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Properties

Label 2-862-1.1-c1-0-4
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 − 1.10·3-s + 4-s − 0.0681·5-s + 1.10·6-s + 2.43·7-s − 8-s − 1.78·9-s + 0.0681·10-s + 5.61·11-s − 1.10·12-s + 1.70·13-s − 2.43·14-s + 0.0751·15-s + 16-s − 3.58·17-s + 1.78·18-s − 4.83·19-s − 0.0681·20-s − 2.68·21-s − 5.61·22-s + 4.79·23-s + 1.10·24-s − 4.99·25-s − 1.70·26-s + 5.27·27-s + 2.43·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.637·3-s + 0.5·4-s − 0.0304·5-s + 0.450·6-s + 0.920·7-s − 0.353·8-s − 0.594·9-s + 0.0215·10-s + 1.69·11-s − 0.318·12-s + 0.471·13-s − 0.651·14-s + 0.0194·15-s + 0.250·16-s − 0.870·17-s + 0.420·18-s − 1.10·19-s − 0.0152·20-s − 0.586·21-s − 1.19·22-s + 0.999·23-s + 0.225·24-s − 0.999·25-s − 0.333·26-s + 1.01·27-s + 0.460·28-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 1.0014508911.001450891
L(12)L(\frac12) ≈\approx 1.0014508911.001450891
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+1.10T+3T2 1 + 1.10T + 3T^{2}
5 1+0.0681T+5T2 1 + 0.0681T + 5T^{2}
7 1−2.43T+7T2 1 - 2.43T + 7T^{2}
11 1−5.61T+11T2 1 - 5.61T + 11T^{2}
13 1−1.70T+13T2 1 - 1.70T + 13T^{2}
17 1+3.58T+17T2 1 + 3.58T + 17T^{2}
19 1+4.83T+19T2 1 + 4.83T + 19T^{2}
23 1−4.79T+23T2 1 - 4.79T + 23T^{2}
29 1−3.94T+29T2 1 - 3.94T + 29T^{2}
31 1−1.53T+31T2 1 - 1.53T + 31T^{2}
37 1−3.27T+37T2 1 - 3.27T + 37T^{2}
41 1−4.85T+41T2 1 - 4.85T + 41T^{2}
43 1+4.76T+43T2 1 + 4.76T + 43T^{2}
47 1−9.21T+47T2 1 - 9.21T + 47T^{2}
53 1−12.7T+53T2 1 - 12.7T + 53T^{2}
59 1+8.98T+59T2 1 + 8.98T + 59T^{2}
61 1−12.9T+61T2 1 - 12.9T + 61T^{2}
67 1+12.1T+67T2 1 + 12.1T + 67T^{2}
71 1+1.99T+71T2 1 + 1.99T + 71T^{2}
73 1−6.42T+73T2 1 - 6.42T + 73T^{2}
79 1−6.23T+79T2 1 - 6.23T + 79T^{2}
83 1−16.8T+83T2 1 - 16.8T + 83T^{2}
89 1−9.89T+89T2 1 - 9.89T + 89T^{2}
97 1−11.8T+97T2 1 - 11.8T + 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

−10.33525570528387829556833574922, −8.936959347578442996907878887889, −8.827231399344623107919346487931, −7.71659126151810509269699017379, −6.56155370613151692581545383749, −6.13926162462006156762536054053, −4.86506488943449905879590807788, −3.87525105139685593427920287695, −2.24791839778951874881219456553, −0.963170104323067709732555299310, 0.963170104323067709732555299310, 2.24791839778951874881219456553, 3.87525105139685593427920287695, 4.86506488943449905879590807788, 6.13926162462006156762536054053, 6.56155370613151692581545383749, 7.71659126151810509269699017379, 8.827231399344623107919346487931, 8.936959347578442996907878887889, 10.33525570528387829556833574922

Graph of the ZZ-function along the critical line