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 − 5·13-s − 14-s + 16-s + 3·17-s + 2·19-s − 9·23-s − 5·26-s − 28-s + 3·29-s + 5·31-s + 32-s + 3·34-s − 2·37-s + 2·38-s + 6·41-s + 43-s − 9·46-s − 6·47-s + 49-s − 5·52-s + 3·53-s − 56-s + 3·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.38·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.458·19-s − 1.87·23-s − 0.980·26-s − 0.188·28-s + 0.557·29-s + 0.898·31-s + 0.176·32-s + 0.514·34-s − 0.328·37-s + 0.324·38-s + 0.937·41-s + 0.152·43-s − 1.32·46-s − 0.875·47-s + 1/7·49-s − 0.693·52-s + 0.412·53-s − 0.133·56-s + 0.393·58-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 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
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

−7.29199281546736967538845772277, −6.62139626687353922215576455203, −5.87287819102938113469707399631, −5.34622096867902893048908248821, −4.50987208135388742371514439705, −3.94841862875190945765095501059, −2.97249796179752463562060054454, −2.46273168262964029176117899456, −1.38076542358485290966418709809, 0, 1.38076542358485290966418709809, 2.46273168262964029176117899456, 2.97249796179752463562060054454, 3.94841862875190945765095501059, 4.50987208135388742371514439705, 5.34622096867902893048908248821, 5.87287819102938113469707399631, 6.62139626687353922215576455203, 7.29199281546736967538845772277

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