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Properties

Label 2-862-1.1-c1-0-19
Degree 22
Conductor 862862
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.74·3-s + 4-s + 2.20·5-s + 1.74·6-s + 0.156·7-s − 8-s + 0.0472·9-s − 2.20·10-s + 0.652·11-s − 1.74·12-s − 6.50·13-s − 0.156·14-s − 3.84·15-s + 16-s − 0.464·17-s − 0.0472·18-s + 5.83·19-s + 2.20·20-s − 0.273·21-s − 0.652·22-s − 8.89·23-s + 1.74·24-s − 0.138·25-s + 6.50·26-s + 5.15·27-s + 0.156·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.00·3-s + 0.5·4-s + 0.986·5-s + 0.712·6-s + 0.0592·7-s − 0.353·8-s + 0.0157·9-s − 0.697·10-s + 0.196·11-s − 0.503·12-s − 1.80·13-s − 0.0418·14-s − 0.993·15-s + 0.250·16-s − 0.112·17-s − 0.0111·18-s + 1.33·19-s + 0.493·20-s − 0.0596·21-s − 0.139·22-s − 1.85·23-s + 0.356·24-s − 0.0277·25-s + 1.27·26-s + 0.991·27-s + 0.0296·28-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: −1-1
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: 11
Selberg data: (2, 862, ( :1/2), −1)(2,\ 862,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+1.74T+3T2 1 + 1.74T + 3T^{2}
5 1−2.20T+5T2 1 - 2.20T + 5T^{2}
7 1−0.156T+7T2 1 - 0.156T + 7T^{2}
11 1−0.652T+11T2 1 - 0.652T + 11T^{2}
13 1+6.50T+13T2 1 + 6.50T + 13T^{2}
17 1+0.464T+17T2 1 + 0.464T + 17T^{2}
19 1−5.83T+19T2 1 - 5.83T + 19T^{2}
23 1+8.89T+23T2 1 + 8.89T + 23T^{2}
29 1−0.0633T+29T2 1 - 0.0633T + 29T^{2}
31 1−1.92T+31T2 1 - 1.92T + 31T^{2}
37 1+2.74T+37T2 1 + 2.74T + 37T^{2}
41 1−0.948T+41T2 1 - 0.948T + 41T^{2}
43 1+5.66T+43T2 1 + 5.66T + 43T^{2}
47 1−13.3T+47T2 1 - 13.3T + 47T^{2}
53 1+12.5T+53T2 1 + 12.5T + 53T^{2}
59 1+11.7T+59T2 1 + 11.7T + 59T^{2}
61 1+0.890T+61T2 1 + 0.890T + 61T^{2}
67 1+11.0T+67T2 1 + 11.0T + 67T^{2}
71 1+0.438T+71T2 1 + 0.438T + 71T^{2}
73 1−3.32T+73T2 1 - 3.32T + 73T^{2}
79 1+6.31T+79T2 1 + 6.31T + 79T^{2}
83 1+10.4T+83T2 1 + 10.4T + 83T^{2}
89 1+15.7T+89T2 1 + 15.7T + 89T^{2}
97 1+2.07T+97T2 1 + 2.07T + 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

−9.881372789015223040305866055191, −9.207747246209854060277454102041, −7.973780224038943052407613768862, −7.16453963526854979932401062916, −6.16498129826795321633908221129, −5.58078163031941193791032685974, −4.65568393819433903031570293196, −2.85688414970249145040367873531, −1.69916984989903885174403769138, 0, 1.69916984989903885174403769138, 2.85688414970249145040367873531, 4.65568393819433903031570293196, 5.58078163031941193791032685974, 6.16498129826795321633908221129, 7.16453963526854979932401062916, 7.973780224038943052407613768862, 9.207747246209854060277454102041, 9.881372789015223040305866055191

Graph of the ZZ-function along the critical line