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  =  $(2,\ 200,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8086102703$
$L(\frac12)$  $\approx$  $0.8086102703$
$L(\frac{3}{2})$   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

−19.14723371044300, −18.42171082828484, −17.69225808592154, −17.02512694970830, −16.27122216278093, −15.47064369798938, −14.27764454251723, −13.20082020156241, −12.07137470450027, −11.57330873285149, −10.73352697394265, −9.870419190423355, −8.343465400986864, −7.122291658517668, −5.985797498466172, −5.262609439011659, −3.982572760536045, −1.248815021250025, 1.248815021250025, 3.982572760536045, 5.262609439011659, 5.985797498466172, 7.122291658517668, 8.343465400986864, 9.870419190423355, 10.73352697394265, 11.57330873285149, 12.07137470450027, 13.20082020156241, 14.27764454251723, 15.47064369798938, 16.27122216278093, 17.02512694970830, 17.69225808592154, 18.42171082828484, 19.14723371044300

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