Properties

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

Origins

Related objects

Downloads

Learn more about

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  =  1
Selberg data  =  $(2,\ 200,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 16 T + p T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.74424283824015, −19.02755508052120, −18.07932355043632, −17.28398850274401, −16.63252564309894, −15.80886445087647, −14.93768116099267, −13.66722426051922, −12.62008217819357, −12.15068586386178, −10.84574214924305, −10.33957279438762, −9.143525842083093, −7.723541441248536, −6.621513112759598, −5.628311540247105, −4.646064207341218, −2.750461748190416, 0, 2.750461748190416, 4.646064207341218, 5.628311540247105, 6.621513112759598, 7.723541441248536, 9.143525842083093, 10.33957279438762, 10.84574214924305, 12.15068586386178, 12.62008217819357, 13.66722426051922, 14.93768116099267, 15.80886445087647, 16.63252564309894, 17.28398850274401, 18.07932355043632, 19.02755508052120, 19.74424283824015

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