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Properties

Label 2-252-1.1-c1-0-1
Degree 22
Conductor 252252
Sign −1-1
Analytic cond. 2.012232.01223
Root an. cond. 1.418531.41853
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  − 4·5-s − 7-s − 2·11-s − 6·13-s + 4·17-s − 4·19-s − 2·23-s + 11·25-s + 2·29-s + 4·35-s + 2·37-s − 4·43-s − 12·47-s + 49-s + 6·53-s + 8·55-s + 8·59-s + 6·61-s + 24·65-s − 8·67-s − 14·71-s − 2·73-s + 2·77-s + 12·79-s + 4·83-s − 16·85-s + 6·91-s + ⋯
L(s)  = 1  − 1.78·5-s − 0.377·7-s − 0.603·11-s − 1.66·13-s + 0.970·17-s − 0.917·19-s − 0.417·23-s + 11/5·25-s + 0.371·29-s + 0.676·35-s + 0.328·37-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.824·53-s + 1.07·55-s + 1.04·59-s + 0.768·61-s + 2.97·65-s − 0.977·67-s − 1.66·71-s − 0.234·73-s + 0.227·77-s + 1.35·79-s + 0.439·83-s − 1.73·85-s + 0.628·91-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: −1-1
Analytic conductor: 2.012232.01223
Root analytic conductor: 1.418531.41853
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 252, ( :1/2), −1)(2,\ 252,\ (\ :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 1
7 1+T 1 + T
good5 1+4T+pT2 1 + 4 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+2T+pT2 1 + 2 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1+14T+pT2 1 + 14 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1−12T+pT2 1 - 12 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+2T+pT2 1 + 2 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

−11.83481765842467689738267568685, −10.65141343664047244023029939050, −9.763849160343204480377812055766, −8.349702149825146102162236283121, −7.69907439436331841244821924582, −6.81660893407171735772005884639, −5.13729108364941390504198607970, −4.08565362497912592132601547351, −2.85749596756875682654277889497, 0, 2.85749596756875682654277889497, 4.08565362497912592132601547351, 5.13729108364941390504198607970, 6.81660893407171735772005884639, 7.69907439436331841244821924582, 8.349702149825146102162236283121, 9.763849160343204480377812055766, 10.65141343664047244023029939050, 11.83481765842467689738267568685

Graph of the ZZ-function along the critical line