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Properties

Label 2-862-1.1-c1-0-13
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 + 2.62·3-s + 4-s + 2.00·5-s − 2.62·6-s + 0.373·7-s − 8-s + 3.90·9-s − 2.00·10-s − 1.73·11-s + 2.62·12-s − 1.83·13-s − 0.373·14-s + 5.27·15-s + 16-s − 1.21·17-s − 3.90·18-s + 7.37·19-s + 2.00·20-s + 0.982·21-s + 1.73·22-s + 8.08·23-s − 2.62·24-s − 0.964·25-s + 1.83·26-s + 2.38·27-s + 0.373·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.51·3-s + 0.5·4-s + 0.898·5-s − 1.07·6-s + 0.141·7-s − 0.353·8-s + 1.30·9-s − 0.635·10-s − 0.521·11-s + 0.758·12-s − 0.509·13-s − 0.0998·14-s + 1.36·15-s + 0.250·16-s − 0.294·17-s − 0.920·18-s + 1.69·19-s + 0.449·20-s + 0.214·21-s + 0.368·22-s + 1.68·23-s − 0.536·24-s − 0.192·25-s + 0.360·26-s + 0.458·27-s + 0.0706·28-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 2.2138982602.213898260
L(12)L(\frac12) ≈\approx 2.2138982602.213898260
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 1−2.62T+3T2 1 - 2.62T + 3T^{2}
5 1−2.00T+5T2 1 - 2.00T + 5T^{2}
7 1−0.373T+7T2 1 - 0.373T + 7T^{2}
11 1+1.73T+11T2 1 + 1.73T + 11T^{2}
13 1+1.83T+13T2 1 + 1.83T + 13T^{2}
17 1+1.21T+17T2 1 + 1.21T + 17T^{2}
19 1−7.37T+19T2 1 - 7.37T + 19T^{2}
23 1−8.08T+23T2 1 - 8.08T + 23T^{2}
29 1−7.03T+29T2 1 - 7.03T + 29T^{2}
31 1+8.79T+31T2 1 + 8.79T + 31T^{2}
37 1+2.45T+37T2 1 + 2.45T + 37T^{2}
41 1−2.74T+41T2 1 - 2.74T + 41T^{2}
43 1−12.4T+43T2 1 - 12.4T + 43T^{2}
47 1+2.86T+47T2 1 + 2.86T + 47T^{2}
53 1+2.22T+53T2 1 + 2.22T + 53T^{2}
59 1+7.28T+59T2 1 + 7.28T + 59T^{2}
61 1−3.58T+61T2 1 - 3.58T + 61T^{2}
67 1+8.67T+67T2 1 + 8.67T + 67T^{2}
71 1+9.48T+71T2 1 + 9.48T + 71T^{2}
73 1+13.2T+73T2 1 + 13.2T + 73T^{2}
79 1+14.9T+79T2 1 + 14.9T + 79T^{2}
83 1+4.24T+83T2 1 + 4.24T + 83T^{2}
89 1−17.4T+89T2 1 - 17.4T + 89T^{2}
97 1−12.1T+97T2 1 - 12.1T + 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

−9.854246965441506830803815933455, −9.197747254150742202007951592306, −8.771025305030022562433349747209, −7.57881940742678820891137873325, −7.29919410717147803557767005143, −5.89444662565118889181734491441, −4.82618173131407098069204345582, −3.22785802056232158671060587440, −2.57731636585566383061267853511, −1.46832194973399051293646783991, 1.46832194973399051293646783991, 2.57731636585566383061267853511, 3.22785802056232158671060587440, 4.82618173131407098069204345582, 5.89444662565118889181734491441, 7.29919410717147803557767005143, 7.57881940742678820891137873325, 8.771025305030022562433349747209, 9.197747254150742202007951592306, 9.854246965441506830803815933455

Graph of the ZZ-function along the critical line