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 − 2·11-s + 3·13-s − 14-s + 16-s + 17-s + 2·22-s + 3·23-s − 3·26-s + 28-s − 3·29-s − 31-s − 32-s − 34-s + 4·37-s − 10·41-s − 43-s − 2·44-s − 3·46-s + 10·47-s + 49-s + 3·52-s − 53-s − 56-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.603·11-s + 0.832·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.426·22-s + 0.625·23-s − 0.588·26-s + 0.188·28-s − 0.557·29-s − 0.179·31-s − 0.176·32-s − 0.171·34-s + 0.657·37-s − 1.56·41-s − 0.152·43-s − 0.301·44-s − 0.442·46-s + 1.45·47-s + 1/7·49-s + 0.416·52-s − 0.137·53-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  =  0
Selberg data  =  $(2,\ 9450,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.484261232$
$L(\frac12)$  $\approx$  $1.484261232$
$L(\frac{3}{2})$   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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
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

−7.934678094637928647151679949471, −7.06747369308095997974190290691, −6.52570024756723649510363126125, −5.61171300767616887934115577993, −5.15154377847649382881394030281, −4.10574338933960328477538929109, −3.34429936437356909372680533789, −2.46973556288306462787255308969, −1.60786010853034356947454881639, −0.66890876518947021032886689250, 0.66890876518947021032886689250, 1.60786010853034356947454881639, 2.46973556288306462787255308969, 3.34429936437356909372680533789, 4.10574338933960328477538929109, 5.15154377847649382881394030281, 5.61171300767616887934115577993, 6.52570024756723649510363126125, 7.06747369308095997974190290691, 7.934678094637928647151679949471

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