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Properties

Label 2-9000-1.1-c1-0-37
Degree 22
Conductor 90009000
Sign 11
Analytic cond. 71.865371.8653
Root an. cond. 8.477348.47734
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  + 3.61·7-s + 11-s − 4.23·13-s − 2.85·17-s − 1.76·19-s + 4.70·23-s + 6.23·29-s − 10.0·31-s − 6.23·37-s − 2.09·41-s + 9.85·43-s − 2.23·47-s + 6.09·49-s + 3.38·53-s + 5.14·59-s − 2.85·61-s + 10.4·67-s + 9.56·71-s + 5.85·73-s + 3.61·77-s + 13.7·79-s + 12.0·83-s + 11·89-s − 15.3·91-s − 7.79·97-s + 16.9·101-s + 7.38·103-s + ⋯
L(s)  = 1  + 1.36·7-s + 0.301·11-s − 1.17·13-s − 0.692·17-s − 0.404·19-s + 0.981·23-s + 1.15·29-s − 1.81·31-s − 1.02·37-s − 0.326·41-s + 1.50·43-s − 0.326·47-s + 0.870·49-s + 0.464·53-s + 0.669·59-s − 0.365·61-s + 1.27·67-s + 1.13·71-s + 0.685·73-s + 0.412·77-s + 1.54·79-s + 1.32·83-s + 1.16·89-s − 1.60·91-s − 0.791·97-s + 1.68·101-s + 0.727·103-s + ⋯

Functional equation

Λ(s)=(9000s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9000s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 90009000    =    23⋅32⋅532^{3} \cdot 3^{2} \cdot 5^{3}
Sign: 11
Analytic conductor: 71.865371.8653
Root analytic conductor: 8.477348.47734
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9000, ( :1/2), 1)(2,\ 9000,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2539385092.253938509
L(12)L(\frac12) ≈\approx 2.2539385092.253938509
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 1
3 1 1
5 1 1
good7 1−3.61T+7T2 1 - 3.61T + 7T^{2}
11 1−T+11T2 1 - T + 11T^{2}
13 1+4.23T+13T2 1 + 4.23T + 13T^{2}
17 1+2.85T+17T2 1 + 2.85T + 17T^{2}
19 1+1.76T+19T2 1 + 1.76T + 19T^{2}
23 1−4.70T+23T2 1 - 4.70T + 23T^{2}
29 1−6.23T+29T2 1 - 6.23T + 29T^{2}
31 1+10.0T+31T2 1 + 10.0T + 31T^{2}
37 1+6.23T+37T2 1 + 6.23T + 37T^{2}
41 1+2.09T+41T2 1 + 2.09T + 41T^{2}
43 1−9.85T+43T2 1 - 9.85T + 43T^{2}
47 1+2.23T+47T2 1 + 2.23T + 47T^{2}
53 1−3.38T+53T2 1 - 3.38T + 53T^{2}
59 1−5.14T+59T2 1 - 5.14T + 59T^{2}
61 1+2.85T+61T2 1 + 2.85T + 61T^{2}
67 1−10.4T+67T2 1 - 10.4T + 67T^{2}
71 1−9.56T+71T2 1 - 9.56T + 71T^{2}
73 1−5.85T+73T2 1 - 5.85T + 73T^{2}
79 1−13.7T+79T2 1 - 13.7T + 79T^{2}
83 1−12.0T+83T2 1 - 12.0T + 83T^{2}
89 1−11T+89T2 1 - 11T + 89T^{2}
97 1+7.79T+97T2 1 + 7.79T + 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

−7.77288916589702870102809899529, −7.04572919765603936892131118034, −6.56912221184652781840814351901, −5.38947060420747885596139063092, −5.02935715778046591858223521669, −4.36883671080791157387114132842, −3.53481967717745791530523888771, −2.39349678620666597099252761861, −1.88111596377499958923950916183, −0.72624096277977009652866925385, 0.72624096277977009652866925385, 1.88111596377499958923950916183, 2.39349678620666597099252761861, 3.53481967717745791530523888771, 4.36883671080791157387114132842, 5.02935715778046591858223521669, 5.38947060420747885596139063092, 6.56912221184652781840814351901, 7.04572919765603936892131118034, 7.77288916589702870102809899529

Graph of the ZZ-function along the critical line