Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \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 − 5-s + 7-s + 8-s − 10-s + 2·13-s + 14-s + 16-s + 6·17-s + 8·19-s − 20-s + 25-s + 2·26-s + 28-s − 6·29-s − 4·31-s + 32-s + 6·34-s − 35-s − 10·37-s + 8·38-s − 40-s + 6·41-s − 4·43-s + 49-s + 50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 1.83·19-s − 0.223·20-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.169·35-s − 1.64·37-s + 1.29·38-s − 0.158·40-s + 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.141·50-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{630} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 630,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.334574167$
$L(\frac12)$  $\approx$  $2.334574167$
$L(\frac{3}{2})$   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 \( 1 + T \)
7 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−19.09443582008748, −18.43568312472132, −17.64944525316685, −16.52275704242733, −16.16389958042461, −15.25424514464404, −14.52711164210978, −13.89087574102980, −13.08823218839446, −12.09611444827710, −11.66604590674684, −10.77509849140856, −9.857402504230162, −8.790922026040486, −7.678455085915795, −7.198013863583408, −5.760935968135276, −5.202720311641407, −3.882731997730706, −3.131255575877541, −1.410061558454018, 1.410061558454018, 3.131255575877541, 3.882731997730706, 5.202720311641407, 5.760935968135276, 7.198013863583408, 7.678455085915795, 8.790922026040486, 9.857402504230162, 10.77509849140856, 11.66604590674684, 12.09611444827710, 13.08823218839446, 13.89087574102980, 14.52711164210978, 15.25424514464404, 16.16389958042461, 16.52275704242733, 17.64944525316685, 18.43568312472132, 19.09443582008748

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