Properties

Degree 2
Conductor $ 2^{3} \cdot 5^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

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

\( d \)  =  \(2\)
\( N \)  =  \(200\)    =    \(2^{3} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 200,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.660108418$
$L(\frac12)$  $\approx$  $1.660108418$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
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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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.63251622909724, −18.58817084229040, −18.08397001745166, −16.94062466836969, −15.79542312752491, −15.13862567943492, −14.25532770975716, −13.53620112286326, −12.73146010216775, −11.33359133996889, −10.53622600687691, −9.301178546757554, −8.212252834472441, −7.930378831312214, −6.253523458436641, −4.826047997090230, −3.418091032726363, −2.087936085249858, 2.087936085249858, 3.418091032726363, 4.826047997090230, 6.253523458436641, 7.930378831312214, 8.212252834472441, 9.301178546757554, 10.53622600687691, 11.33359133996889, 12.73146010216775, 13.53620112286326, 14.25532770975716, 15.13862567943492, 15.79542312752491, 16.94062466836969, 18.08397001745166, 18.58817084229040, 19.63251622909724

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