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Properties

Label 2-600-1.1-c3-0-26
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 + 9·9-s + 4·11-s − 54·13-s − 114·17-s + 44·19-s − 96·23-s + 27·27-s + 134·29-s − 272·31-s + 12·33-s + 98·37-s − 162·39-s − 6·41-s − 12·43-s + 200·47-s − 343·49-s − 342·51-s − 654·53-s + 132·57-s + 36·59-s − 442·61-s + 188·67-s − 288·69-s − 632·71-s + 390·73-s + 688·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 0.109·11-s − 1.15·13-s − 1.62·17-s + 0.531·19-s − 0.870·23-s + 0.192·27-s + 0.858·29-s − 1.57·31-s + 0.0633·33-s + 0.435·37-s − 0.665·39-s − 0.0228·41-s − 0.0425·43-s + 0.620·47-s − 49-s − 0.939·51-s − 1.69·53-s + 0.306·57-s + 0.0794·59-s − 0.927·61-s + 0.342·67-s − 0.502·69-s − 1.05·71-s + 0.625·73-s + 0.979·79-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+p3T2 1 + p^{3} T^{2}
11 1−4T+p3T2 1 - 4 T + p^{3} T^{2}
13 1+54T+p3T2 1 + 54 T + p^{3} T^{2}
17 1+114T+p3T2 1 + 114 T + p^{3} T^{2}
19 1−44T+p3T2 1 - 44 T + p^{3} T^{2}
23 1+96T+p3T2 1 + 96 T + p^{3} T^{2}
29 1−134T+p3T2 1 - 134 T + p^{3} T^{2}
31 1+272T+p3T2 1 + 272 T + p^{3} T^{2}
37 1−98T+p3T2 1 - 98 T + p^{3} T^{2}
41 1+6T+p3T2 1 + 6 T + p^{3} T^{2}
43 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
47 1−200T+p3T2 1 - 200 T + p^{3} T^{2}
53 1+654T+p3T2 1 + 654 T + p^{3} T^{2}
59 1−36T+p3T2 1 - 36 T + p^{3} T^{2}
61 1+442T+p3T2 1 + 442 T + p^{3} T^{2}
67 1−188T+p3T2 1 - 188 T + p^{3} T^{2}
71 1+632T+p3T2 1 + 632 T + p^{3} T^{2}
73 1−390T+p3T2 1 - 390 T + p^{3} T^{2}
79 1−688T+p3T2 1 - 688 T + p^{3} T^{2}
83 1+1188T+p3T2 1 + 1188 T + p^{3} T^{2}
89 1+694T+p3T2 1 + 694 T + p^{3} T^{2}
97 1−1726T+p3T2 1 - 1726 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

−9.655062054452861470103965400382, −9.058631997660681547640582988283, −8.043773473711925219962196281201, −7.23777748193364407539882315937, −6.32264660827690651554660580118, −5.01577188495701691962861012071, −4.12588707083103503084398181007, −2.85265198802496213377326143370, −1.83481155734267114601777652366, 0, 1.83481155734267114601777652366, 2.85265198802496213377326143370, 4.12588707083103503084398181007, 5.01577188495701691962861012071, 6.32264660827690651554660580118, 7.23777748193364407539882315937, 8.043773473711925219962196281201, 9.058631997660681547640582988283, 9.655062054452861470103965400382

Graph of the ZZ-function along the critical line