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Properties

Label 2-862-1.1-c1-0-7
Degree 22
Conductor 862862
Sign 11
Analytic cond. 6.883106.88310
Root an. cond. 2.623562.62356
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 − 2.23·3-s + 4-s + 5-s − 2.23·6-s + 3.23·7-s + 8-s + 2.00·9-s + 10-s − 0.236·11-s − 2.23·12-s − 3.23·13-s + 3.23·14-s − 2.23·15-s + 16-s + 4.47·17-s + 2.00·18-s + 4.23·19-s + 20-s − 7.23·21-s − 0.236·22-s − 1.76·23-s − 2.23·24-s − 4·25-s − 3.23·26-s + 2.23·27-s + 3.23·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.29·3-s + 0.5·4-s + 0.447·5-s − 0.912·6-s + 1.22·7-s + 0.353·8-s + 0.666·9-s + 0.316·10-s − 0.0711·11-s − 0.645·12-s − 0.897·13-s + 0.864·14-s − 0.577·15-s + 0.250·16-s + 1.08·17-s + 0.471·18-s + 0.971·19-s + 0.223·20-s − 1.57·21-s − 0.0503·22-s − 0.367·23-s − 0.456·24-s − 0.800·25-s − 0.634·26-s + 0.430·27-s + 0.611·28-s + ⋯

Functional equation

Λ(s)=(862s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(862s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 862862    =    2⋅4312 \cdot 431
Sign: 11
Analytic conductor: 6.883106.88310
Root analytic conductor: 2.623562.62356
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 862, ( :1/2), 1)(2,\ 862,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9344191421.934419142
L(12)L(\frac12) ≈\approx 1.9344191421.934419142
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
431 1+T 1 + T
good3 1+2.23T+3T2 1 + 2.23T + 3T^{2}
5 1−T+5T2 1 - T + 5T^{2}
7 1−3.23T+7T2 1 - 3.23T + 7T^{2}
11 1+0.236T+11T2 1 + 0.236T + 11T^{2}
13 1+3.23T+13T2 1 + 3.23T + 13T^{2}
17 1−4.47T+17T2 1 - 4.47T + 17T^{2}
19 1−4.23T+19T2 1 - 4.23T + 19T^{2}
23 1+1.76T+23T2 1 + 1.76T + 23T^{2}
29 1−5T+29T2 1 - 5T + 29T^{2}
31 1−2T+31T2 1 - 2T + 31T^{2}
37 1−6T+37T2 1 - 6T + 37T^{2}
41 1+4.47T+41T2 1 + 4.47T + 41T^{2}
43 1−3.23T+43T2 1 - 3.23T + 43T^{2}
47 1−4T+47T2 1 - 4T + 47T^{2}
53 1−1.47T+53T2 1 - 1.47T + 53T^{2}
59 1−11.1T+59T2 1 - 11.1T + 59T^{2}
61 1−4.47T+61T2 1 - 4.47T + 61T^{2}
67 1−8T+67T2 1 - 8T + 67T^{2}
71 1−2.76T+71T2 1 - 2.76T + 71T^{2}
73 1+6.47T+73T2 1 + 6.47T + 73T^{2}
79 1+7.52T+79T2 1 + 7.52T + 79T^{2}
83 1−8.47T+83T2 1 - 8.47T + 83T^{2}
89 1+4T+89T2 1 + 4T + 89T^{2}
97 1+5.94T+97T2 1 + 5.94T + 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

−10.32481702134480335745756151991, −9.723605817208830015907629702348, −8.204352759652178116214598202661, −7.44953280414046437937874287737, −6.44116444032137020041614035301, −5.40668935333792939185731574652, −5.22964447690082351168695655388, −4.15617552990097400756556311261, −2.57529465246915388061748919956, −1.17917255355302963741113192973, 1.17917255355302963741113192973, 2.57529465246915388061748919956, 4.15617552990097400756556311261, 5.22964447690082351168695655388, 5.40668935333792939185731574652, 6.44116444032137020041614035301, 7.44953280414046437937874287737, 8.204352759652178116214598202661, 9.723605817208830015907629702348, 10.32481702134480335745756151991

Graph of the ZZ-function along the critical line