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 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 7-s + 8-s − 5·11-s + 14-s + 16-s + 2·17-s − 19-s − 5·22-s − 23-s + 28-s − 4·29-s − 9·31-s + 32-s + 2·34-s − 5·37-s − 38-s + 9·41-s + 10·43-s − 5·44-s − 46-s + 6·47-s + 49-s + 12·53-s + 56-s − 4·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.50·11-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.229·19-s − 1.06·22-s − 0.208·23-s + 0.188·28-s − 0.742·29-s − 1.61·31-s + 0.176·32-s + 0.342·34-s − 0.821·37-s − 0.162·38-s + 1.40·41-s + 1.52·43-s − 0.753·44-s − 0.147·46-s + 0.875·47-s + 1/7·49-s + 1.64·53-s + 0.133·56-s − 0.525·58-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  =  0
Selberg data  =  $(2,\ 9450,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.013104356$
$L(\frac12)$  $\approx$  $3.013104356$
$L(\frac{3}{2})$   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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−7.46782089600524145924378084994, −7.23121471355869503316580940451, −6.11146364437453685304727079614, −5.38728005132442870577236549443, −5.22933389027289580544317892179, −4.10619494891641938149588170500, −3.61821594379271561802584596428, −2.50798084154612067735059981675, −2.09937771523637851330725269426, −0.72299960170725188223213743879, 0.72299960170725188223213743879, 2.09937771523637851330725269426, 2.50798084154612067735059981675, 3.61821594379271561802584596428, 4.10619494891641938149588170500, 5.22933389027289580544317892179, 5.38728005132442870577236549443, 6.11146364437453685304727079614, 7.23121471355869503316580940451, 7.46782089600524145924378084994

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