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Properties

Label 2-226512-1.1-c1-0-122
Degree 22
Conductor 226512226512
Sign −1-1
Analytic cond. 1808.701808.70
Root an. cond. 42.528942.5289
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  − 4·7-s + 13-s + 5·17-s + 6·19-s − 23-s − 5·25-s − 29-s + 6·37-s − 4·41-s − 43-s + 6·47-s + 9·49-s + 53-s + 4·59-s − 61-s + 10·71-s − 2·73-s − 3·79-s + 14·83-s − 4·91-s − 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 20·119-s + ⋯
L(s)  = 1  − 1.51·7-s + 0.277·13-s + 1.21·17-s + 1.37·19-s − 0.208·23-s − 25-s − 0.185·29-s + 0.986·37-s − 0.624·41-s − 0.152·43-s + 0.875·47-s + 9/7·49-s + 0.137·53-s + 0.520·59-s − 0.128·61-s + 1.18·71-s − 0.234·73-s − 0.337·79-s + 1.53·83-s − 0.419·91-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.83·119-s + ⋯

Functional equation

Λ(s)=(226512s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(226512s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 226512226512    =    24⋅32⋅112⋅132^{4} \cdot 3^{2} \cdot 11^{2} \cdot 13
Sign: −1-1
Analytic conductor: 1808.701808.70
Root analytic conductor: 42.528942.5289
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 226512, ( :1/2), −1)(2,\ 226512,\ (\ :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
11 1 1
13 1−T 1 - T
good5 1+pT2 1 + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
17 1−5T+pT2 1 - 5 T + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1+T+pT2 1 + T + p T^{2}
29 1+T+pT2 1 + T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1+4T+pT2 1 + 4 T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1−T+pT2 1 - T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1+T+pT2 1 + T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1−10T+pT2 1 - 10 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+3T+pT2 1 + 3 T + p T^{2}
83 1−14T+pT2 1 - 14 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+12T+pT2 1 + 12 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

−13.28761542500111, −12.58381713791788, −12.33525629252626, −11.84352145292042, −11.43121983402815, −10.80965955998316, −10.18283902683267, −9.922103420975933, −9.412169620409273, −9.256676830581096, −8.429032531418749, −7.885125682843656, −7.547273619910381, −6.955241837533491, −6.469982051089845, −5.994305023131523, −5.457497190349627, −5.206804057382435, −4.167484099301685, −3.813054876598937, −3.281461092893725, −2.874758333142114, −2.227317866469202, −1.339980737755165, −0.7886701747709631, 0, 0.7886701747709631, 1.339980737755165, 2.227317866469202, 2.874758333142114, 3.281461092893725, 3.813054876598937, 4.167484099301685, 5.206804057382435, 5.457497190349627, 5.994305023131523, 6.469982051089845, 6.955241837533491, 7.547273619910381, 7.885125682843656, 8.429032531418749, 9.256676830581096, 9.412169620409273, 9.922103420975933, 10.18283902683267, 10.80965955998316, 11.43121983402815, 11.84352145292042, 12.33525629252626, 12.58381713791788, 13.28761542500111

Graph of the ZZ-function along the critical line