Properties

Label 2-200-1.1-c1-0-4
Degree $2$
Conductor $200$
Sign $-1$
Analytic cond. $1.59700$
Root an. cond. $1.26372$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·7-s + 9-s − 4·11-s − 4·13-s − 4·19-s + 4·21-s + 2·23-s + 4·27-s + 2·29-s + 8·33-s − 4·37-s + 8·39-s + 2·41-s + 6·43-s + 6·47-s − 3·49-s + 4·53-s + 8·57-s − 12·59-s − 10·61-s − 2·63-s − 14·67-s − 4·69-s + 8·71-s − 8·73-s + 8·77-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s − 0.917·19-s + 0.872·21-s + 0.417·23-s + 0.769·27-s + 0.371·29-s + 1.39·33-s − 0.657·37-s + 1.28·39-s + 0.312·41-s + 0.914·43-s + 0.875·47-s − 3/7·49-s + 0.549·53-s + 1.05·57-s − 1.56·59-s − 1.28·61-s − 0.251·63-s − 1.71·67-s − 0.481·69-s + 0.949·71-s − 0.936·73-s + 0.911·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.15068586386177913384059280623, −10.84574214924305286935175238342, −10.33957279438761642704116829921, −9.143525842083093285719526383981, −7.72354144124853582700669789995, −6.62151311275959826478778900954, −5.62831154024710482280783911304, −4.64606420734121812424941498850, −2.75046174819041571949427472141, 0, 2.75046174819041571949427472141, 4.64606420734121812424941498850, 5.62831154024710482280783911304, 6.62151311275959826478778900954, 7.72354144124853582700669789995, 9.143525842083093285719526383981, 10.33957279438761642704116829921, 10.84574214924305286935175238342, 12.15068586386177913384059280623

Graph of the $Z$-function along the critical line