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Properties

Label 2-208725-1.1-c1-0-14
Degree 22
Conductor 208725208725
Sign −1-1
Analytic cond. 1666.671666.67
Root an. cond. 40.824940.8249
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s + 2·6-s − 5·7-s + 9-s − 2·12-s − 6·13-s + 10·14-s − 4·16-s + 17-s − 2·18-s − 2·19-s + 5·21-s + 23-s + 12·26-s − 27-s − 10·28-s + 29-s − 5·31-s + 8·32-s − 2·34-s + 2·36-s + 7·37-s + 4·38-s + 6·39-s + 7·41-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s − 1.88·7-s + 1/3·9-s − 0.577·12-s − 1.66·13-s + 2.67·14-s − 16-s + 0.242·17-s − 0.471·18-s − 0.458·19-s + 1.09·21-s + 0.208·23-s + 2.35·26-s − 0.192·27-s − 1.88·28-s + 0.185·29-s − 0.898·31-s + 1.41·32-s − 0.342·34-s + 1/3·36-s + 1.15·37-s + 0.648·38-s + 0.960·39-s + 1.09·41-s + ⋯

Functional equation

Λ(s)=(208725s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(208725s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 208725208725    =    3⋅52⋅112⋅233 \cdot 5^{2} \cdot 11^{2} \cdot 23
Sign: −1-1
Analytic conductor: 1666.671666.67
Root analytic conductor: 40.824940.8249
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 208725, ( :1/2), −1)(2,\ 208725,\ (\ :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+T 1 + T
5 1 1
11 1 1
23 1−T 1 - T
good2 1+pT+pT2 1 + p T + p T^{2}
7 1+5T+pT2 1 + 5 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1−T+pT2 1 - T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
29 1−T+pT2 1 - T + p T^{2}
31 1+5T+pT2 1 + 5 T + p T^{2}
37 1−7T+pT2 1 - 7 T + p T^{2}
41 1−7T+pT2 1 - 7 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1+9T+pT2 1 + 9 T + p T^{2}
59 1−3T+pT2 1 - 3 T + p T^{2}
61 1+12T+pT2 1 + 12 T + p T^{2}
67 1−T+pT2 1 - T + p T^{2}
71 1−5T+pT2 1 - 5 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−3T+pT2 1 - 3 T + p T^{2}
89 1−8T+pT2 1 - 8 T + p T^{2}
97 1+14T+pT2 1 + 14 T + p T^{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

−13.06284212792602, −12.68165504942170, −12.25288414920217, −11.92329346603056, −11.13106971259212, −10.75147340453941, −10.32412946485382, −9.814469451550098, −9.541750345402851, −9.253902604003219, −8.723526752503129, −7.980156188337239, −7.422269552730471, −7.243202899285689, −6.648326835160661, −6.200472918362076, −5.756370125496938, −4.969241692509509, −4.514459858717073, −3.830172927057493, −3.167106148656592, −2.502846871531705, −2.144977206642708, −1.150057823625039, −0.4874063002971954, 0, 0.4874063002971954, 1.150057823625039, 2.144977206642708, 2.502846871531705, 3.167106148656592, 3.830172927057493, 4.514459858717073, 4.969241692509509, 5.756370125496938, 6.200472918362076, 6.648326835160661, 7.243202899285689, 7.422269552730471, 7.980156188337239, 8.723526752503129, 9.253902604003219, 9.541750345402851, 9.814469451550098, 10.32412946485382, 10.75147340453941, 11.13106971259212, 11.92329346603056, 12.25288414920217, 12.68165504942170, 13.06284212792602

Graph of the ZZ-function along the critical line