Properties

Degree 2
Conductor $ 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 $
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 − 4·11-s − 13-s − 14-s + 16-s − 17-s + 5·19-s − 4·22-s + 4·23-s − 26-s − 28-s + 3·29-s − 8·31-s + 32-s − 34-s − 4·37-s + 5·38-s − 2·41-s − 4·44-s + 4·46-s + 7·47-s + 49-s − 52-s − 9·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.20·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 1.14·19-s − 0.852·22-s + 0.834·23-s − 0.196·26-s − 0.188·28-s + 0.557·29-s − 1.43·31-s + 0.176·32-s − 0.171·34-s − 0.657·37-s + 0.811·38-s − 0.312·41-s − 0.603·44-s + 0.589·46-s + 1.02·47-s + 1/7·49-s − 0.138·52-s − 1.23·53-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9450\)    =    \(2 \cdot 3^{3} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{9450} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 9450,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−7.22979888626344936993475953323, −6.75799033316513875922597800244, −5.74606213093277194864170128887, −5.31789084372976327916693451415, −4.71104562984737045916236722757, −3.76771319597827775689689658891, −3.03529154497781957346835911555, −2.47602735585761364876867214308, −1.36178182186921226633814507701, 0, 1.36178182186921226633814507701, 2.47602735585761364876867214308, 3.03529154497781957346835911555, 3.76771319597827775689689658891, 4.71104562984737045916236722757, 5.31789084372976327916693451415, 5.74606213093277194864170128887, 6.75799033316513875922597800244, 7.22979888626344936993475953323

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