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Properties

Label 2-232050-1.1-c1-0-129
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 + 6·11-s − 12-s − 13-s − 14-s + 16-s + 17-s + 18-s − 6·19-s + 21-s + 6·22-s − 4·23-s − 24-s − 26-s − 27-s − 28-s + 4·29-s − 4·31-s + 32-s − 6·33-s + 34-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 + 1.80·11-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 − 1.37·19-s + 0.218·21-s + 1.27·22-s − 0.834·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.742·29-s − 0.718·31-s + 0.176·32-s − 1.04·33-s + 0.171·34-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 1−6T+pT2 1 - 6 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−4T+pT2 1 - 4 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1+10T+pT2 1 + 10 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1−12T+pT2 1 - 12 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1−4T+pT2 1 - 4 T + p T^{2}
73 1+14T+pT2 1 + 14 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1−14T+pT2 1 - 14 T + p T^{2}
89 1+6T+pT2 1 + 6 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

−13.09036239104660, −12.51503268320737, −12.23611009281631, −11.91203828362150, −11.48571567025650, −10.84777571999092, −10.47001340722299, −10.05764337202630, −9.418526569033141, −8.998672942775841, −8.489167422752406, −7.920323178624142, −7.155369995106719, −6.831484924033622, −6.423460813578107, −6.051309641606902, −5.454292180837073, −4.965096954778533, −4.236596038949827, −3.975370328713279, −3.548374582176091, −2.785805872867200, −2.016581887145315, −1.624595137183882, −0.8339765139779465, 0, 0.8339765139779465, 1.624595137183882, 2.016581887145315, 2.785805872867200, 3.548374582176091, 3.975370328713279, 4.236596038949827, 4.965096954778533, 5.454292180837073, 6.051309641606902, 6.423460813578107, 6.831484924033622, 7.155369995106719, 7.920323178624142, 8.489167422752406, 8.998672942775841, 9.418526569033141, 10.05764337202630, 10.47001340722299, 10.84777571999092, 11.48571567025650, 11.91203828362150, 12.23611009281631, 12.51503268320737, 13.09036239104660

Graph of the ZZ-function along the critical line