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  + 2·7-s + 4·13-s − 6·17-s − 4·19-s + 6·23-s − 5·25-s + 6·29-s − 8·31-s − 10·37-s + 6·41-s + 8·43-s − 6·47-s − 3·49-s − 8·61-s + 4·67-s + 6·71-s − 2·73-s + 14·79-s + 12·83-s + 6·89-s + 8·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.755·7-s + 1.10·13-s − 1.45·17-s − 0.917·19-s + 1.25·23-s − 25-s + 1.11·29-s − 1.43·31-s − 1.64·37-s + 0.937·41-s + 1.21·43-s − 0.875·47-s − 3/7·49-s − 1.02·61-s + 0.488·67-s + 0.712·71-s − 0.234·73-s + 1.57·79-s + 1.31·83-s + 0.635·89-s + 0.838·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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  =  $(2,\ 17424,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.177391420$
$L(\frac12)$  $\approx$  $2.177391420$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.88342769722021, −15.28533571690598, −14.81840275643099, −14.15743613929795, −13.66466407013410, −13.10891609761953, −12.62433066648123, −11.91998717462277, −11.17242276236749, −10.89777266217461, −10.53923495598034, −9.484270532742746, −8.917540314219934, −8.583847778553026, −7.891234820719873, −7.231601065149410, −6.490474830453645, −6.094116011627113, −5.155508555289785, −4.661055942954321, −3.944093233419409, −3.287891751966839, −2.204326501668277, −1.720503610179550, −0.6238940091137152, 0.6238940091137152, 1.720503610179550, 2.204326501668277, 3.287891751966839, 3.944093233419409, 4.661055942954321, 5.155508555289785, 6.094116011627113, 6.490474830453645, 7.231601065149410, 7.891234820719873, 8.583847778553026, 8.917540314219934, 9.484270532742746, 10.53923495598034, 10.89777266217461, 11.17242276236749, 11.91998717462277, 12.62433066648123, 13.10891609761953, 13.66466407013410, 14.15743613929795, 14.81840275643099, 15.28533571690598, 15.88342769722021

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