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Properties

Label 2-1125-1.1-c1-0-22
Degree 22
Conductor 11251125
Sign −1-1
Analytic cond. 8.983178.98317
Root an. cond. 2.997192.99719
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  − 0.618·2-s − 1.61·4-s − 3·7-s + 2.23·8-s + 3·11-s + 1.85·13-s + 1.85·14-s + 1.85·16-s + 0.236·17-s − 1.38·19-s − 1.85·22-s − 3.23·23-s − 1.14·26-s + 4.85·28-s + 6.70·29-s − 6.09·31-s − 5.61·32-s − 0.145·34-s − 9.70·37-s + 0.854·38-s + 3·41-s − 9·43-s − 4.85·44-s + 2.00·46-s − 7.32·47-s + 2·49-s − 3·52-s + ⋯
L(s)  = 1  − 0.437·2-s − 0.809·4-s − 1.13·7-s + 0.790·8-s + 0.904·11-s + 0.514·13-s + 0.495·14-s + 0.463·16-s + 0.0572·17-s − 0.317·19-s − 0.395·22-s − 0.674·23-s − 0.224·26-s + 0.917·28-s + 1.24·29-s − 1.09·31-s − 0.993·32-s − 0.0250·34-s − 1.59·37-s + 0.138·38-s + 0.468·41-s − 1.37·43-s − 0.731·44-s + 0.294·46-s − 1.06·47-s + 0.285·49-s − 0.416·52-s + ⋯

Functional equation

Λ(s)=(1125s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1125s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11251125    =    32⋅533^{2} \cdot 5^{3}
Sign: −1-1
Analytic conductor: 8.983178.98317
Root analytic conductor: 2.997192.99719
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1125, ( :1/2), −1)(2,\ 1125,\ (\ :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)
bad3 1 1
5 1 1
good2 1+0.618T+2T2 1 + 0.618T + 2T^{2}
7 1+3T+7T2 1 + 3T + 7T^{2}
11 1−3T+11T2 1 - 3T + 11T^{2}
13 1−1.85T+13T2 1 - 1.85T + 13T^{2}
17 1−0.236T+17T2 1 - 0.236T + 17T^{2}
19 1+1.38T+19T2 1 + 1.38T + 19T^{2}
23 1+3.23T+23T2 1 + 3.23T + 23T^{2}
29 1−6.70T+29T2 1 - 6.70T + 29T^{2}
31 1+6.09T+31T2 1 + 6.09T + 31T^{2}
37 1+9.70T+37T2 1 + 9.70T + 37T^{2}
41 1−3T+41T2 1 - 3T + 41T^{2}
43 1+9T+43T2 1 + 9T + 43T^{2}
47 1+7.32T+47T2 1 + 7.32T + 47T^{2}
53 1+2.38T+53T2 1 + 2.38T + 53T^{2}
59 1+10.8T+59T2 1 + 10.8T + 59T^{2}
61 1−5.09T+61T2 1 - 5.09T + 61T^{2}
67 1+7.14T+67T2 1 + 7.14T + 67T^{2}
71 1−3T+71T2 1 - 3T + 71T^{2}
73 1+4.85T+73T2 1 + 4.85T + 73T^{2}
79 1−9.47T+79T2 1 - 9.47T + 79T^{2}
83 1−8.47T+83T2 1 - 8.47T + 83T^{2}
89 1+13.4T+89T2 1 + 13.4T + 89T^{2}
97 1−1.14T+97T2 1 - 1.14T + 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

−9.364548413281520664964006656047, −8.767499683874449884153001477285, −7.982644500631273399449482634357, −6.83618840359182586655778819742, −6.20349528195766851787401906927, −5.06964172214320127073450208450, −3.99124950656939240773113677567, −3.28553482112215943120812418959, −1.53511278705354164363266331654, 0, 1.53511278705354164363266331654, 3.28553482112215943120812418959, 3.99124950656939240773113677567, 5.06964172214320127073450208450, 6.20349528195766851787401906927, 6.83618840359182586655778819742, 7.982644500631273399449482634357, 8.767499683874449884153001477285, 9.364548413281520664964006656047

Graph of the ZZ-function along the critical line