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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 7-s + 8-s + 2·11-s − 14-s + 16-s + 4·17-s − 19-s + 2·22-s + 3·23-s − 28-s − 2·29-s + 4·31-s + 32-s + 4·34-s − 10·37-s − 38-s − 7·41-s + 12·43-s + 2·44-s + 3·46-s − 8·47-s − 6·49-s − 9·53-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 + 0.603·11-s − 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.229·19-s + 0.426·22-s + 0.625·23-s − 0.188·28-s − 0.371·29-s + 0.718·31-s + 0.176·32-s + 0.685·34-s − 1.64·37-s − 0.162·38-s − 1.09·41-s + 1.82·43-s + 0.301·44-s + 0.442·46-s − 1.16·47-s − 6/7·49-s − 1.23·53-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  =  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 \( 1 + T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−14.30911775475891, −13.79663347011373, −13.39003465399007, −12.76113533805331, −12.32275993859507, −12.03084484458998, −11.36119652318151, −10.89676014136648, −10.39681998326199, −9.714755557189016, −9.435820671336252, −8.673000370362778, −8.168029018223443, −7.569072064988548, −6.951335770807577, −6.539460339521378, −6.018849044447756, −5.375808594723829, −4.894849096918229, −4.290123876208060, −3.485660134276313, −3.310567757510899, −2.509470221672082, −1.686028057203916, −1.105766469531810, 0, 1.105766469531810, 1.686028057203916, 2.509470221672082, 3.310567757510899, 3.485660134276313, 4.290123876208060, 4.894849096918229, 5.375808594723829, 6.018849044447756, 6.539460339521378, 6.951335770807577, 7.569072064988548, 8.168029018223443, 8.673000370362778, 9.435820671336252, 9.714755557189016, 10.39681998326199, 10.89676014136648, 11.36119652318151, 12.03084484458998, 12.32275993859507, 12.76113533805331, 13.39003465399007, 13.79663347011373, 14.30911775475891

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