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Properties

Label 2-2156-1.1-c1-0-16
Degree 22
Conductor 21562156
Sign −1-1
Analytic cond. 17.215717.2157
Root an. cond. 4.149184.14918
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  − 1.65·3-s − 2.91·5-s − 0.255·9-s + 11-s − 0.343·13-s + 4.82·15-s + 1.25·17-s + 4.39·19-s + 8.70·23-s + 3.48·25-s + 5.39·27-s + 5.22·29-s − 9.73·31-s − 1.65·33-s + 2.17·37-s + 0.568·39-s − 9.99·41-s − 3.79·43-s + 0.744·45-s − 9.70·47-s − 2.08·51-s − 4.34·53-s − 2.91·55-s − 7.27·57-s + 4.99·59-s + 0.511·61-s + 65-s + ⋯
L(s)  = 1  − 0.956·3-s − 1.30·5-s − 0.0852·9-s + 0.301·11-s − 0.0952·13-s + 1.24·15-s + 0.304·17-s + 1.00·19-s + 1.81·23-s + 0.696·25-s + 1.03·27-s + 0.970·29-s − 1.74·31-s − 0.288·33-s + 0.357·37-s + 0.0910·39-s − 1.56·41-s − 0.578·43-s + 0.110·45-s − 1.41·47-s − 0.291·51-s − 0.596·53-s − 0.392·55-s − 0.964·57-s + 0.649·59-s + 0.0654·61-s + 0.124·65-s + ⋯

Functional equation

Λ(s)=(2156s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2156s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21562156    =    22⋅72⋅112^{2} \cdot 7^{2} \cdot 11
Sign: −1-1
Analytic conductor: 17.215717.2157
Root analytic conductor: 4.149184.14918
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2156, ( :1/2), −1)(2,\ 2156,\ (\ :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
7 1 1
11 1−T 1 - T
good3 1+1.65T+3T2 1 + 1.65T + 3T^{2}
5 1+2.91T+5T2 1 + 2.91T + 5T^{2}
13 1+0.343T+13T2 1 + 0.343T + 13T^{2}
17 1−1.25T+17T2 1 - 1.25T + 17T^{2}
19 1−4.39T+19T2 1 - 4.39T + 19T^{2}
23 1−8.70T+23T2 1 - 8.70T + 23T^{2}
29 1−5.22T+29T2 1 - 5.22T + 29T^{2}
31 1+9.73T+31T2 1 + 9.73T + 31T^{2}
37 1−2.17T+37T2 1 - 2.17T + 37T^{2}
41 1+9.99T+41T2 1 + 9.99T + 41T^{2}
43 1+3.79T+43T2 1 + 3.79T + 43T^{2}
47 1+9.70T+47T2 1 + 9.70T + 47T^{2}
53 1+4.34T+53T2 1 + 4.34T + 53T^{2}
59 1−4.99T+59T2 1 - 4.99T + 59T^{2}
61 1−0.511T+61T2 1 - 0.511T + 61T^{2}
67 1+10.3T+67T2 1 + 10.3T + 67T^{2}
71 1−4.45T+71T2 1 - 4.45T + 71T^{2}
73 1−7.28T+73T2 1 - 7.28T + 73T^{2}
79 1−10.9T+79T2 1 - 10.9T + 79T^{2}
83 1+4.11T+83T2 1 + 4.11T + 83T^{2}
89 1+2.26T+89T2 1 + 2.26T + 89T^{2}
97 1+1.14T+97T2 1 + 1.14T + 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

−8.614711496269650076393770884659, −7.84571927772915964006822975864, −7.06156892111758091911946536898, −6.46144291448875524985003605157, −5.25285577588910344806110232343, −4.91676274859652781926695113938, −3.69916054482749458103697846082, −3.02668771153405704820279894340, −1.19802951458921137436160928554, 0, 1.19802951458921137436160928554, 3.02668771153405704820279894340, 3.69916054482749458103697846082, 4.91676274859652781926695113938, 5.25285577588910344806110232343, 6.46144291448875524985003605157, 7.06156892111758091911946536898, 7.84571927772915964006822975864, 8.614711496269650076393770884659

Graph of the ZZ-function along the critical line