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  + 3·3-s − 2·7-s + 6·9-s + 11-s − 4·13-s − 5·17-s + 19-s − 6·21-s + 2·23-s + 9·27-s − 8·29-s + 10·31-s + 3·33-s + 6·37-s − 12·39-s − 3·41-s − 4·43-s − 4·47-s − 3·49-s − 15·51-s − 6·53-s + 3·57-s + 8·59-s + 10·61-s − 12·63-s + 67-s + 6·69-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.755·7-s + 2·9-s + 0.301·11-s − 1.10·13-s − 1.21·17-s + 0.229·19-s − 1.30·21-s + 0.417·23-s + 1.73·27-s − 1.48·29-s + 1.79·31-s + 0.522·33-s + 0.986·37-s − 1.92·39-s − 0.468·41-s − 0.609·43-s − 0.583·47-s − 3/7·49-s − 2.10·51-s − 0.824·53-s + 0.397·57-s + 1.04·59-s + 1.28·61-s − 1.51·63-s + 0.122·67-s + 0.722·69-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  =  $(2,\ 200,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.808107531$
$L(\frac12)$  $\approx$  $1.808107531$
$L(\frac{3}{2})$   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 \( 1 - p T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 13 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−19.36954592679050, −19.18958311068676, −17.96207099754466, −16.85188931656429, −15.77571313967724, −15.04689914359411, −14.38903813273620, −13.33985381275135, −12.91670582621016, −11.57805678356630, −10.02244514988645, −9.447015792876853, −8.567472421484776, −7.521563296833928, −6.578588287613274, −4.602350272962787, −3.343596275664506, −2.269170031363564, 2.269170031363564, 3.343596275664506, 4.602350272962787, 6.578588287613274, 7.521563296833928, 8.567472421484776, 9.447015792876853, 10.02244514988645, 11.57805678356630, 12.91670582621016, 13.33985381275135, 14.38903813273620, 15.04689914359411, 15.77571313967724, 16.85188931656429, 17.96207099754466, 19.18958311068676, 19.36954592679050

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