Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 11 $
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6336\)    =    \(2^{6} \cdot 3^{2} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6336} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6336,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 - T \)
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 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
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

−17.65221049858975, −17.11259398960728, −16.37796980762905, −16.00376124085785, −15.29843114217148, −14.67647147521212, −13.84120083372553, −13.48662407900080, −13.07809612422232, −12.23128023940267, −11.65221779986876, −10.84988722159881, −10.35952518261161, −9.424206056642155, −9.294553418111752, −8.568611997358709, −7.470470030075972, −6.991875193217050, −6.179390629134390, −5.653609224764382, −4.982990835398060, −3.784530641342405, −3.338009261009588, −2.183743486754941, −1.475485487509576, 0, 1.475485487509576, 2.183743486754941, 3.338009261009588, 3.784530641342405, 4.982990835398060, 5.653609224764382, 6.179390629134390, 6.991875193217050, 7.470470030075972, 8.568611997358709, 9.294553418111752, 9.424206056642155, 10.35952518261161, 10.84988722159881, 11.65221779986876, 12.23128023940267, 13.07809612422232, 13.48662407900080, 13.84120083372553, 14.67647147521212, 15.29843114217148, 16.00376124085785, 16.37796980762905, 17.11259398960728, 17.65221049858975

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