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Properties

Label 2-4275-1.1-c1-0-136
Degree 22
Conductor 42754275
Sign −1-1
Analytic cond. 34.136034.1360
Root an. cond. 5.842605.84260
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  + 1.95·2-s + 1.82·4-s + 3.56·7-s − 0.340·8-s − 5.56·11-s − 5.26·13-s + 6.96·14-s − 4.31·16-s + 1.40·17-s + 19-s − 10.8·22-s − 6.96·23-s − 10.3·26-s + 6.50·28-s − 1.40·29-s + 1.75·31-s − 7.76·32-s + 2.75·34-s − 3.61·37-s + 1.95·38-s − 4.34·41-s + 3.56·43-s − 10.1·44-s − 13.6·46-s − 8.26·47-s + 5.69·49-s − 9.61·52-s + ⋯
L(s)  = 1  + 1.38·2-s + 0.912·4-s + 1.34·7-s − 0.120·8-s − 1.67·11-s − 1.46·13-s + 1.86·14-s − 1.07·16-s + 0.341·17-s + 0.229·19-s − 2.31·22-s − 1.45·23-s − 2.02·26-s + 1.22·28-s − 0.261·29-s + 0.315·31-s − 1.37·32-s + 0.471·34-s − 0.594·37-s + 0.317·38-s − 0.679·41-s + 0.543·43-s − 1.53·44-s − 2.00·46-s − 1.20·47-s + 0.813·49-s − 1.33·52-s + ⋯

Functional equation

Λ(s)=(4275s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4275s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 42754275    =    32⋅52⋅193^{2} \cdot 5^{2} \cdot 19
Sign: −1-1
Analytic conductor: 34.136034.1360
Root analytic conductor: 5.842605.84260
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4275, ( :1/2), −1)(2,\ 4275,\ (\ :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 1
19 1−T 1 - T
good2 1−1.95T+2T2 1 - 1.95T + 2T^{2}
7 1−3.56T+7T2 1 - 3.56T + 7T^{2}
11 1+5.56T+11T2 1 + 5.56T + 11T^{2}
13 1+5.26T+13T2 1 + 5.26T + 13T^{2}
17 1−1.40T+17T2 1 - 1.40T + 17T^{2}
23 1+6.96T+23T2 1 + 6.96T + 23T^{2}
29 1+1.40T+29T2 1 + 1.40T + 29T^{2}
31 1−1.75T+31T2 1 - 1.75T + 31T^{2}
37 1+3.61T+37T2 1 + 3.61T + 37T^{2}
41 1+4.34T+41T2 1 + 4.34T + 41T^{2}
43 1−3.56T+43T2 1 - 3.56T + 43T^{2}
47 1+8.26T+47T2 1 + 8.26T + 47T^{2}
53 1+7.61T+53T2 1 + 7.61T + 53T^{2}
59 1+9.47T+59T2 1 + 9.47T + 59T^{2}
61 1−9.21T+61T2 1 - 9.21T + 61T^{2}
67 1−4.76T+67T2 1 - 4.76T + 67T^{2}
71 1−14.0T+71T2 1 - 14.0T + 71T^{2}
73 1+6.59T+73T2 1 + 6.59T + 73T^{2}
79 1−5.47T+79T2 1 - 5.47T + 79T^{2}
83 1+4.15T+83T2 1 + 4.15T + 83T^{2}
89 1−9.23T+89T2 1 - 9.23T + 89T^{2}
97 1+11.5T+97T2 1 + 11.5T + 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

−7.86889999139568826536470957187, −7.34938715083010108415824573510, −6.29819399791246483203111757531, −5.37278531923744420303442245109, −5.04777032744290377370965783458, −4.52587171547591653368613668987, −3.49063016580615184669288296634, −2.55223325632033865107478474355, −1.93431037323643681256023393008, 0, 1.93431037323643681256023393008, 2.55223325632033865107478474355, 3.49063016580615184669288296634, 4.52587171547591653368613668987, 5.04777032744290377370965783458, 5.37278531923744420303442245109, 6.29819399791246483203111757531, 7.34938715083010108415824573510, 7.86889999139568826536470957187

Graph of the ZZ-function along the critical line