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Properties

Label 2-1850-1.1-c1-0-45
Degree 22
Conductor 18501850
Sign −1-1
Analytic cond. 14.772314.7723
Root an. cond. 3.843473.84347
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 + 4-s + 2·7-s − 8-s − 3·9-s − 2·13-s − 2·14-s + 16-s + 6·17-s + 3·18-s − 6·19-s − 4·23-s + 2·26-s + 2·28-s − 4·31-s − 32-s − 6·34-s − 3·36-s + 37-s + 6·38-s − 10·41-s + 4·43-s + 4·46-s + 2·47-s − 3·49-s − 2·52-s − 2·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 9-s − 0.554·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.707·18-s − 1.37·19-s − 0.834·23-s + 0.392·26-s + 0.377·28-s − 0.718·31-s − 0.176·32-s − 1.02·34-s − 1/2·36-s + 0.164·37-s + 0.973·38-s − 1.56·41-s + 0.609·43-s + 0.589·46-s + 0.291·47-s − 3/7·49-s − 0.277·52-s − 0.274·53-s + ⋯

Functional equation

Λ(s)=(1850s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1850s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18501850    =    2⋅52⋅372 \cdot 5^{2} \cdot 37
Sign: −1-1
Analytic conductor: 14.772314.7723
Root analytic conductor: 3.843473.84347
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1850, ( :1/2), −1)(2,\ 1850,\ (\ :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
5 1 1
37 1−T 1 - T
good3 1+pT2 1 + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 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+pT2 1 + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
41 1+10T+pT2 1 + 10 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−2T+pT2 1 - 2 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+8T+pT2 1 + 8 T + p T^{2}
79 1−4T+pT2 1 - 4 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 1−10T+pT2 1 - 10 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

−8.776464432479204247972797831801, −8.062677850001302204457856503347, −7.61843478274367654964811575512, −6.48983703143827331517471330308, −5.71184317840727757217829967484, −4.89537956640686110087732549653, −3.67934155458315654719216021516, −2.59022325384696380905670929706, −1.59496787970180487849418213093, 0, 1.59496787970180487849418213093, 2.59022325384696380905670929706, 3.67934155458315654719216021516, 4.89537956640686110087732549653, 5.71184317840727757217829967484, 6.48983703143827331517471330308, 7.61843478274367654964811575512, 8.062677850001302204457856503347, 8.776464432479204247972797831801

Graph of the ZZ-function along the critical line