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Properties

Label 2-43560-1.1-c1-0-31
Degree 22
Conductor 4356043560
Sign 11
Analytic cond. 347.828347.828
Root an. cond. 18.650118.6501
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 4·7-s + 6·13-s + 3·17-s + 4·19-s + 23-s + 25-s − 8·29-s + 5·31-s − 4·35-s − 4·37-s − 2·41-s − 5·47-s + 9·49-s − 13·53-s + 8·59-s − 11·61-s − 6·65-s + 10·67-s − 6·71-s + 4·73-s + 5·79-s + 4·83-s − 3·85-s − 12·89-s + 24·91-s − 4·95-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-s + 1.66·13-s + 0.727·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s − 1.48·29-s + 0.898·31-s − 0.676·35-s − 0.657·37-s − 0.312·41-s − 0.729·47-s + 9/7·49-s − 1.78·53-s + 1.04·59-s − 1.40·61-s − 0.744·65-s + 1.22·67-s − 0.712·71-s + 0.468·73-s + 0.562·79-s + 0.439·83-s − 0.325·85-s − 1.27·89-s + 2.51·91-s − 0.410·95-s + ⋯

Functional equation

Λ(s)=(43560s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(43560s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4356043560    =    23⋅32⋅5⋅1122^{3} \cdot 3^{2} \cdot 5 \cdot 11^{2}
Sign: 11
Analytic conductor: 347.828347.828
Root analytic conductor: 18.650118.6501
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 43560, ( :1/2), 1)(2,\ 43560,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.3886763043.388676304
L(12)L(\frac12) ≈\approx 3.3886763043.388676304
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
5 1+T 1 + T
11 1 1
good7 1−4T+pT2 1 - 4 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−T+pT2 1 - T + p T^{2}
29 1+8T+pT2 1 + 8 T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+5T+pT2 1 + 5 T + p T^{2}
53 1+13T+pT2 1 + 13 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1+11T+pT2 1 + 11 T + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1−5T+pT2 1 - 5 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+12T+pT2 1 + 12 T + p T^{2}
97 1−2T+pT2 1 - 2 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

−14.61239537952978, −14.15708208772392, −13.79273264815407, −13.18942660071874, −12.59720976467712, −11.94212391662280, −11.42789927808009, −11.19739250527568, −10.70689151401868, −10.00866213486594, −9.365093861805828, −8.713609345749764, −8.278857816802699, −7.811454814605926, −7.402772590138352, −6.604561058684720, −5.955062214422765, −5.337415646483482, −4.882472183719536, −4.184263290762184, −3.518761368565563, −3.088971045653812, −1.881243646928165, −1.446313140224272, −0.7158511945983851, 0.7158511945983851, 1.446313140224272, 1.881243646928165, 3.088971045653812, 3.518761368565563, 4.184263290762184, 4.882472183719536, 5.337415646483482, 5.955062214422765, 6.604561058684720, 7.402772590138352, 7.811454814605926, 8.278857816802699, 8.713609345749764, 9.365093861805828, 10.00866213486594, 10.70689151401868, 11.19739250527568, 11.42789927808009, 11.94212391662280, 12.59720976467712, 13.18942660071874, 13.79273264815407, 14.15708208772392, 14.61239537952978

Graph of the ZZ-function along the critical line