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Properties

Label 4-41472-1.1-c1e2-0-2
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s + 6·25-s + 4·29-s − 16·43-s + 16·47-s + 2·49-s + 4·53-s − 16·67-s − 4·73-s − 4·97-s − 12·101-s + 10·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 1.78·5-s + 6/5·25-s + 0.742·29-s − 2.43·43-s + 2.33·47-s + 2/7·49-s + 0.549·53-s − 1.95·67-s − 0.468·73-s − 0.406·97-s − 1.19·101-s + 0.909·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.461·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-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: \(0\)
Selberg data: \((4,\ 41472,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.787257678\)
\(L(\frac12)\) \(\approx\) \(1.787257678\)
\(L(\frac{3}{2})\) 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$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.5.ae_k
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.ae_ba
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.31.a_o
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.41.a_abu
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.q_fe
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.47.aq_fm
53$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.53.ae_ec
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.61.a_ak
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.q_ha
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.e_di
79$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.79.a_aek
83$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.83.a_g
89$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.89.a_ek
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−10.20112785355525978021686420612, −9.793874630729459108321058730438, −9.318884771790808129088719714626, −8.728225827780637437596320661999, −8.397481846764847666787406523046, −7.49657678256045028484353447998, −7.01544420731205789665335181084, −6.32380917769720974578055853068, −5.94195423883222522173962982434, −5.39251352388973439705582823019, −4.80906308846480705289436249292, −3.98581372221839079065980647652, −2.99136609483359382134054205460, −2.28090584644293007601298990383, −1.43579895495645945319504330384, 1.43579895495645945319504330384, 2.28090584644293007601298990383, 2.99136609483359382134054205460, 3.98581372221839079065980647652, 4.80906308846480705289436249292, 5.39251352388973439705582823019, 5.94195423883222522173962982434, 6.32380917769720974578055853068, 7.01544420731205789665335181084, 7.49657678256045028484353447998, 8.397481846764847666787406523046, 8.728225827780637437596320661999, 9.318884771790808129088719714626, 9.793874630729459108321058730438, 10.20112785355525978021686420612

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