Properties

Label 2-6080-1.1-c1-0-21
Degree 22
Conductor 60806080
Sign 11
Analytic cond. 48.549048.5490
Root an. cond. 6.967716.96771
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.585·3-s − 5-s + 4.82·7-s − 2.65·9-s − 2·11-s − 2.24·13-s + 0.585·15-s − 4.82·17-s − 19-s − 2.82·21-s + 6·23-s + 25-s + 3.31·27-s − 10.4·29-s + 1.17·31-s + 1.17·33-s − 4.82·35-s + 10.2·37-s + 1.31·39-s − 7.65·41-s − 0.828·43-s + 2.65·45-s − 0.828·47-s + 16.3·49-s + 2.82·51-s + 5.07·53-s + 2·55-s + ⋯
L(s)  = 1  − 0.338·3-s − 0.447·5-s + 1.82·7-s − 0.885·9-s − 0.603·11-s − 0.621·13-s + 0.151·15-s − 1.17·17-s − 0.229·19-s − 0.617·21-s + 1.25·23-s + 0.200·25-s + 0.637·27-s − 1.94·29-s + 0.210·31-s + 0.203·33-s − 0.816·35-s + 1.68·37-s + 0.210·39-s − 1.19·41-s − 0.126·43-s + 0.396·45-s − 0.120·47-s + 2.33·49-s + 0.396·51-s + 0.696·53-s + 0.269·55-s + ⋯

Functional equation

Λ(s)=(6080s/2ΓC(s)L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6080s/2ΓC(s+1/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 60806080    =    265192^{6} \cdot 5 \cdot 19
Sign: 11
Analytic conductor: 48.549048.5490
Root analytic conductor: 6.967716.96771
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 6080, ( :1/2), 1)(2,\ 6080,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) \approx 1.3871963861.387196386
L(12)L(\frac12) \approx 1.3871963861.387196386
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1+T 1 + T
19 1+T 1 + T
good3 1+0.585T+3T2 1 + 0.585T + 3T^{2}
7 14.82T+7T2 1 - 4.82T + 7T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
13 1+2.24T+13T2 1 + 2.24T + 13T^{2}
17 1+4.82T+17T2 1 + 4.82T + 17T^{2}
23 16T+23T2 1 - 6T + 23T^{2}
29 1+10.4T+29T2 1 + 10.4T + 29T^{2}
31 11.17T+31T2 1 - 1.17T + 31T^{2}
37 110.2T+37T2 1 - 10.2T + 37T^{2}
41 1+7.65T+41T2 1 + 7.65T + 41T^{2}
43 1+0.828T+43T2 1 + 0.828T + 43T^{2}
47 1+0.828T+47T2 1 + 0.828T + 47T^{2}
53 15.07T+53T2 1 - 5.07T + 53T^{2}
59 12.34T+59T2 1 - 2.34T + 59T^{2}
61 12.34T+61T2 1 - 2.34T + 61T^{2}
67 1+0.585T+67T2 1 + 0.585T + 67T^{2}
71 1+10.8T+71T2 1 + 10.8T + 71T^{2}
73 114.4T+73T2 1 - 14.4T + 73T^{2}
79 19.65T+79T2 1 - 9.65T + 79T^{2}
83 19.31T+83T2 1 - 9.31T + 83T^{2}
89 110.4T+89T2 1 - 10.4T + 89T^{2}
97 1+1.75T+97T2 1 + 1.75T + 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−7.947861460022356608331365400666, −7.56816008764371744985267929475, −6.73293544784854002754262875083, −5.76684419830009614012899265730, −4.98504766596448602259716081332, −4.77616891203080074289430267124, −3.75088571733376059673956504636, −2.59335884053265231702517697252, −1.93929606032299414233200589226, −0.61401219380394009344543808522, 0.61401219380394009344543808522, 1.93929606032299414233200589226, 2.59335884053265231702517697252, 3.75088571733376059673956504636, 4.77616891203080074289430267124, 4.98504766596448602259716081332, 5.76684419830009614012899265730, 6.73293544784854002754262875083, 7.56816008764371744985267929475, 7.947861460022356608331365400666

Graph of the ZZ-function along the critical line