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Properties

Label 2-226512-1.1-c1-0-116
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  + 5-s + 7-s + 13-s − 17-s − 4·19-s − 8·23-s − 4·25-s − 8·29-s + 35-s + 7·37-s − 8·41-s + 11·43-s − 47-s − 6·49-s − 2·53-s + 14·59-s + 8·61-s + 65-s − 8·67-s + 9·71-s + 4·73-s + 2·79-s − 85-s + 4·89-s + 91-s − 4·95-s + 8·97-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 0.277·13-s − 0.242·17-s − 0.917·19-s − 1.66·23-s − 4/5·25-s − 1.48·29-s + 0.169·35-s + 1.15·37-s − 1.24·41-s + 1.67·43-s − 0.145·47-s − 6/7·49-s − 0.274·53-s + 1.82·59-s + 1.02·61-s + 0.124·65-s − 0.977·67-s + 1.06·71-s + 0.468·73-s + 0.225·79-s − 0.108·85-s + 0.423·89-s + 0.104·91-s − 0.410·95-s + 0.812·97-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−T+pT2 1 - T + p T^{2}
7 1−T+pT2 1 - T + p T^{2}
17 1+T+pT2 1 + T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1+8T+pT2 1 + 8 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−7T+pT2 1 - 7 T + p T^{2}
41 1+8T+pT2 1 + 8 T + p T^{2}
43 1−11T+pT2 1 - 11 T + p T^{2}
47 1+T+pT2 1 + T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1−14T+pT2 1 - 14 T + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1−9T+pT2 1 - 9 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1−2T+pT2 1 - 2 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−4T+pT2 1 - 4 T + p T^{2}
97 1−8T+pT2 1 - 8 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

−13.17098518765178, −12.75519316863724, −12.34674127214759, −11.57717894421839, −11.39847998876301, −10.96470768342082, −10.20746594067268, −10.01543994628227, −9.498217262590306, −8.917965741466379, −8.511772633190076, −7.886507154471373, −7.681550076385156, −6.958802318300737, −6.380684859146945, −5.921320361507268, −5.659661732959502, −4.896013067282820, −4.416026373162720, −3.766594087372089, −3.551538351357170, −2.420480939417923, −2.136002470696981, −1.684191688323479, −0.7668307246592781, 0, 0.7668307246592781, 1.684191688323479, 2.136002470696981, 2.420480939417923, 3.551538351357170, 3.766594087372089, 4.416026373162720, 4.896013067282820, 5.659661732959502, 5.921320361507268, 6.380684859146945, 6.958802318300737, 7.681550076385156, 7.886507154471373, 8.511772633190076, 8.917965741466379, 9.498217262590306, 10.01543994628227, 10.20746594067268, 10.96470768342082, 11.39847998876301, 11.57717894421839, 12.34674127214759, 12.75519316863724, 13.17098518765178

Graph of the ZZ-function along the critical line