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 − 13-s + 16-s − 2·17-s + 5·19-s − 2·22-s − 6·23-s − 26-s + 5·29-s + 8·31-s + 32-s − 2·34-s − 7·37-s + 5·38-s + 3·41-s + 8·43-s − 2·44-s − 6·46-s + 4·47-s − 7·49-s − 52-s − 8·53-s + 5·58-s + 9·59-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.603·11-s − 0.277·13-s + 1/4·16-s − 0.485·17-s + 1.14·19-s − 0.426·22-s − 1.25·23-s − 0.196·26-s + 0.928·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s − 1.15·37-s + 0.811·38-s + 0.468·41-s + 1.21·43-s − 0.301·44-s − 0.884·46-s + 0.583·47-s − 49-s − 0.138·52-s − 1.09·53-s + 0.656·58-s + 1.17·59-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 \( 1 + T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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

−14.09410436402507, −13.81199799204114, −13.55306398572525, −12.67755938927357, −12.40316635330535, −11.94372356967603, −11.40025099226867, −10.90308844947039, −10.29159121726664, −9.900728244815315, −9.391104160930403, −8.607682295570679, −8.073998334979667, −7.688434403986264, −6.989319474281580, −6.569404182809934, −5.861872849321662, −5.478684440182131, −4.789086546250320, −4.364452927617436, −3.738251753269743, −2.898935218702742, −2.642101473460416, −1.795103471130764, −1.015860633203888, 0, 1.015860633203888, 1.795103471130764, 2.642101473460416, 2.898935218702742, 3.738251753269743, 4.364452927617436, 4.789086546250320, 5.478684440182131, 5.861872849321662, 6.569404182809934, 6.989319474281580, 7.688434403986264, 8.073998334979667, 8.607682295570679, 9.391104160930403, 9.900728244815315, 10.29159121726664, 10.90308844947039, 11.40025099226867, 11.94372356967603, 12.40316635330535, 12.67755938927357, 13.55306398572525, 13.81199799204114, 14.09410436402507

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