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 + 2·19-s + 20-s + 25-s + 2·26-s + 28-s − 6·29-s + 8·31-s + 32-s + 35-s − 4·37-s + 2·38-s + 40-s − 6·41-s + 2·43-s + 6·47-s + 49-s + 50-s + 2·52-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 + 0.458·19-s + 0.223·20-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.169·35-s − 0.657·37-s + 0.324·38-s + 0.158·40-s − 0.937·41-s + 0.304·43-s + 0.875·47-s + 1/7·49-s + 0.141·50-s + 0.277·52-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  :  $\chi_{630} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.58229\)
\(L(\frac12)\)  \(\approx\)  \(2.58229\)
\(L(\frac{3}{2})\)   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−10.72404098936852919349512142802, −9.859293451209498331903316737933, −8.837533638096091066090521817753, −7.87266301798245480225791296096, −6.87293396816033774027000812050, −5.93829815350757951860423599864, −5.12156168153122389070395792926, −4.05964914153577511754110235204, −2.88888582302515315869877464652, −1.54625076555069290237365702205, 1.54625076555069290237365702205, 2.88888582302515315869877464652, 4.05964914153577511754110235204, 5.12156168153122389070395792926, 5.93829815350757951860423599864, 6.87293396816033774027000812050, 7.87266301798245480225791296096, 8.837533638096091066090521817753, 9.859293451209498331903316737933, 10.72404098936852919349512142802

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