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Properties

Label 2-232050-1.1-c1-0-112
Degree 22
Conductor 232050232050
Sign −1-1
Analytic cond. 1852.921852.92
Root an. cond. 43.045643.0456
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-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 4·11-s + 12-s − 13-s + 14-s + 16-s + 17-s − 18-s − 8·19-s − 21-s − 4·22-s − 8·23-s − 24-s + 26-s + 27-s − 28-s + 10·29-s − 32-s + 4·33-s − 34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.83·19-s − 0.218·21-s − 0.852·22-s − 1.66·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + 1.85·29-s − 0.176·32-s + 0.696·33-s − 0.171·34-s + 1/6·36-s + ⋯

Functional equation

Λ(s)=(232050s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 232050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(232050s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 232050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 232050232050    =    2⋅3⋅52⋅7⋅13⋅172 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13 \cdot 17
Sign: −1-1
Analytic conductor: 1852.921852.92
Root analytic conductor: 43.045643.0456
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 232050, ( :1/2), −1)(2,\ 232050,\ (\ :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
3 1−T 1 - T
5 1 1
7 1+T 1 + T
13 1+T 1 + T
17 1−T 1 - T
good11 1−4T+pT2 1 - 4 T + p T^{2}
19 1+8T+pT2 1 + 8 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1−10T+pT2 1 - 10 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−6T+pT2 1 - 6 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

−12.97982840060254, −12.73899852335185, −12.11965736964478, −11.81159139546891, −11.31968372269433, −10.62365825701667, −10.24729808453068, −9.890872686978359, −9.325948856737764, −9.028992355457339, −8.402422104466634, −8.087039847733934, −7.702965418323037, −6.891974773634842, −6.577986037816194, −6.126520443739190, −5.790309214360887, −4.567252196038217, −4.422109254723799, −3.881663416367338, −3.085493903683687, −2.727275542909220, −1.967925800643941, −1.582437788526761, −0.7758474443153678, 0, 0.7758474443153678, 1.582437788526761, 1.967925800643941, 2.727275542909220, 3.085493903683687, 3.881663416367338, 4.422109254723799, 4.567252196038217, 5.790309214360887, 6.126520443739190, 6.577986037816194, 6.891974773634842, 7.702965418323037, 8.087039847733934, 8.402422104466634, 9.028992355457339, 9.325948856737764, 9.890872686978359, 10.24729808453068, 10.62365825701667, 11.31968372269433, 11.81159139546891, 12.11965736964478, 12.73899852335185, 12.97982840060254

Graph of the ZZ-function along the critical line