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Properties

Label 2-862-1.1-c1-0-2
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 − 0.705·3-s + 4-s + 0.580·5-s + 0.705·6-s − 4.68·7-s − 8-s − 2.50·9-s − 0.580·10-s − 4.95·11-s − 0.705·12-s + 1.86·13-s + 4.68·14-s − 0.409·15-s + 16-s + 6.13·17-s + 2.50·18-s + 5.02·19-s + 0.580·20-s + 3.30·21-s + 4.95·22-s + 1.58·23-s + 0.705·24-s − 4.66·25-s − 1.86·26-s + 3.88·27-s − 4.68·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.407·3-s + 0.5·4-s + 0.259·5-s + 0.287·6-s − 1.77·7-s − 0.353·8-s − 0.834·9-s − 0.183·10-s − 1.49·11-s − 0.203·12-s + 0.515·13-s + 1.25·14-s − 0.105·15-s + 0.250·16-s + 1.48·17-s + 0.589·18-s + 1.15·19-s + 0.129·20-s + 0.721·21-s + 1.05·22-s + 0.330·23-s + 0.143·24-s − 0.932·25-s − 0.364·26-s + 0.746·27-s − 0.885·28-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 0.63474344770.6347434477
L(12)L(\frac12) ≈\approx 0.63474344770.6347434477
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+0.705T+3T2 1 + 0.705T + 3T^{2}
5 1−0.580T+5T2 1 - 0.580T + 5T^{2}
7 1+4.68T+7T2 1 + 4.68T + 7T^{2}
11 1+4.95T+11T2 1 + 4.95T + 11T^{2}
13 1−1.86T+13T2 1 - 1.86T + 13T^{2}
17 1−6.13T+17T2 1 - 6.13T + 17T^{2}
19 1−5.02T+19T2 1 - 5.02T + 19T^{2}
23 1−1.58T+23T2 1 - 1.58T + 23T^{2}
29 1−10.7T+29T2 1 - 10.7T + 29T^{2}
31 1−4.46T+31T2 1 - 4.46T + 31T^{2}
37 1−5.89T+37T2 1 - 5.89T + 37T^{2}
41 1+10.5T+41T2 1 + 10.5T + 41T^{2}
43 1+4.45T+43T2 1 + 4.45T + 43T^{2}
47 1−0.121T+47T2 1 - 0.121T + 47T^{2}
53 1−13.5T+53T2 1 - 13.5T + 53T^{2}
59 1+11.4T+59T2 1 + 11.4T + 59T^{2}
61 1+3.30T+61T2 1 + 3.30T + 61T^{2}
67 1−0.357T+67T2 1 - 0.357T + 67T^{2}
71 1−5.06T+71T2 1 - 5.06T + 71T^{2}
73 1+1.97T+73T2 1 + 1.97T + 73T^{2}
79 1+2.37T+79T2 1 + 2.37T + 79T^{2}
83 1−15.4T+83T2 1 - 15.4T + 83T^{2}
89 1−10.1T+89T2 1 - 10.1T + 89T^{2}
97 1+2.76T+97T2 1 + 2.76T + 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

−10.09305017133137957179507555355, −9.581974918194055275186273259019, −8.472388505733419348611063415350, −7.73868174526617357018100207316, −6.64133837541557826828257554048, −5.92953068727667598029361917153, −5.22547004904049135923894834855, −3.26697712106623518898772594406, −2.79552848452139372375724727939, −0.69731509541020513539611564516, 0.69731509541020513539611564516, 2.79552848452139372375724727939, 3.26697712106623518898772594406, 5.22547004904049135923894834855, 5.92953068727667598029361917153, 6.64133837541557826828257554048, 7.73868174526617357018100207316, 8.472388505733419348611063415350, 9.581974918194055275186273259019, 10.09305017133137957179507555355

Graph of the ZZ-function along the critical line