Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 11 \cdot 31^{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  − 3-s + 2·5-s − 2·7-s + 9-s − 11-s + 2·13-s − 2·15-s − 4·17-s − 6·19-s + 2·21-s − 25-s − 27-s + 8·29-s + 33-s − 4·35-s − 10·37-s − 2·39-s + 8·41-s + 2·43-s + 2·45-s − 8·47-s − 3·49-s + 4·51-s + 2·53-s − 2·55-s + 6·57-s + 12·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.516·15-s − 0.970·17-s − 1.37·19-s + 0.436·21-s − 1/5·25-s − 0.192·27-s + 1.48·29-s + 0.174·33-s − 0.676·35-s − 1.64·37-s − 0.320·39-s + 1.24·41-s + 0.304·43-s + 0.298·45-s − 1.16·47-s − 3/7·49-s + 0.560·51-s + 0.274·53-s − 0.269·55-s + 0.794·57-s + 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126852 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126852 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(126852\)    =    \(2^{2} \cdot 3 \cdot 11 \cdot 31^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{126852} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 126852,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;31\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
11 \( 1 + T \)
31 \( 1 \)
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 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
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

−13.58760955879774, −13.20612221575989, −12.89747814583004, −12.35893228952460, −11.93873383427208, −11.09837253327688, −10.93101642135562, −10.32013497363073, −9.976257892195560, −9.464813950278765, −8.905757586323797, −8.416795465839633, −7.981819704406012, −6.970340636155780, −6.713587744547758, −6.325946770537594, −5.801191687013291, −5.339740204490234, −4.615251568740218, −4.201634964226513, −3.489796757220532, −2.792059785695254, −2.165431026587630, −1.679891094468205, −0.7409721318191995, 0, 0.7409721318191995, 1.679891094468205, 2.165431026587630, 2.792059785695254, 3.489796757220532, 4.201634964226513, 4.615251568740218, 5.339740204490234, 5.801191687013291, 6.325946770537594, 6.713587744547758, 6.970340636155780, 7.981819704406012, 8.416795465839633, 8.905757586323797, 9.464813950278765, 9.976257892195560, 10.32013497363073, 10.93101642135562, 11.09837253327688, 11.93873383427208, 12.35893228952460, 12.89747814583004, 13.20612221575989, 13.58760955879774

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