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Properties

Label 2-9680-1.1-c1-0-216
Degree 22
Conductor 96809680
Sign −1-1
Analytic cond. 77.295177.2951
Root an. cond. 8.791768.79176
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  + 2.82·3-s − 5-s + 2.09·7-s + 4.99·9-s − 4.99·13-s − 2.82·15-s + 2.80·19-s + 5.92·21-s − 8.45·23-s + 25-s + 5.62·27-s − 4·29-s − 9.18·31-s − 2.09·35-s − 9.98·37-s − 14.1·39-s + 2.53·41-s + 0.233·43-s − 4.99·45-s − 9.15·47-s − 2.61·49-s + 1.33·53-s + 7.91·57-s − 12.1·59-s + 1.12·61-s + 10.4·63-s + 4.99·65-s + ⋯
L(s)  = 1  + 1.63·3-s − 0.447·5-s + 0.791·7-s + 1.66·9-s − 1.38·13-s − 0.729·15-s + 0.642·19-s + 1.29·21-s − 1.76·23-s + 0.200·25-s + 1.08·27-s − 0.742·29-s − 1.64·31-s − 0.354·35-s − 1.64·37-s − 2.25·39-s + 0.395·41-s + 0.0356·43-s − 0.744·45-s − 1.33·47-s − 0.373·49-s + 0.183·53-s + 1.04·57-s − 1.57·59-s + 0.143·61-s + 1.31·63-s + 0.619·65-s + ⋯

Functional equation

Λ(s)=(9680s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9680s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 96809680    =    24⋅5⋅1122^{4} \cdot 5 \cdot 11^{2}
Sign: −1-1
Analytic conductor: 77.295177.2951
Root analytic conductor: 8.791768.79176
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9680, ( :1/2), −1)(2,\ 9680,\ (\ :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
5 1+T 1 + T
11 1 1
good3 1−2.82T+3T2 1 - 2.82T + 3T^{2}
7 1−2.09T+7T2 1 - 2.09T + 7T^{2}
13 1+4.99T+13T2 1 + 4.99T + 13T^{2}
17 1+17T2 1 + 17T^{2}
19 1−2.80T+19T2 1 - 2.80T + 19T^{2}
23 1+8.45T+23T2 1 + 8.45T + 23T^{2}
29 1+4T+29T2 1 + 4T + 29T^{2}
31 1+9.18T+31T2 1 + 9.18T + 31T^{2}
37 1+9.98T+37T2 1 + 9.98T + 37T^{2}
41 1−2.53T+41T2 1 - 2.53T + 41T^{2}
43 1−0.233T+43T2 1 - 0.233T + 43T^{2}
47 1+9.15T+47T2 1 + 9.15T + 47T^{2}
53 1−1.33T+53T2 1 - 1.33T + 53T^{2}
59 1+12.1T+59T2 1 + 12.1T + 59T^{2}
61 1−1.12T+61T2 1 - 1.12T + 61T^{2}
67 1−1.50T+67T2 1 - 1.50T + 67T^{2}
71 1−7.72T+71T2 1 - 7.72T + 71T^{2}
73 1+8T+73T2 1 + 8T + 73T^{2}
79 1+10.1T+79T2 1 + 10.1T + 79T^{2}
83 1+13.7T+83T2 1 + 13.7T + 83T^{2}
89 1−8.60T+89T2 1 - 8.60T + 89T^{2}
97 1+3.04T+97T2 1 + 3.04T + 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

−7.39978757008716278154453316097, −7.23754813431319223230198700393, −5.94676033882238045205428003437, −5.05480355771892218049849775572, −4.44102239991572346583896289794, −3.62896965604636943885027106008, −3.12530124079580747050494071797, −2.03650857308754127556879998884, −1.76363794217171689242095748292, 0, 1.76363794217171689242095748292, 2.03650857308754127556879998884, 3.12530124079580747050494071797, 3.62896965604636943885027106008, 4.44102239991572346583896289794, 5.05480355771892218049849775572, 5.94676033882238045205428003437, 7.23754813431319223230198700393, 7.39978757008716278154453316097

Graph of the ZZ-function along the critical line