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Properties

Label 2-862-1.1-c1-0-5
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 − 1.94·3-s + 4-s + 3.69·5-s + 1.94·6-s − 2.11·7-s − 8-s + 0.782·9-s − 3.69·10-s + 1.68·11-s − 1.94·12-s + 4.38·13-s + 2.11·14-s − 7.19·15-s + 16-s + 1.07·17-s − 0.782·18-s + 0.525·19-s + 3.69·20-s + 4.10·21-s − 1.68·22-s − 0.463·23-s + 1.94·24-s + 8.67·25-s − 4.38·26-s + 4.31·27-s − 2.11·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.12·3-s + 0.5·4-s + 1.65·5-s + 0.793·6-s − 0.797·7-s − 0.353·8-s + 0.260·9-s − 1.16·10-s + 0.506·11-s − 0.561·12-s + 1.21·13-s + 0.564·14-s − 1.85·15-s + 0.250·16-s + 0.259·17-s − 0.184·18-s + 0.120·19-s + 0.826·20-s + 0.895·21-s − 0.358·22-s − 0.0967·23-s + 0.396·24-s + 1.73·25-s − 0.859·26-s + 0.830·27-s − 0.398·28-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 0.97672152600.9767215260
L(12)L(\frac12) ≈\approx 0.97672152600.9767215260
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+1.94T+3T2 1 + 1.94T + 3T^{2}
5 1−3.69T+5T2 1 - 3.69T + 5T^{2}
7 1+2.11T+7T2 1 + 2.11T + 7T^{2}
11 1−1.68T+11T2 1 - 1.68T + 11T^{2}
13 1−4.38T+13T2 1 - 4.38T + 13T^{2}
17 1−1.07T+17T2 1 - 1.07T + 17T^{2}
19 1−0.525T+19T2 1 - 0.525T + 19T^{2}
23 1+0.463T+23T2 1 + 0.463T + 23T^{2}
29 1+9.66T+29T2 1 + 9.66T + 29T^{2}
31 1+9.57T+31T2 1 + 9.57T + 31T^{2}
37 1−9.09T+37T2 1 - 9.09T + 37T^{2}
41 1−8.45T+41T2 1 - 8.45T + 41T^{2}
43 1−10.3T+43T2 1 - 10.3T + 43T^{2}
47 1+3.30T+47T2 1 + 3.30T + 47T^{2}
53 1−3.73T+53T2 1 - 3.73T + 53T^{2}
59 1−6.67T+59T2 1 - 6.67T + 59T^{2}
61 1−0.444T+61T2 1 - 0.444T + 61T^{2}
67 1−13.2T+67T2 1 - 13.2T + 67T^{2}
71 1−6.94T+71T2 1 - 6.94T + 71T^{2}
73 1−2.36T+73T2 1 - 2.36T + 73T^{2}
79 1+10.9T+79T2 1 + 10.9T + 79T^{2}
83 1−8.78T+83T2 1 - 8.78T + 83T^{2}
89 1−9.82T+89T2 1 - 9.82T + 89T^{2}
97 1−8.50T+97T2 1 - 8.50T + 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

−10.07821065773379726394383621695, −9.399244311390401173441220164486, −8.908602897429034142208258397766, −7.44970352868895002490540754243, −6.36332400333869620016754066348, −6.01050858668370635061648891260, −5.39848228313376513312921754988, −3.68029316881676265954060674277, −2.21953324977738327032463579201, −0.966330481356713775991703942349, 0.966330481356713775991703942349, 2.21953324977738327032463579201, 3.68029316881676265954060674277, 5.39848228313376513312921754988, 6.01050858668370635061648891260, 6.36332400333869620016754066348, 7.44970352868895002490540754243, 8.908602897429034142208258397766, 9.399244311390401173441220164486, 10.07821065773379726394383621695

Graph of the ZZ-function along the critical line