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 − 6·13-s + 14-s + 16-s − 4·17-s − 4·22-s − 4·23-s + 6·26-s − 28-s − 3·29-s + 7·31-s − 32-s + 4·34-s + 37-s + 7·41-s + 10·43-s + 4·44-s + 4·46-s + 13·47-s + 49-s − 6·52-s − 6·53-s + 56-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.20·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.852·22-s − 0.834·23-s + 1.17·26-s − 0.188·28-s − 0.557·29-s + 1.25·31-s − 0.176·32-s + 0.685·34-s + 0.164·37-s + 1.09·41-s + 1.52·43-s + 0.603·44-s + 0.589·46-s + 1.89·47-s + 1/7·49-s − 0.832·52-s − 0.824·53-s + 0.133·56-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−7.44938209328880730387539177172, −6.71565645067984072568496716858, −6.24862150302888314053698385583, −5.42627756526030903880835373264, −4.39590641910139547691191321645, −3.94819450407876239546169643226, −2.68665003226096656161243988692, −2.25726713756267008942843236200, −1.08171226124613748582138172247, 0, 1.08171226124613748582138172247, 2.25726713756267008942843236200, 2.68665003226096656161243988692, 3.94819450407876239546169643226, 4.39590641910139547691191321645, 5.42627756526030903880835373264, 6.24862150302888314053698385583, 6.71565645067984072568496716858, 7.44938209328880730387539177172

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