Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 11^{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 − 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 + 12·59-s + 10·61-s − 4·65-s − 12·67-s − 8·71-s − 6·73-s + 2·79-s − 16·83-s + 8·85-s + 14·89-s − 4·91-s − 12·95-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.755·7-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 + 1.56·59-s + 1.28·61-s − 0.496·65-s − 1.46·67-s − 0.949·71-s − 0.702·73-s + 0.225·79-s − 1.75·83-s + 0.867·85-s + 1.48·89-s − 0.419·91-s − 1.23·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(69696\)    =    \(2^{6} \cdot 3^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{69696} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 69696,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 \)
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

−14.38227283306540, −14.14312381638523, −13.26418446362410, −13.04877112602439, −12.28960642639212, −12.04799364403350, −11.35807937145454, −10.78019826774671, −10.22738489803651, −10.06366797436295, −9.302575667640989, −8.670721471723700, −8.473587248237279, −7.561887722649132, −7.316580586810413, −6.520624637549553, −6.020331846436753, −5.424445709979690, −5.046023736961219, −4.329122974347763, −3.780103610380525, −2.918141752344186, −2.262332492860653, −1.779831797588810, −1.075130590742313, 0, 1.075130590742313, 1.779831797588810, 2.262332492860653, 2.918141752344186, 3.780103610380525, 4.329122974347763, 5.046023736961219, 5.424445709979690, 6.020331846436753, 6.520624637549553, 7.316580586810413, 7.561887722649132, 8.473587248237279, 8.670721471723700, 9.302575667640989, 10.06366797436295, 10.22738489803651, 10.78019826774671, 11.35807937145454, 12.04799364403350, 12.28960642639212, 13.04877112602439, 13.26418446362410, 14.14312381638523, 14.38227283306540

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