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Properties

Label 2-483-1.1-c1-0-10
Degree 22
Conductor 483483
Sign 11
Analytic cond. 3.856773.85677
Root an. cond. 1.963861.96386
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  + 0.618·2-s + 3-s − 1.61·4-s + 3.61·5-s + 0.618·6-s + 7-s − 2.23·8-s + 9-s + 2.23·10-s + 11-s − 1.61·12-s + 0.618·13-s + 0.618·14-s + 3.61·15-s + 1.85·16-s − 5.47·17-s + 0.618·18-s + 4.23·19-s − 5.85·20-s + 21-s + 0.618·22-s + 23-s − 2.23·24-s + 8.09·25-s + 0.381·26-s + 27-s − 1.61·28-s + ⋯
L(s)  = 1  + 0.437·2-s + 0.577·3-s − 0.809·4-s + 1.61·5-s + 0.252·6-s + 0.377·7-s − 0.790·8-s + 0.333·9-s + 0.707·10-s + 0.301·11-s − 0.467·12-s + 0.171·13-s + 0.165·14-s + 0.934·15-s + 0.463·16-s − 1.32·17-s + 0.145·18-s + 0.971·19-s − 1.30·20-s + 0.218·21-s + 0.131·22-s + 0.208·23-s − 0.456·24-s + 1.61·25-s + 0.0749·26-s + 0.192·27-s − 0.305·28-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: 11
Analytic conductor: 3.856773.85677
Root analytic conductor: 1.963861.96386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 483, ( :1/2), 1)(2,\ 483,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2928149952.292814995
L(12)L(\frac12) ≈\approx 2.2928149952.292814995
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−T 1 - T
7 1−T 1 - T
23 1−T 1 - T
good2 1−0.618T+2T2 1 - 0.618T + 2T^{2}
5 1−3.61T+5T2 1 - 3.61T + 5T^{2}
11 1−T+11T2 1 - T + 11T^{2}
13 1−0.618T+13T2 1 - 0.618T + 13T^{2}
17 1+5.47T+17T2 1 + 5.47T + 17T^{2}
19 1−4.23T+19T2 1 - 4.23T + 19T^{2}
29 1−1.76T+29T2 1 - 1.76T + 29T^{2}
31 1+8.70T+31T2 1 + 8.70T + 31T^{2}
37 1−0.236T+37T2 1 - 0.236T + 37T^{2}
41 1+3.47T+41T2 1 + 3.47T + 41T^{2}
43 1+3.85T+43T2 1 + 3.85T + 43T^{2}
47 1−11.7T+47T2 1 - 11.7T + 47T^{2}
53 1+0.0901T+53T2 1 + 0.0901T + 53T^{2}
59 1+3.61T+59T2 1 + 3.61T + 59T^{2}
61 1+7.85T+61T2 1 + 7.85T + 61T^{2}
67 1+8.09T+67T2 1 + 8.09T + 67T^{2}
71 1+10.3T+71T2 1 + 10.3T + 71T^{2}
73 1−1.76T+73T2 1 - 1.76T + 73T^{2}
79 1+14.2T+79T2 1 + 14.2T + 79T^{2}
83 1+17.9T+83T2 1 + 17.9T + 83T^{2}
89 1−13.5T+89T2 1 - 13.5T + 89T^{2}
97 1−6.70T+97T2 1 - 6.70T + 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

−10.81527660649945414976973685101, −9.883853675703476396976187449017, −9.102666662268302116038186060727, −8.718857592445549267898173247998, −7.25858215220100463093501573265, −6.09140108542775404097550834575, −5.28527823728134241165798506356, −4.29191789780391757303113677118, −2.94766514301093849313159544074, −1.64314884892201059797517287380, 1.64314884892201059797517287380, 2.94766514301093849313159544074, 4.29191789780391757303113677118, 5.28527823728134241165798506356, 6.09140108542775404097550834575, 7.25858215220100463093501573265, 8.718857592445549267898173247998, 9.102666662268302116038186060727, 9.883853675703476396976187449017, 10.81527660649945414976973685101

Graph of the ZZ-function along the critical line