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 − 11-s − 3·13-s + 14-s + 16-s + 8·17-s − 3·19-s − 22-s − 6·23-s − 3·26-s + 28-s − 6·29-s − 4·31-s + 32-s + 8·34-s − 2·37-s − 3·38-s − 11·41-s − 43-s − 44-s − 6·46-s − 47-s + 49-s − 3·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 0.301·11-s − 0.832·13-s + 0.267·14-s + 1/4·16-s + 1.94·17-s − 0.688·19-s − 0.213·22-s − 1.25·23-s − 0.588·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 1.37·34-s − 0.328·37-s − 0.486·38-s − 1.71·41-s − 0.152·43-s − 0.150·44-s − 0.884·46-s − 0.145·47-s + 1/7·49-s − 0.416·52-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 \( 1 + T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
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

−7.50497819389855751237713837020, −6.59788487551126754548506984690, −5.77681567302171082156520036288, −5.33258509131763330613282269062, −4.65379605198411918849061820626, −3.77897272297251269969679810219, −3.21272968600666125911987979012, −2.19652260487469449650208200845, −1.51864247643044039261478452422, 0, 1.51864247643044039261478452422, 2.19652260487469449650208200845, 3.21272968600666125911987979012, 3.77897272297251269969679810219, 4.65379605198411918849061820626, 5.33258509131763330613282269062, 5.77681567302171082156520036288, 6.59788487551126754548506984690, 7.50497819389855751237713837020

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