Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 11 \cdot 23^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 2·7-s + 11-s − 2·13-s + 4·17-s + 6·19-s − 25-s + 8·29-s − 8·31-s + 4·35-s − 10·37-s − 8·41-s + 2·43-s + 8·47-s − 3·49-s − 2·53-s + 2·55-s − 12·59-s − 10·61-s − 4·65-s − 12·67-s − 8·71-s + 6·73-s + 2·77-s + 2·79-s + 16·83-s + 8·85-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.755·7-s + 0.301·11-s − 0.554·13-s + 0.970·17-s + 1.37·19-s − 1/5·25-s + 1.48·29-s − 1.43·31-s + 0.676·35-s − 1.64·37-s − 1.24·41-s + 0.304·43-s + 1.16·47-s − 3/7·49-s − 0.274·53-s + 0.269·55-s − 1.56·59-s − 1.28·61-s − 0.496·65-s − 1.46·67-s − 0.949·71-s + 0.702·73-s + 0.227·77-s + 0.225·79-s + 1.75·83-s + 0.867·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(209484\)    =    \(2^{2} \cdot 3^{2} \cdot 11 \cdot 23^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{209484} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 209484,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 - T \)
23 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−13.49756647339383, −12.57824204315807, −12.37460675008760, −11.72646692536463, −11.67668004386852, −10.64886608867218, −10.53169607631148, −9.998174915198371, −9.449766668597832, −9.117726848031068, −8.609373602325271, −7.952367954523135, −7.459499579644307, −7.231480775902780, −6.421717969507280, −5.989840035370064, −5.421162920282435, −5.038365870717675, −4.664665844621078, −3.814308261815697, −3.231282777751319, −2.821603802935422, −1.889939817793374, −1.613600492560256, −1.011175000314495, 0, 1.011175000314495, 1.613600492560256, 1.889939817793374, 2.821603802935422, 3.231282777751319, 3.814308261815697, 4.664665844621078, 5.038365870717675, 5.421162920282435, 5.989840035370064, 6.421717969507280, 7.231480775902780, 7.459499579644307, 7.952367954523135, 8.609373602325271, 9.117726848031068, 9.449766668597832, 9.998174915198371, 10.53169607631148, 10.64886608867218, 11.67668004386852, 11.72646692536463, 12.37460675008760, 12.57824204315807, 13.49756647339383

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