Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \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  − 2·5-s + 7-s − 4·11-s + 2·13-s + 6·17-s + 4·19-s − 25-s − 2·29-s − 2·35-s − 6·37-s − 2·41-s − 4·43-s + 49-s + 6·53-s + 8·55-s − 12·59-s + 2·61-s − 4·65-s + 4·67-s − 6·73-s − 4·77-s + 16·79-s + 12·83-s − 12·85-s + 14·89-s + 2·91-s − 8·95-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s − 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s − 1/5·25-s − 0.371·29-s − 0.338·35-s − 0.986·37-s − 0.312·41-s − 0.609·43-s + 1/7·49-s + 0.824·53-s + 1.07·55-s − 1.56·59-s + 0.256·61-s − 0.496·65-s + 0.488·67-s − 0.702·73-s − 0.455·77-s + 1.80·79-s + 1.31·83-s − 1.30·85-s + 1.48·89-s + 0.209·91-s − 0.820·95-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.473336744$
$L(\frac12)$  $\approx$  $1.473336744$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−18.28882051162758, −17.56412888727163, −16.73404337763195, −16.20283349376123, −15.63112999906630, −15.18130576687009, −14.42451995678790, −13.73446231903253, −13.19508063302946, −12.27044893044395, −11.91433295586737, −11.20992491334543, −10.50494844898212, −9.970888617518094, −9.042140676903253, −8.229992153219431, −7.706911835010117, −7.324419367497448, −6.145216521425822, −5.346417456975608, −4.807224522687250, −3.622561990430718, −3.227465062771161, −1.934808584887592, −0.6957792438310278, 0.6957792438310278, 1.934808584887592, 3.227465062771161, 3.622561990430718, 4.807224522687250, 5.346417456975608, 6.145216521425822, 7.324419367497448, 7.706911835010117, 8.229992153219431, 9.042140676903253, 9.970888617518094, 10.50494844898212, 11.20992491334543, 11.91433295586737, 12.27044893044395, 13.19508063302946, 13.73446231903253, 14.42451995678790, 15.18130576687009, 15.63112999906630, 16.20283349376123, 16.73404337763195, 17.56412888727163, 18.28882051162758

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