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Properties

Label 2-147-21.20-c1-0-0
Degree 22
Conductor 147147
Sign −0.907+0.419i-0.907 + 0.419i
Analytic cond. 1.173801.17380
Root an. cond. 1.083421.08342
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.10i·2-s + (−1.13 + 1.30i)3-s − 2.41·4-s − 1.60·5-s + (−2.74 − 2.38i)6-s − 0.870i·8-s + (−0.414 − 2.97i)9-s − 3.37i·10-s + 2.97i·11-s + (2.74 − 3.15i)12-s + 0.317i·13-s + (1.82 − 2.10i)15-s − 2.99·16-s + 3.88·17-s + (6.24 − 0.870i)18-s + 5.22i·19-s + ⋯
L(s)  = 1  + 1.48i·2-s + (−0.656 + 0.754i)3-s − 1.20·4-s − 0.719·5-s + (−1.12 − 0.975i)6-s − 0.307i·8-s + (−0.138 − 0.990i)9-s − 1.06i·10-s + 0.895i·11-s + (0.792 − 0.910i)12-s + 0.0879i·13-s + (0.472 − 0.542i)15-s − 0.749·16-s + 0.941·17-s + (1.47 − 0.205i)18-s + 1.19i·19-s + ⋯

Functional equation

Λ(s)=(147s/2 ΓC(s) L(s)=((−0.907+0.419i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(147s/2 ΓC(s+1/2) L(s)=((−0.907+0.419i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 147147    =    3⋅723 \cdot 7^{2}
Sign: −0.907+0.419i-0.907 + 0.419i
Analytic conductor: 1.173801.17380
Root analytic conductor: 1.083421.08342
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ147(146,⋅)\chi_{147} (146, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 147, ( :1/2), −0.907+0.419i)(2,\ 147,\ (\ :1/2),\ -0.907 + 0.419i)

Particular Values

L(1)L(1) ≈\approx 0.148401−0.674424i0.148401 - 0.674424i
L(12)L(\frac12) ≈\approx 0.148401−0.674424i0.148401 - 0.674424i
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)
bad3 1+(1.13−1.30i)T 1 + (1.13 - 1.30i)T
7 1 1
good2 1−2.10iT−2T2 1 - 2.10iT - 2T^{2}
5 1+1.60T+5T2 1 + 1.60T + 5T^{2}
11 1−2.97iT−11T2 1 - 2.97iT - 11T^{2}
13 1−0.317iT−13T2 1 - 0.317iT - 13T^{2}
17 1−3.88T+17T2 1 - 3.88T + 17T^{2}
19 1−5.22iT−19T2 1 - 5.22iT - 19T^{2}
23 1+1.23iT−23T2 1 + 1.23iT - 23T^{2}
29 1−4.71iT−29T2 1 - 4.71iT - 29T^{2}
31 1−2.61iT−31T2 1 - 2.61iT - 31T^{2}
37 1−9.07T+37T2 1 - 9.07T + 37T^{2}
41 1+6.15T+41T2 1 + 6.15T + 41T^{2}
43 1−2T+43T2 1 - 2T + 43T^{2}
47 1−13.2T+47T2 1 - 13.2T + 47T^{2}
53 1+1.74iT−53T2 1 + 1.74iT - 53T^{2}
59 1−6.43T+59T2 1 - 6.43T + 59T^{2}
61 1+7.70iT−61T2 1 + 7.70iT - 61T^{2}
67 1−2.48T+67T2 1 - 2.48T + 67T^{2}
71 1+4.71iT−71T2 1 + 4.71iT - 71T^{2}
73 1+10.3iT−73T2 1 + 10.3iT - 73T^{2}
79 1+0.828T+79T2 1 + 0.828T + 79T^{2}
83 1−3.60T+83T2 1 - 3.60T + 83T^{2}
89 1+10.3T+89T2 1 + 10.3T + 89T^{2}
97 1+7.25iT−97T2 1 + 7.25iT - 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

−14.19120089691844708489097101548, −12.52500651003821949549821875671, −11.71578978331332446008011809252, −10.44145232429624283406771439238, −9.381365971137434766553906444379, −8.135361291874329589043016370562, −7.20882592285636863191267396531, −6.05191121758066991199647938355, −5.02973151913301200869107537277, −3.90433293438228257629142751242, 0.77338498070463203181121372422, 2.67391096067655859329598825529, 4.14626243564433909347861378325, 5.72756207853992555131417474541, 7.22124737852037633303357433759, 8.368728828773948765938371544100, 9.773620194693128426659457887197, 10.95123487317507977165094015691, 11.50755661072854572913973266899, 12.16909210526343022814400333145

Graph of the ZZ-function along the critical line