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 − 2·7-s − 8-s − 2·11-s + 2·13-s + 2·14-s + 16-s + 4·19-s + 2·22-s + 4·23-s − 2·26-s − 2·28-s + 6·29-s − 32-s + 10·37-s − 4·38-s − 2·41-s + 4·43-s − 2·44-s − 4·46-s + 8·47-s − 3·49-s + 2·52-s + 2·53-s + 2·56-s − 6·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.603·11-s + 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.917·19-s + 0.426·22-s + 0.834·23-s − 0.392·26-s − 0.377·28-s + 1.11·29-s − 0.176·32-s + 1.64·37-s − 0.648·38-s − 0.312·41-s + 0.609·43-s − 0.301·44-s − 0.589·46-s + 1.16·47-s − 3/7·49-s + 0.277·52-s + 0.274·53-s + 0.267·56-s − 0.787·58-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.31128185056241, −13.69390089006357, −13.33594187897190, −12.80317212101968, −12.27200613144141, −11.79926705972207, −11.15626714282017, −10.78912751127997, −10.19368709667589, −9.812891749991187, −9.209397087189867, −8.854924593473505, −8.235448255892189, −7.597844311368922, −7.322911390367875, −6.536174427659912, −6.170400521156252, −5.574817073339641, −4.933484706125071, −4.225759512733907, −3.508805490519659, −2.748663638289470, −2.641074389313036, −1.402919705490135, −0.9148682085326425, 0, 0.9148682085326425, 1.402919705490135, 2.641074389313036, 2.748663638289470, 3.508805490519659, 4.225759512733907, 4.933484706125071, 5.574817073339641, 6.170400521156252, 6.536174427659912, 7.322911390367875, 7.597844311368922, 8.235448255892189, 8.854924593473505, 9.209397087189867, 9.812891749991187, 10.19368709667589, 10.78912751127997, 11.15626714282017, 11.79926705972207, 12.27200613144141, 12.80317212101968, 13.33594187897190, 13.69390089006357, 14.31128185056241

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