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Properties

Label 2-252-1.1-c3-0-6
Degree 22
Conductor 252252
Sign −1-1
Analytic cond. 14.868414.8684
Root an. cond. 3.855963.85596
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  − 6·5-s + 7·7-s + 12·11-s − 82·13-s + 30·17-s + 68·19-s − 216·23-s − 89·25-s − 246·29-s − 112·31-s − 42·35-s + 110·37-s + 246·41-s − 172·43-s − 192·47-s + 49·49-s − 558·53-s − 72·55-s − 540·59-s + 110·61-s + 492·65-s + 140·67-s + 840·71-s − 550·73-s + 84·77-s − 208·79-s − 516·83-s + ⋯
L(s)  = 1  − 0.536·5-s + 0.377·7-s + 0.328·11-s − 1.74·13-s + 0.428·17-s + 0.821·19-s − 1.95·23-s − 0.711·25-s − 1.57·29-s − 0.648·31-s − 0.202·35-s + 0.488·37-s + 0.937·41-s − 0.609·43-s − 0.595·47-s + 1/7·49-s − 1.44·53-s − 0.176·55-s − 1.19·59-s + 0.230·61-s + 0.938·65-s + 0.255·67-s + 1.40·71-s − 0.881·73-s + 0.124·77-s − 0.296·79-s − 0.682·83-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, 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: 14.868414.8684
Root analytic conductor: 3.855963.85596
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 252, ( :3/2), −1)(2,\ 252,\ (\ :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 1
7 1−pT 1 - p T
good5 1+6T+p3T2 1 + 6 T + p^{3} T^{2}
11 1−12T+p3T2 1 - 12 T + p^{3} T^{2}
13 1+82T+p3T2 1 + 82 T + p^{3} T^{2}
17 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
19 1−68T+p3T2 1 - 68 T + p^{3} T^{2}
23 1+216T+p3T2 1 + 216 T + p^{3} T^{2}
29 1+246T+p3T2 1 + 246 T + p^{3} T^{2}
31 1+112T+p3T2 1 + 112 T + p^{3} T^{2}
37 1−110T+p3T2 1 - 110 T + p^{3} T^{2}
41 1−6pT+p3T2 1 - 6 p T + p^{3} T^{2}
43 1+4pT+p3T2 1 + 4 p T + p^{3} T^{2}
47 1+192T+p3T2 1 + 192 T + p^{3} T^{2}
53 1+558T+p3T2 1 + 558 T + p^{3} T^{2}
59 1+540T+p3T2 1 + 540 T + p^{3} T^{2}
61 1−110T+p3T2 1 - 110 T + p^{3} T^{2}
67 1−140T+p3T2 1 - 140 T + p^{3} T^{2}
71 1−840T+p3T2 1 - 840 T + p^{3} T^{2}
73 1+550T+p3T2 1 + 550 T + p^{3} T^{2}
79 1+208T+p3T2 1 + 208 T + p^{3} T^{2}
83 1+516T+p3T2 1 + 516 T + p^{3} T^{2}
89 1−1398T+p3T2 1 - 1398 T + p^{3} T^{2}
97 1−1586T+p3T2 1 - 1586 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

−11.36769045390804808334607189627, −10.01949038908630846224976367690, −9.364713950783082040274656552290, −7.83925777564651194973630289705, −7.48693625443598578444463711977, −5.94302480272967733000104894711, −4.78931704989477250882495591841, −3.61943208330552797867352883915, −2.00270477810762360828902519984, 0, 2.00270477810762360828902519984, 3.61943208330552797867352883915, 4.78931704989477250882495591841, 5.94302480272967733000104894711, 7.48693625443598578444463711977, 7.83925777564651194973630289705, 9.364713950783082040274656552290, 10.01949038908630846224976367690, 11.36769045390804808334607189627

Graph of the ZZ-function along the critical line