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 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 7-s − 8-s − 4·13-s + 14-s + 16-s + 8·17-s − 7·19-s + 7·23-s + 4·26-s − 28-s + 2·29-s − 2·31-s − 32-s − 8·34-s − 12·37-s + 7·38-s − 7·41-s − 4·43-s − 7·46-s − 2·47-s − 6·49-s − 4·52-s + 5·53-s + 56-s − 2·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.94·17-s − 1.60·19-s + 1.45·23-s + 0.784·26-s − 0.188·28-s + 0.371·29-s − 0.359·31-s − 0.176·32-s − 1.37·34-s − 1.97·37-s + 1.13·38-s − 1.09·41-s − 0.609·43-s − 1.03·46-s − 0.291·47-s − 6/7·49-s − 0.554·52-s + 0.686·53-s + 0.133·56-s − 0.262·58-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  =  0
Selberg data  =  $(2,\ 75150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7796023626$
$L(\frac12)$  $\approx$  $0.7796023626$
$L(\frac{3}{2})$   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 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−14.28380364817437, −13.48066588643007, −12.94803997720866, −12.47509917802926, −12.05682929048922, −11.65936992522737, −10.84876793686998, −10.42051793453403, −10.06733051589950, −9.605305202857866, −8.927247117058491, −8.559092853309532, −7.980590321851974, −7.367442992996961, −6.959298882094228, −6.468328441115500, −5.803201319539262, −5.039794284690694, −4.861139101976751, −3.701389982186071, −3.313110695721657, −2.668593551901906, −1.916940230699050, −1.280424638622738, −0.3333014408165975, 0.3333014408165975, 1.280424638622738, 1.916940230699050, 2.668593551901906, 3.313110695721657, 3.701389982186071, 4.861139101976751, 5.039794284690694, 5.803201319539262, 6.468328441115500, 6.959298882094228, 7.367442992996961, 7.980590321851974, 8.559092853309532, 8.927247117058491, 9.605305202857866, 10.06733051589950, 10.42051793453403, 10.84876793686998, 11.65936992522737, 12.05682929048922, 12.47509917802926, 12.94803997720866, 13.48066588643007, 14.28380364817437

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