0.0725·23-s − 0.192·27-s − 0.883·29-s + 0.463·31-s + 0.443·33-s − 0.133·37-s − 0.911·39-s + 1.07·41-s − 0.0141·43-s − 0.744·47-s + 0.679·49-s + 0.675·51-s + 0.336·53-s − 0.641·57-s + 1.31·59-s − 0.457·61-s + 0.431·63-s + 0.795·67-s + 0.0418·69-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: 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 1.9282080681.928208068
L(12)L(\frac12) ≈\approx 1.9282080681.928208068
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 1−24T+p3T2 1 - 24 T + p^{3} T^{2}
11 1+28T+p3T2 1 + 28 T + p^{3} T^{2}
13 1−74T+p3T2 1 - 74 T + p^{3} T^{2}
17 1+82T+p3T2 1 + 82 T + p^{3} T^{2}
19 1−92T+p3T2 1 - 92 T + p^{3} T^{2}
23 1+8T+p3T2 1 + 8 T + p^{3} T^{2}
29 1+138T+p3T2 1 + 138 T + p^{3} T^{2}
31 1−80T+p3T2 1 - 80 T + p^{3} T^{2}
37 1+30T+p3T2 1 + 30 T + p^{3} T^{2}
41 1−282T+p3T2 1 - 282 T + p^{3} T^{2}
43 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
47 1+240T+p3T2 1 + 240 T + p^{3} T^{2}
53 1−130T+p3T2 1 - 130 T + p^{3} T^{2}
59 1−596T+p3T2 1 - 596 T + p^{3} T^{2}
61 1+218T+p3T2 1 + 218 T + p^{3} T^{2}
67 1−436T+p3T2 1 - 436 T + p^{3} T^{2}
71 1−856T+p3T2 1 - 856 T + p^{3} T^{2}
73 1−998T+p3T2 1 - 998 T + p^{3} T^{2}
79 1+32T+p3T2 1 + 32 T + p^{3} T^{2}
83 1−1508T+p3T2 1 - 1508 T + p^{3} T^{2}
89 1+246T+p3T2 1 + 246 T + p^{3} T^{2}
97 1+866T+p3T2 1 + 866 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

−10.57019342725302538270023012424, −9.370127133152483918207890587415, −8.380065520221327281264298396348, −7.71289718518656898619825265009, −6.58725397791577455460013079240, −5.57084276231146608224523748329, −4.82366423283596447473635762626, −3.72300527212293946505667702472, −2.09444546061579446526958323399, −0.890704268512275218823788069009, 0.890704268512275218823788069009, 2.09444546061579446526958323399, 3.72300527212293946505667702472, 4.82366423283596447473635762626, 5.57084276231146608224523748329, 6.58725397791577455460013079240, 7.71289718518656898619825265009, 8.380065520221327281264298396348, 9.370127133152483918207890587415, 10.57019342725302538270023012424

Graph of the ZZ-function along the critical line