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 2

Origins

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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  =  2
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 \( 1 + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
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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

−14.47573258140753, −14.09852015399591, −13.56058409667924, −12.88488610188543, −12.76957765987189, −12.19167406376312, −11.65737310164533, −11.05315402830027, −10.69068024532009, −10.09231193576852, −9.625434453983959, −8.910351668391808, −8.523150721984903, −7.872380753287046, −7.179914394999867, −6.785004243621004, −6.256652152618399, −5.825856883595197, −4.955584249139207, −4.610254619727646, −4.011306729612293, −3.522138118029200, −2.570856799310268, −2.184666358366814, −1.670345148872581, 0, 0, 1.670345148872581, 2.184666358366814, 2.570856799310268, 3.522138118029200, 4.011306729612293, 4.610254619727646, 4.955584249139207, 5.825856883595197, 6.256652152618399, 6.785004243621004, 7.179914394999867, 7.872380753287046, 8.523150721984903, 8.910351668391808, 9.625434453983959, 10.09231193576852, 10.69068024532009, 11.05315402830027, 11.65737310164533, 12.19167406376312, 12.76957765987189, 12.88488610188543, 13.56058409667924, 14.09852015399591, 14.47573258140753

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