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Properties

Label 2-405-1.1-c1-0-14
Degree 22
Conductor 405405
Sign −1-1
Analytic cond. 3.233943.23394
Root an. cond. 1.798311.79831
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.732·2-s − 1.46·4-s + 5-s − 4.73·7-s − 2.53·8-s + 0.732·10-s − 5.73·11-s + 1.46·13-s − 3.46·14-s + 1.07·16-s − 2.73·17-s + 4.46·19-s − 1.46·20-s − 4.19·22-s − 3.46·23-s + 25-s + 1.07·26-s + 6.92·28-s + 3.19·29-s − 3·31-s + 5.85·32-s − 2·34-s − 4.73·35-s − 2.73·37-s + 3.26·38-s − 2.53·40-s − 7.19·41-s + ⋯
L(s)  = 1  + 0.517·2-s − 0.732·4-s + 0.447·5-s − 1.78·7-s − 0.896·8-s + 0.231·10-s − 1.72·11-s + 0.406·13-s − 0.925·14-s + 0.267·16-s − 0.662·17-s + 1.02·19-s − 0.327·20-s − 0.894·22-s − 0.722·23-s + 0.200·25-s + 0.210·26-s + 1.30·28-s + 0.593·29-s − 0.538·31-s + 1.03·32-s − 0.342·34-s − 0.799·35-s − 0.449·37-s + 0.530·38-s − 0.400·40-s − 1.12·41-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: −1-1
Analytic conductor: 3.233943.23394
Root analytic conductor: 1.798311.79831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 405, ( :1/2), −1)(2,\ 405,\ (\ :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)
bad3 1 1
5 1−T 1 - T
good2 1−0.732T+2T2 1 - 0.732T + 2T^{2}
7 1+4.73T+7T2 1 + 4.73T + 7T^{2}
11 1+5.73T+11T2 1 + 5.73T + 11T^{2}
13 1−1.46T+13T2 1 - 1.46T + 13T^{2}
17 1+2.73T+17T2 1 + 2.73T + 17T^{2}
19 1−4.46T+19T2 1 - 4.46T + 19T^{2}
23 1+3.46T+23T2 1 + 3.46T + 23T^{2}
29 1−3.19T+29T2 1 - 3.19T + 29T^{2}
31 1+3T+31T2 1 + 3T + 31T^{2}
37 1+2.73T+37T2 1 + 2.73T + 37T^{2}
41 1+7.19T+41T2 1 + 7.19T + 41T^{2}
43 1−0.196T+43T2 1 - 0.196T + 43T^{2}
47 1+8.73T+47T2 1 + 8.73T + 47T^{2}
53 1−6.73T+53T2 1 - 6.73T + 53T^{2}
59 1+8.26T+59T2 1 + 8.26T + 59T^{2}
61 1−4T+61T2 1 - 4T + 61T^{2}
67 1−3.46T+67T2 1 - 3.46T + 67T^{2}
71 1+3.73T+71T2 1 + 3.73T + 71T^{2}
73 1+7.66T+73T2 1 + 7.66T + 73T^{2}
79 1−15.4T+79T2 1 - 15.4T + 79T^{2}
83 1−2.19T+83T2 1 - 2.19T + 83T^{2}
89 1−5.19T+89T2 1 - 5.19T + 89T^{2}
97 1+9.66T+97T2 1 + 9.66T + 97T^{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

−10.53395455354729580116053469386, −9.891659987097994186587409226244, −9.160920829589346233820277538329, −8.113279120425039489990623479181, −6.77562535273111037724955155460, −5.84110997200682682606964101848, −5.05003835867806828695676183738, −3.61864026751503349114017566624, −2.75549156342552231674726398882, 0, 2.75549156342552231674726398882, 3.61864026751503349114017566624, 5.05003835867806828695676183738, 5.84110997200682682606964101848, 6.77562535273111037724955155460, 8.113279120425039489990623479181, 9.160920829589346233820277538329, 9.891659987097994186587409226244, 10.53395455354729580116053469386

Graph of the ZZ-function along the critical line