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Properties

Label 2-4230-1.1-c1-0-0
Degree 22
Conductor 42304230
Sign 11
Analytic cond. 33.776733.7767
Root an. cond. 5.811775.81177
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  − 2-s + 4-s − 5-s − 2.28·7-s − 8-s + 10-s − 5.31·11-s − 5.59·13-s + 2.28·14-s + 16-s − 4.76·17-s + 1.81·19-s − 20-s + 5.31·22-s − 1.81·23-s + 25-s + 5.59·26-s − 2.28·28-s − 4.76·29-s − 0.833·31-s − 32-s + 4.76·34-s + 2.28·35-s + 1.52·37-s − 1.81·38-s + 40-s − 9.12·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.864·7-s − 0.353·8-s + 0.316·10-s − 1.60·11-s − 1.55·13-s + 0.611·14-s + 0.250·16-s − 1.15·17-s + 0.415·19-s − 0.223·20-s + 1.13·22-s − 0.377·23-s + 0.200·25-s + 1.09·26-s − 0.432·28-s − 0.884·29-s − 0.149·31-s − 0.176·32-s + 0.817·34-s + 0.386·35-s + 0.250·37-s − 0.293·38-s + 0.158·40-s − 1.42·41-s + ⋯

Functional equation

Λ(s)=(4230s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4230s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 42304230    =    2⋅32⋅5⋅472 \cdot 3^{2} \cdot 5 \cdot 47
Sign: 11
Analytic conductor: 33.776733.7767
Root analytic conductor: 5.811775.81177
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4230, ( :1/2), 1)(2,\ 4230,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.20357962950.2035796295
L(12)L(\frac12) ≈\approx 0.20357962950.2035796295
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+T 1 + T
3 1 1
5 1+T 1 + T
47 1+T 1 + T
good7 1+2.28T+7T2 1 + 2.28T + 7T^{2}
11 1+5.31T+11T2 1 + 5.31T + 11T^{2}
13 1+5.59T+13T2 1 + 5.59T + 13T^{2}
17 1+4.76T+17T2 1 + 4.76T + 17T^{2}
19 1−1.81T+19T2 1 - 1.81T + 19T^{2}
23 1+1.81T+23T2 1 + 1.81T + 23T^{2}
29 1+4.76T+29T2 1 + 4.76T + 29T^{2}
31 1+0.833T+31T2 1 + 0.833T + 31T^{2}
37 1−1.52T+37T2 1 - 1.52T + 37T^{2}
41 1+9.12T+41T2 1 + 9.12T + 41T^{2}
43 1−0.476T+43T2 1 - 0.476T + 43T^{2}
53 1+4.76T+53T2 1 + 4.76T + 53T^{2}
59 1+1.81T+59T2 1 + 1.81T + 59T^{2}
61 1−12.7T+61T2 1 - 12.7T + 61T^{2}
67 1+4.09T+67T2 1 + 4.09T + 67T^{2}
71 1−6.86T+71T2 1 - 6.86T + 71T^{2}
73 1+6.09T+73T2 1 + 6.09T + 73T^{2}
79 1−15.1T+79T2 1 - 15.1T + 79T^{2}
83 1−8T+83T2 1 - 8T + 83T^{2}
89 1−8.62T+89T2 1 - 8.62T + 89T^{2}
97 1−16.2T+97T2 1 - 16.2T + 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

−8.251664159803859009443287971722, −7.74567189112693695114841345755, −7.08635597091231373926120623417, −6.46261116250379047016845327529, −5.38154290108312139239686917998, −4.80367228353826690856478237482, −3.61702248263725479266290864634, −2.73425666963857501696261937854, −2.09064740265861626043080291762, −0.25978703992295786536531629789, 0.25978703992295786536531629789, 2.09064740265861626043080291762, 2.73425666963857501696261937854, 3.61702248263725479266290864634, 4.80367228353826690856478237482, 5.38154290108312139239686917998, 6.46261116250379047016845327529, 7.08635597091231373926120623417, 7.74567189112693695114841345755, 8.251664159803859009443287971722

Graph of the ZZ-function along the critical line