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Properties

Label 2-867-1.1-c1-0-9
Degree 22
Conductor 867867
Sign 11
Analytic cond. 6.923026.92302
Root an. cond. 2.631162.63116
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  + 0.222·2-s + 3-s − 1.95·4-s − 0.636·5-s + 0.222·6-s − 1.72·7-s − 0.877·8-s + 9-s − 0.141·10-s + 4.95·11-s − 1.95·12-s + 2.50·13-s − 0.384·14-s − 0.636·15-s + 3.70·16-s + 0.222·18-s − 0.950·19-s + 1.24·20-s − 1.72·21-s + 1.09·22-s − 2.12·23-s − 0.877·24-s − 4.59·25-s + 0.556·26-s + 27-s + 3.37·28-s + 9.58·29-s + ⋯
L(s)  = 1  + 0.157·2-s + 0.577·3-s − 0.975·4-s − 0.284·5-s + 0.0907·6-s − 0.653·7-s − 0.310·8-s + 0.333·9-s − 0.0447·10-s + 1.49·11-s − 0.563·12-s + 0.695·13-s − 0.102·14-s − 0.164·15-s + 0.926·16-s + 0.0523·18-s − 0.218·19-s + 0.277·20-s − 0.377·21-s + 0.234·22-s − 0.442·23-s − 0.179·24-s − 0.918·25-s + 0.109·26-s + 0.192·27-s + 0.637·28-s + 1.78·29-s + ⋯

Functional equation

Λ(s)=(867s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(867s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 867867    =    3⋅1723 \cdot 17^{2}
Sign: 11
Analytic conductor: 6.923026.92302
Root analytic conductor: 2.631162.63116
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 867, ( :1/2), 1)(2,\ 867,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6018494501.601849450
L(12)L(\frac12) ≈\approx 1.6018494501.601849450
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)
bad3 1−T 1 - T
17 1 1
good2 1−0.222T+2T2 1 - 0.222T + 2T^{2}
5 1+0.636T+5T2 1 + 0.636T + 5T^{2}
7 1+1.72T+7T2 1 + 1.72T + 7T^{2}
11 1−4.95T+11T2 1 - 4.95T + 11T^{2}
13 1−2.50T+13T2 1 - 2.50T + 13T^{2}
19 1+0.950T+19T2 1 + 0.950T + 19T^{2}
23 1+2.12T+23T2 1 + 2.12T + 23T^{2}
29 1−9.58T+29T2 1 - 9.58T + 29T^{2}
31 1−5.27T+31T2 1 - 5.27T + 31T^{2}
37 1−8.48T+37T2 1 - 8.48T + 37T^{2}
41 1−6.92T+41T2 1 - 6.92T + 41T^{2}
43 1−7.15T+43T2 1 - 7.15T + 43T^{2}
47 1−8.10T+47T2 1 - 8.10T + 47T^{2}
53 1+6.44T+53T2 1 + 6.44T + 53T^{2}
59 1+10.1T+59T2 1 + 10.1T + 59T^{2}
61 1−2.96T+61T2 1 - 2.96T + 61T^{2}
67 1−7.70T+67T2 1 - 7.70T + 67T^{2}
71 1+0.384T+71T2 1 + 0.384T + 71T^{2}
73 1−6.51T+73T2 1 - 6.51T + 73T^{2}
79 1+2.37T+79T2 1 + 2.37T + 79T^{2}
83 1+13.3T+83T2 1 + 13.3T + 83T^{2}
89 1+1.98T+89T2 1 + 1.98T + 89T^{2}
97 1−3.04T+97T2 1 - 3.04T + 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

−9.764464811886636511142975033575, −9.396628236284825040726095301219, −8.529264463988429356438407167125, −7.87761658656161682931531229925, −6.56691151042302356558644670277, −5.93932232992331387427846798535, −4.35454511804715088539774576545, −3.98666856142688385078573779875, −2.88260412275820003463458089772, −1.03989225069973365425781144792, 1.03989225069973365425781144792, 2.88260412275820003463458089772, 3.98666856142688385078573779875, 4.35454511804715088539774576545, 5.93932232992331387427846798535, 6.56691151042302356558644670277, 7.87761658656161682931531229925, 8.529264463988429356438407167125, 9.396628236284825040726095301219, 9.764464811886636511142975033575

Graph of the ZZ-function along the critical line