Properties

Label 2-200-1.1-c5-0-13
Degree $2$
Conductor $200$
Sign $-1$
Analytic cond. $32.0767$
Root an. cond. $5.66363$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 20·3-s + 24·7-s + 157·9-s + 124·11-s − 478·13-s + 1.19e3·17-s + 3.04e3·19-s − 480·21-s − 184·23-s + 1.72e3·27-s − 3.28e3·29-s − 5.72e3·31-s − 2.48e3·33-s − 1.03e4·37-s + 9.56e3·39-s − 8.88e3·41-s + 9.18e3·43-s − 2.36e4·47-s − 1.62e4·49-s − 2.39e4·51-s − 1.16e4·53-s − 6.08e4·57-s + 1.68e4·59-s − 1.84e4·61-s + 3.76e3·63-s + 1.55e4·67-s + 3.68e3·69-s + ⋯
L(s)  = 1  − 1.28·3-s + 0.185·7-s + 0.646·9-s + 0.308·11-s − 0.784·13-s + 1.00·17-s + 1.93·19-s − 0.237·21-s − 0.0725·23-s + 0.454·27-s − 0.724·29-s − 1.07·31-s − 0.396·33-s − 1.24·37-s + 1.00·39-s − 0.825·41-s + 0.757·43-s − 1.56·47-s − 0.965·49-s − 1.28·51-s − 0.571·53-s − 2.48·57-s + 0.631·59-s − 0.635·61-s + 0.119·63-s + 0.422·67-s + 0.0930·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(32.0767\)
Root analytic conductor: \(5.66363\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: $\chi_{200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 200,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 20 T + p^{5} T^{2} \)
7 \( 1 - 24 T + p^{5} T^{2} \)
11 \( 1 - 124 T + p^{5} T^{2} \)
13 \( 1 + 478 T + p^{5} T^{2} \)
17 \( 1 - 1198 T + p^{5} T^{2} \)
19 \( 1 - 3044 T + p^{5} T^{2} \)
23 \( 1 + 8 p T + p^{5} T^{2} \)
29 \( 1 + 3282 T + p^{5} T^{2} \)
31 \( 1 + 5728 T + p^{5} T^{2} \)
37 \( 1 + 10326 T + p^{5} T^{2} \)
41 \( 1 + 8886 T + p^{5} T^{2} \)
43 \( 1 - 9188 T + p^{5} T^{2} \)
47 \( 1 + 23664 T + p^{5} T^{2} \)
53 \( 1 + 11686 T + p^{5} T^{2} \)
59 \( 1 - 16876 T + p^{5} T^{2} \)
61 \( 1 + 18482 T + p^{5} T^{2} \)
67 \( 1 - 15532 T + p^{5} T^{2} \)
71 \( 1 + 31960 T + p^{5} T^{2} \)
73 \( 1 - 4886 T + p^{5} T^{2} \)
79 \( 1 - 44560 T + p^{5} T^{2} \)
83 \( 1 + 67364 T + p^{5} T^{2} \)
89 \( 1 - 71994 T + p^{5} T^{2} \)
97 \( 1 + 48866 T + p^{5} T^{2} \)
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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

−11.31613980150516750687028441410, −10.20683900646814095076681785596, −9.356129762461071123269514243454, −7.80200954890915609293932100697, −6.87094927563451275048400262545, −5.59134558525468200560540041988, −5.01582418677113003460207233046, −3.35886491162595454477621142342, −1.38861605357133654847665905758, 0, 1.38861605357133654847665905758, 3.35886491162595454477621142342, 5.01582418677113003460207233046, 5.59134558525468200560540041988, 6.87094927563451275048400262545, 7.80200954890915609293932100697, 9.356129762461071123269514243454, 10.20683900646814095076681785596, 11.31613980150516750687028441410

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