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Properties

Label 4-41472-1.1-c1e2-0-6
Degree $4$
Conductor $41472$
Sign $-1$
Analytic cond. $2.64429$
Root an. cond. $1.27519$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·11-s − 4·13-s − 8·23-s − 2·25-s − 4·37-s + 8·47-s + 2·49-s − 4·61-s − 8·71-s + 12·73-s − 8·83-s + 12·97-s − 16·107-s + 12·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 32·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯
L(s)  = 1  − 2.41·11-s − 1.10·13-s − 1.66·23-s − 2/5·25-s − 0.657·37-s + 1.16·47-s + 2/7·49-s − 0.512·61-s − 0.949·71-s + 1.40·73-s − 0.878·83-s + 1.21·97-s − 1.54·107-s + 1.14·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.67·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41472 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(41472\)    =    \(2^{9} \cdot 3^{4}\)
Sign: $-1$
Analytic conductor: \(2.64429\)
Root analytic conductor: \(1.27519\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 41472,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.11.i_bm
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.e_o
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.i_bu
29$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.29.a_c
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.31.a_o
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.e_o
41$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.41.a_co
43$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.43.a_aba
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.47.ai_dq
53$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.53.a_ada
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.67.a_cc
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.i_fm
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.am_eo
79$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.79.a_ade
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.i_eo
89$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.89.a_de
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.am_gk
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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

−10.04239490327039020804834100895, −9.671137631678653310194703102301, −8.947220995210199516220363970479, −8.235856561928500880487725685498, −7.919149368288401056596003007042, −7.43954650029933091969931577343, −6.97479060152944198107820505538, −5.96645258709776056472230677282, −5.63174825127725680977095102702, −4.98232615845313737476367504219, −4.45409902508691909602462330470, −3.52353181965425220918934352536, −2.60736399031644305889760904369, −2.13395303201845493039367507448, 0, 2.13395303201845493039367507448, 2.60736399031644305889760904369, 3.52353181965425220918934352536, 4.45409902508691909602462330470, 4.98232615845313737476367504219, 5.63174825127725680977095102702, 5.96645258709776056472230677282, 6.97479060152944198107820505538, 7.43954650029933091969931577343, 7.919149368288401056596003007042, 8.235856561928500880487725685498, 8.947220995210199516220363970479, 9.671137631678653310194703102301, 10.04239490327039020804834100895

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