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Properties

Label 2-88e2-1.1-c1-0-133
Degree 22
Conductor 77447744
Sign −1-1
Analytic cond. 61.836161.8361
Root an. cond. 7.863597.86359
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  + 0.0643·3-s − 2.31·5-s + 1.63·7-s − 2.99·9-s + 1.15·13-s − 0.148·15-s + 3.36·17-s + 5.83·19-s + 0.104·21-s − 7.86·23-s + 0.350·25-s − 0.385·27-s + 5.92·29-s − 3.61·31-s − 3.77·35-s + 0.362·37-s + 0.0743·39-s − 4.40·41-s − 5.15·43-s + 6.92·45-s − 1.15·47-s − 4.34·49-s + 0.216·51-s − 6.14·53-s + 0.375·57-s + 8.12·59-s + 8.82·61-s + ⋯
L(s)  = 1  + 0.0371·3-s − 1.03·5-s + 0.616·7-s − 0.998·9-s + 0.320·13-s − 0.0384·15-s + 0.815·17-s + 1.33·19-s + 0.0228·21-s − 1.63·23-s + 0.0700·25-s − 0.0742·27-s + 1.10·29-s − 0.648·31-s − 0.637·35-s + 0.0595·37-s + 0.0119·39-s − 0.688·41-s − 0.786·43-s + 1.03·45-s − 0.168·47-s − 0.620·49-s + 0.0302·51-s − 0.843·53-s + 0.0497·57-s + 1.05·59-s + 1.13·61-s + ⋯

Functional equation

Λ(s)=(7744s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(7744s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 77447744    =    26⋅1122^{6} \cdot 11^{2}
Sign: −1-1
Analytic conductor: 61.836161.8361
Root analytic conductor: 7.863597.86359
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 7744, ( :1/2), −1)(2,\ 7744,\ (\ :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
11 1 1
good3 1−0.0643T+3T2 1 - 0.0643T + 3T^{2}
5 1+2.31T+5T2 1 + 2.31T + 5T^{2}
7 1−1.63T+7T2 1 - 1.63T + 7T^{2}
13 1−1.15T+13T2 1 - 1.15T + 13T^{2}
17 1−3.36T+17T2 1 - 3.36T + 17T^{2}
19 1−5.83T+19T2 1 - 5.83T + 19T^{2}
23 1+7.86T+23T2 1 + 7.86T + 23T^{2}
29 1−5.92T+29T2 1 - 5.92T + 29T^{2}
31 1+3.61T+31T2 1 + 3.61T + 31T^{2}
37 1−0.362T+37T2 1 - 0.362T + 37T^{2}
41 1+4.40T+41T2 1 + 4.40T + 41T^{2}
43 1+5.15T+43T2 1 + 5.15T + 43T^{2}
47 1+1.15T+47T2 1 + 1.15T + 47T^{2}
53 1+6.14T+53T2 1 + 6.14T + 53T^{2}
59 1−8.12T+59T2 1 - 8.12T + 59T^{2}
61 1−8.82T+61T2 1 - 8.82T + 61T^{2}
67 1−12.6T+67T2 1 - 12.6T + 67T^{2}
71 1+15.0T+71T2 1 + 15.0T + 71T^{2}
73 1−13.0T+73T2 1 - 13.0T + 73T^{2}
79 1−2.07T+79T2 1 - 2.07T + 79T^{2}
83 1+8.94T+83T2 1 + 8.94T + 83T^{2}
89 1−13.9T+89T2 1 - 13.9T + 89T^{2}
97 1−0.00798T+97T2 1 - 0.00798T + 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

−7.73030519142363399042660603829, −6.94221961929963925995740483810, −6.04478043605088898077845986480, −5.37591723460893189669061890469, −4.74336263606720155163104282139, −3.69762718790402184484569366385, −3.37180485772636108063604241743, −2.28524569727403331293985629422, −1.16716977753463265865640322435, 0, 1.16716977753463265865640322435, 2.28524569727403331293985629422, 3.37180485772636108063604241743, 3.69762718790402184484569366385, 4.74336263606720155163104282139, 5.37591723460893189669061890469, 6.04478043605088898077845986480, 6.94221961929963925995740483810, 7.73030519142363399042660603829

Graph of the ZZ-function along the critical line