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Properties

Label 2-4304-1.1-c1-0-129
Degree 22
Conductor 43044304
Sign −1-1
Analytic cond. 34.367634.3676
Root an. cond. 5.862385.86238
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  + 1.39·3-s + 1.13·5-s + 0.122·7-s − 1.05·9-s + 1.04·11-s + 1.87·13-s + 1.58·15-s − 4.65·17-s − 4.43·19-s + 0.170·21-s − 5.95·23-s − 3.71·25-s − 5.65·27-s − 4.89·29-s − 2.03·31-s + 1.45·33-s + 0.138·35-s − 5.04·37-s + 2.60·39-s + 10.8·41-s − 1.42·43-s − 1.20·45-s − 8.58·47-s − 6.98·49-s − 6.48·51-s − 1.94·53-s + 1.18·55-s + ⋯
L(s)  = 1  + 0.804·3-s + 0.507·5-s + 0.0462·7-s − 0.352·9-s + 0.314·11-s + 0.518·13-s + 0.408·15-s − 1.12·17-s − 1.01·19-s + 0.0371·21-s − 1.24·23-s − 0.742·25-s − 1.08·27-s − 0.909·29-s − 0.365·31-s + 0.252·33-s + 0.0234·35-s − 0.829·37-s + 0.417·39-s + 1.69·41-s − 0.217·43-s − 0.179·45-s − 1.25·47-s − 0.997·49-s − 0.907·51-s − 0.267·53-s + 0.159·55-s + ⋯

Functional equation

Λ(s)=(4304s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4304s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 43044304    =    24⋅2692^{4} \cdot 269
Sign: −1-1
Analytic conductor: 34.367634.3676
Root analytic conductor: 5.862385.86238
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4304, ( :1/2), −1)(2,\ 4304,\ (\ :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
269 1+T 1 + T
good3 1−1.39T+3T2 1 - 1.39T + 3T^{2}
5 1−1.13T+5T2 1 - 1.13T + 5T^{2}
7 1−0.122T+7T2 1 - 0.122T + 7T^{2}
11 1−1.04T+11T2 1 - 1.04T + 11T^{2}
13 1−1.87T+13T2 1 - 1.87T + 13T^{2}
17 1+4.65T+17T2 1 + 4.65T + 17T^{2}
19 1+4.43T+19T2 1 + 4.43T + 19T^{2}
23 1+5.95T+23T2 1 + 5.95T + 23T^{2}
29 1+4.89T+29T2 1 + 4.89T + 29T^{2}
31 1+2.03T+31T2 1 + 2.03T + 31T^{2}
37 1+5.04T+37T2 1 + 5.04T + 37T^{2}
41 1−10.8T+41T2 1 - 10.8T + 41T^{2}
43 1+1.42T+43T2 1 + 1.42T + 43T^{2}
47 1+8.58T+47T2 1 + 8.58T + 47T^{2}
53 1+1.94T+53T2 1 + 1.94T + 53T^{2}
59 1+10.6T+59T2 1 + 10.6T + 59T^{2}
61 1−5.47T+61T2 1 - 5.47T + 61T^{2}
67 1+7.37T+67T2 1 + 7.37T + 67T^{2}
71 1−5.54T+71T2 1 - 5.54T + 71T^{2}
73 1−9.82T+73T2 1 - 9.82T + 73T^{2}
79 1−6.01T+79T2 1 - 6.01T + 79T^{2}
83 1−8.42T+83T2 1 - 8.42T + 83T^{2}
89 1−1.48T+89T2 1 - 1.48T + 89T^{2}
97 1−1.99T+97T2 1 - 1.99T + 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

−8.119605213142832267197526484703, −7.48079628722363559102276454466, −6.24917647625324350732725807115, −6.15413749882031133590867248805, −4.98714384577896301644551623807, −4.03092270233291571272096262871, −3.42960782874919352302498067565, −2.23467820946110499880276457755, −1.84206838083076597171661082562, 0, 1.84206838083076597171661082562, 2.23467820946110499880276457755, 3.42960782874919352302498067565, 4.03092270233291571272096262871, 4.98714384577896301644551623807, 6.15413749882031133590867248805, 6.24917647625324350732725807115, 7.48079628722363559102276454466, 8.119605213142832267197526484703

Graph of the ZZ-function along the critical line