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 + 4·7-s + 11-s + 2·13-s + 2·17-s − 8·23-s − 25-s − 6·29-s − 8·31-s − 8·35-s − 6·37-s + 2·41-s − 8·47-s + 9·49-s + 6·53-s − 2·55-s − 4·59-s − 6·61-s − 4·65-s + 4·67-s − 14·73-s + 4·77-s − 4·79-s + 12·83-s − 4·85-s + 6·89-s + 8·91-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.51·7-s + 0.301·11-s + 0.554·13-s + 0.485·17-s − 1.66·23-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 1.35·35-s − 0.986·37-s + 0.312·41-s − 1.16·47-s + 9/7·49-s + 0.824·53-s − 0.269·55-s − 0.520·59-s − 0.768·61-s − 0.496·65-s + 0.488·67-s − 1.63·73-s + 0.455·77-s − 0.450·79-s + 1.31·83-s − 0.433·85-s + 0.635·89-s + 0.838·91-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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−17.76038673773152, −17.06353872639512, −16.36440832952840, −15.92581420013421, −15.17604423878997, −14.67970220532197, −14.22336418972633, −13.58099657340895, −12.77266912894848, −11.92113554398929, −11.73466494635702, −11.04920290267837, −10.54760585469901, −9.664785531073104, −8.832735428994558, −8.283207628441610, −7.660515096277707, −7.331283768844487, −6.163663752714437, −5.513226133325592, −4.736182170912008, −3.932853773746884, −3.524539184527543, −2.062484114062448, −1.444229616593001, 0, 1.444229616593001, 2.062484114062448, 3.524539184527543, 3.932853773746884, 4.736182170912008, 5.513226133325592, 6.163663752714437, 7.331283768844487, 7.660515096277707, 8.283207628441610, 8.832735428994558, 9.664785531073104, 10.54760585469901, 11.04920290267837, 11.73466494635702, 11.92113554398929, 12.77266912894848, 13.58099657340895, 14.22336418972633, 14.67970220532197, 15.17604423878997, 15.92581420013421, 16.36440832952840, 17.06353872639512, 17.76038673773152

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