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 + 5·13-s − 14-s + 16-s + 6·17-s + 6·19-s + 2·22-s − 8·23-s − 5·26-s + 28-s + 5·29-s + 7·31-s − 32-s − 6·34-s + 4·37-s − 6·38-s − 41-s + 4·43-s − 2·44-s + 8·46-s − 2·47-s + 49-s + 5·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.603·11-s + 1.38·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 1.37·19-s + 0.426·22-s − 1.66·23-s − 0.980·26-s + 0.188·28-s + 0.928·29-s + 1.25·31-s − 0.176·32-s − 1.02·34-s + 0.657·37-s − 0.973·38-s − 0.156·41-s + 0.609·43-s − 0.301·44-s + 1.17·46-s − 0.291·47-s + 1/7·49-s + 0.693·52-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  =  0
Selberg data  =  $(2,\ 9450,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.885603744$
$L(\frac12)$  $\approx$  $1.885603744$
$L(\frac{3}{2})$   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 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 9 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 5 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
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

−7.955306972882156411046145235557, −7.25897008423325185889933050977, −6.22909204676621175697952521133, −5.84898447064486668557891821784, −5.08245920273153004138635153784, −4.09912831009139447668038383358, −3.30645760049096373522973532262, −2.57562071040025305858443625141, −1.44495762941669005269705758245, −0.814341757215904798544444927775, 0.814341757215904798544444927775, 1.44495762941669005269705758245, 2.57562071040025305858443625141, 3.30645760049096373522973532262, 4.09912831009139447668038383358, 5.08245920273153004138635153784, 5.84898447064486668557891821784, 6.22909204676621175697952521133, 7.25897008423325185889933050977, 7.955306972882156411046145235557

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