Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 11^{2} $
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 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(17424\)    =    \(2^{4} \cdot 3^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{17424} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 17424,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.058475574$
$L(\frac12)$  $\approx$  $1.058475574$
$L(\frac{3}{2})$   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

−15.95177301129892, −15.25780421491458, −14.82694306291888, −14.39195413583055, −13.38534255279198, −13.15763199999155, −12.53317587492180, −11.98823079121174, −11.32915958314514, −10.99850035890980, −10.12190311787466, −9.753028853473163, −9.024873633289371, −8.362169533304468, −7.832925507795445, −7.370321570925424, −6.402593969730050, −6.157175272252493, −5.344371274936327, −4.320513120100667, −4.017860740718832, −3.236367739813282, −2.584513658397476, −1.487524585815824, −0.4449890013012034, 0.4449890013012034, 1.487524585815824, 2.584513658397476, 3.236367739813282, 4.017860740718832, 4.320513120100667, 5.344371274936327, 6.157175272252493, 6.402593969730050, 7.370321570925424, 7.832925507795445, 8.362169533304468, 9.024873633289371, 9.753028853473163, 10.12190311787466, 10.99850035890980, 11.32915958314514, 11.98823079121174, 12.53317587492180, 13.15763199999155, 13.38534255279198, 14.39195413583055, 14.82694306291888, 15.25780421491458, 15.95177301129892

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