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Properties

Label 2-6080-1.1-c1-0-121
Degree 22
Conductor 60806080
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.642·3-s − 5-s + 3.58·7-s − 2.58·9-s − 1.35·13-s − 0.642·15-s + 5.58·17-s − 19-s + 2.30·21-s − 4.87·23-s + 25-s − 3.58·27-s − 9.58·29-s − 7.17·31-s − 3.58·35-s + 0.945·37-s − 0.871·39-s + 10.4·41-s + 2.71·43-s + 2.58·45-s − 5.89·47-s + 5.87·49-s + 3.58·51-s − 9.81·53-s − 0.642·57-s − 10.1·59-s − 3.28·61-s + ⋯
L(s)  = 1  + 0.370·3-s − 0.447·5-s + 1.35·7-s − 0.862·9-s − 0.376·13-s − 0.165·15-s + 1.35·17-s − 0.229·19-s + 0.502·21-s − 1.01·23-s + 0.200·25-s − 0.690·27-s − 1.78·29-s − 1.28·31-s − 0.606·35-s + 0.155·37-s − 0.139·39-s + 1.63·41-s + 0.414·43-s + 0.385·45-s − 0.859·47-s + 0.838·49-s + 0.502·51-s − 1.34·53-s − 0.0850·57-s − 1.32·59-s − 0.420·61-s + ⋯

Functional equation

Λ(s)=(6080s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(6080s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\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    =    26⋅5⋅192^{6} \cdot 5 \cdot 19
Sign: −1-1
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: 11
Selberg data: (2, 6080, ( :1/2), −1)(2,\ 6080,\ (\ :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
19 1+T 1 + T
good3 1−0.642T+3T2 1 - 0.642T + 3T^{2}
7 1−3.58T+7T2 1 - 3.58T + 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+1.35T+13T2 1 + 1.35T + 13T^{2}
17 1−5.58T+17T2 1 - 5.58T + 17T^{2}
23 1+4.87T+23T2 1 + 4.87T + 23T^{2}
29 1+9.58T+29T2 1 + 9.58T + 29T^{2}
31 1+7.17T+31T2 1 + 7.17T + 31T^{2}
37 1−0.945T+37T2 1 - 0.945T + 37T^{2}
41 1−10.4T+41T2 1 - 10.4T + 41T^{2}
43 1−2.71T+43T2 1 - 2.71T + 43T^{2}
47 1+5.89T+47T2 1 + 5.89T + 47T^{2}
53 1+9.81T+53T2 1 + 9.81T + 53T^{2}
59 1+10.1T+59T2 1 + 10.1T + 59T^{2}
61 1+3.28T+61T2 1 + 3.28T + 61T^{2}
67 1+10.3T+67T2 1 + 10.3T + 67T^{2}
71 1−14.3T+71T2 1 - 14.3T + 71T^{2}
73 1+4.15T+73T2 1 + 4.15T + 73T^{2}
79 1−1.28T+79T2 1 - 1.28T + 79T^{2}
83 1−11.1T+83T2 1 - 11.1T + 83T^{2}
89 1+6.45T+89T2 1 + 6.45T + 89T^{2}
97 1+13.4T+97T2 1 + 13.4T + 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

−7.76594381346805826000289704132, −7.45962995331873622906375596586, −6.11369829933117776505833896818, −5.55074601115719497442076162762, −4.84967987306799960429711169877, −3.97125873834504043717718322731, −3.28742650705999088550005517886, −2.26073590758127434110668982171, −1.47421181480811057726730724861, 0, 1.47421181480811057726730724861, 2.26073590758127434110668982171, 3.28742650705999088550005517886, 3.97125873834504043717718322731, 4.84967987306799960429711169877, 5.55074601115719497442076162762, 6.11369829933117776505833896818, 7.45962995331873622906375596586, 7.76594381346805826000289704132

Graph of the ZZ-function along the critical line