Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 167 $
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-s + 4-s − 8-s − 2·11-s − 6·13-s + 16-s + 4·17-s − 8·19-s + 2·22-s + 6·26-s + 6·29-s + 8·31-s − 32-s − 4·34-s + 2·37-s + 8·38-s − 10·41-s − 4·43-s − 2·44-s − 8·47-s − 7·49-s − 6·52-s + 4·53-s − 6·58-s + 12·59-s − 10·61-s − 8·62-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.603·11-s − 1.66·13-s + 1/4·16-s + 0.970·17-s − 1.83·19-s + 0.426·22-s + 1.17·26-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.685·34-s + 0.328·37-s + 1.29·38-s − 1.56·41-s − 0.609·43-s − 0.301·44-s − 1.16·47-s − 49-s − 0.832·52-s + 0.549·53-s − 0.787·58-s + 1.56·59-s − 1.28·61-s − 1.01·62-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(75150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 167\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{75150} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 75150,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;167\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
167 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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

−14.40486140708599, −13.92546822806996, −13.19193429710113, −12.66046356362962, −12.36222444792563, −11.72168510745536, −11.40335838269235, −10.48430513004232, −10.28969367470897, −9.832446511423950, −9.446919176974855, −8.518169751632115, −8.111631734309948, −8.021604881587667, −7.014935374449216, −6.738804467257493, −6.228550340866989, −5.340250275874235, −4.934953932108035, −4.402081965600890, −3.508136418935859, −2.816688791556347, −2.349072318783601, −1.715119822766701, −0.7229536094002055, 0, 0.7229536094002055, 1.715119822766701, 2.349072318783601, 2.816688791556347, 3.508136418935859, 4.402081965600890, 4.934953932108035, 5.340250275874235, 6.228550340866989, 6.738804467257493, 7.014935374449216, 8.021604881587667, 8.111631734309948, 8.518169751632115, 9.446919176974855, 9.832446511423950, 10.28969367470897, 10.48430513004232, 11.40335838269235, 11.72168510745536, 12.36222444792563, 12.66046356362962, 13.19193429710113, 13.92546822806996, 14.40486140708599

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