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Properties

Label 2-226512-1.1-c1-0-115
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 − 2·7-s + 13-s − 4·17-s + 2·19-s + 7·23-s − 4·25-s − 2·29-s + 3·31-s − 2·35-s − 11·37-s + 10·41-s − 4·43-s − 4·47-s − 3·49-s − 2·53-s − 59-s + 2·61-s + 65-s + 67-s − 9·71-s + 16·73-s + 8·79-s − 4·85-s + 7·89-s − 2·91-s + 2·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s + 0.277·13-s − 0.970·17-s + 0.458·19-s + 1.45·23-s − 4/5·25-s − 0.371·29-s + 0.538·31-s − 0.338·35-s − 1.80·37-s + 1.56·41-s − 0.609·43-s − 0.583·47-s − 3/7·49-s − 0.274·53-s − 0.130·59-s + 0.256·61-s + 0.124·65-s + 0.122·67-s − 1.06·71-s + 1.87·73-s + 0.900·79-s − 0.433·85-s + 0.741·89-s − 0.209·91-s + 0.205·95-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+2T+pT2 1 + 2 T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1−7T+pT2 1 - 7 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1−3T+pT2 1 - 3 T + p T^{2}
37 1+11T+pT2 1 + 11 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1+T+pT2 1 + T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−T+pT2 1 - T + p T^{2}
71 1+9T+pT2 1 + 9 T + p T^{2}
73 1−16T+pT2 1 - 16 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−7T+pT2 1 - 7 T + p T^{2}
97 1+13T+pT2 1 + 13 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

−13.17229928775125, −12.89773917210681, −12.23759182303610, −11.84405902601713, −11.24092372526028, −10.82450843045894, −10.46370974138635, −9.790915856005430, −9.343015661540156, −9.205540734438780, −8.475605348638811, −8.082336206689725, −7.359064025777045, −6.942967720268963, −6.454935258051028, −6.125646129454446, −5.383400244029357, −5.073895001712893, −4.415765143102789, −3.770509601440700, −3.284524852629678, −2.770573339394819, −2.108840581088215, −1.541376523474898, −0.7616885825412951, 0, 0.7616885825412951, 1.541376523474898, 2.108840581088215, 2.770573339394819, 3.284524852629678, 3.770509601440700, 4.415765143102789, 5.073895001712893, 5.383400244029357, 6.125646129454446, 6.454935258051028, 6.942967720268963, 7.359064025777045, 8.082336206689725, 8.475605348638811, 9.205540734438780, 9.343015661540156, 9.790915856005430, 10.46370974138635, 10.82450843045894, 11.24092372526028, 11.84405902601713, 12.23759182303610, 12.89773917210681, 13.17229928775125

Graph of the ZZ-function along the critical line