Properties

Label 4-147e2-1.1-c5e2-0-4
Degree $4$
Conductor $21609$
Sign $1$
Analytic cond. $555.847$
Root an. cond. $4.85555$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 64·4-s − 243·9-s + 3.07e3·16-s − 6.25e3·25-s − 1.55e4·36-s + 1.33e4·37-s − 4.49e4·43-s + 1.31e5·64-s − 7.58e4·67-s + 1.81e5·79-s + 5.90e4·81-s − 4.00e5·100-s + 4.95e5·109-s + 3.22e5·121-s + 127-s + 131-s + 137-s + 139-s − 7.46e5·144-s + 8.52e5·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 5.60e5·169-s − 2.87e6·172-s + ⋯
L(s)  = 1  + 2·4-s − 9-s + 3·16-s − 2·25-s − 2·36-s + 1.59·37-s − 3.70·43-s + 4·64-s − 2.06·67-s + 3.27·79-s + 81-s − 4·100-s + 3.99·109-s + 2·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 3·144-s + 3.19·148-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 1.50·169-s − 7.41·172-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21609\)    =    \(3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(555.847\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{147} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21609,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.790590819\)
\(L(\frac12)\) \(\approx\) \(3.790590819\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( 1 + p^{5} T^{2} \)
7 \( 1 \)
good2$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
5$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
11$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 427 T + p^{5} T^{2} )( 1 + 427 T + p^{5} T^{2} ) \)
17$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 3143 T + p^{5} T^{2} )( 1 + 3143 T + p^{5} T^{2} ) \)
23$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
29$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2723 T + p^{5} T^{2} )( 1 + 2723 T + p^{5} T^{2} ) \)
37$C_2$ \( ( 1 - 6661 T + p^{5} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 22475 T + p^{5} T^{2} )^{2} \)
47$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
53$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
59$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 38626 T + p^{5} T^{2} )( 1 + 38626 T + p^{5} T^{2} ) \)
67$C_2$ \( ( 1 + 37939 T + p^{5} T^{2} )^{2} \)
71$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 78127 T + p^{5} T^{2} )( 1 + 78127 T + p^{5} T^{2} ) \)
79$C_2$ \( ( 1 - 90857 T + p^{5} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
89$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 134386 T + p^{5} T^{2} )( 1 + 134386 T + p^{5} T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.05209667589065974776508992130, −11.80383264807515841549356492128, −11.49734065026930261320698469912, −11.00677069816009224198216718914, −10.52176348900673877468694182112, −9.871762248068828598134417346377, −9.568641317143513811998128737777, −8.487735949416690918645448220058, −8.128821895189613885053504448728, −7.61818635230342723819810588003, −7.05305830452699616447645318448, −6.36362963533110108365532922463, −6.02292028923262869382215871363, −5.53488486678782408201045875137, −4.61818483850676143900332679746, −3.33781404314844089602731556852, −3.24023912076418603559457399214, −2.10094004331203923642477091921, −1.86243862887864202761484032072, −0.61319383602466123003363729961, 0.61319383602466123003363729961, 1.86243862887864202761484032072, 2.10094004331203923642477091921, 3.24023912076418603559457399214, 3.33781404314844089602731556852, 4.61818483850676143900332679746, 5.53488486678782408201045875137, 6.02292028923262869382215871363, 6.36362963533110108365532922463, 7.05305830452699616447645318448, 7.61818635230342723819810588003, 8.128821895189613885053504448728, 8.487735949416690918645448220058, 9.568641317143513811998128737777, 9.871762248068828598134417346377, 10.52176348900673877468694182112, 11.00677069816009224198216718914, 11.49734065026930261320698469912, 11.80383264807515841549356492128, 12.05209667589065974776508992130

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