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Properties

Label 2-226512-1.1-c1-0-140
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  + 4·5-s − 2·7-s − 13-s + 3·17-s − 23-s + 11·25-s − 5·29-s − 4·31-s − 8·35-s − 2·41-s − 43-s + 12·47-s − 3·49-s − 9·53-s + 6·59-s − 5·61-s − 4·65-s + 6·67-s + 12·71-s − 6·73-s + 3·79-s + 4·83-s + 12·85-s + 2·91-s − 12·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.755·7-s − 0.277·13-s + 0.727·17-s − 0.208·23-s + 11/5·25-s − 0.928·29-s − 0.718·31-s − 1.35·35-s − 0.312·41-s − 0.152·43-s + 1.75·47-s − 3/7·49-s − 1.23·53-s + 0.781·59-s − 0.640·61-s − 0.496·65-s + 0.733·67-s + 1.42·71-s − 0.702·73-s + 0.337·79-s + 0.439·83-s + 1.30·85-s + 0.209·91-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-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 1−4T+pT2 1 - 4 T + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+T+pT2 1 + T + p T^{2}
29 1+5T+pT2 1 + 5 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1+pT2 1 + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1+9T+pT2 1 + 9 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1+5T+pT2 1 + 5 T + p T^{2}
67 1−6T+pT2 1 - 6 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1−3T+pT2 1 - 3 T + p T^{2}
83 1−4T+pT2 1 - 4 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

−13.26186472884007, −12.72238061491220, −12.43604191286832, −11.94272746619187, −11.06822560741868, −10.88422412058710, −10.18639326803160, −9.848539809664274, −9.559336570202599, −9.120533571074823, −8.656296168151827, −7.982875615121153, −7.345212577519393, −6.962639181527372, −6.336634235426341, −5.979714719708479, −5.569996290051190, −5.103919016829113, −4.555645422429341, −3.616686121102064, −3.367765740283387, −2.525131895019615, −2.199243732010420, −1.556450563232726, −0.9272887672005608, 0, 0.9272887672005608, 1.556450563232726, 2.199243732010420, 2.525131895019615, 3.367765740283387, 3.616686121102064, 4.555645422429341, 5.103919016829113, 5.569996290051190, 5.979714719708479, 6.336634235426341, 6.962639181527372, 7.345212577519393, 7.982875615121153, 8.656296168151827, 9.120533571074823, 9.559336570202599, 9.848539809664274, 10.18639326803160, 10.88422412058710, 11.06822560741868, 11.94272746619187, 12.43604191286832, 12.72238061491220, 13.26186472884007

Graph of the ZZ-function along the critical line