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Properties

Label 4-1184e2-1.1-c1e2-0-0
Degree $4$
Conductor $1401856$
Sign $1$
Analytic cond. $89.3835$
Root an. cond. $3.07478$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·9-s − 6·25-s + 2·37-s − 20·41-s − 14·49-s − 28·53-s + 12·73-s + 27·81-s + 4·101-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 2·9-s − 6/5·25-s + 0.328·37-s − 3.12·41-s − 2·49-s − 3.84·53-s + 1.40·73-s + 3·81-s + 0.398·101-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1401856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1401856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1401856\)    =    \(2^{10} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(89.3835\)
Root analytic conductor: \(3.07478\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1401856,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5651367930\)
\(L(\frac12)\) \(\approx\) \(0.5651367930\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.41.u_ha
43$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.43.a_adi
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.53.bc_lq
59$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.59.a_aeo
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.67.a_fe
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.79.a_agc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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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

−10.50948119523858406985507295777, −9.463178162563749165658667546576, −9.329791819739600874960477252523, −8.627403054786748208048664375201, −8.248493668774160775968796430180, −8.001799447951367241330174869746, −7.82940300944463022937953364262, −6.88174859611327535839881687718, −6.63852211821145231821294354239, −6.17053948069340967271182253313, −5.82067515382681722509550802376, −5.26203522072472271435850933361, −4.95098240246475257944088347505, −4.46607885112938751201893603231, −3.61987018759147037392354195495, −3.16066683988079724792026570252, −3.04316555091601773050747185261, −2.03548828488710953314284376557, −1.68261929239363820604726831753, −0.30734160942371176695811949259, 0.30734160942371176695811949259, 1.68261929239363820604726831753, 2.03548828488710953314284376557, 3.04316555091601773050747185261, 3.16066683988079724792026570252, 3.61987018759147037392354195495, 4.46607885112938751201893603231, 4.95098240246475257944088347505, 5.26203522072472271435850933361, 5.82067515382681722509550802376, 6.17053948069340967271182253313, 6.63852211821145231821294354239, 6.88174859611327535839881687718, 7.82940300944463022937953364262, 8.001799447951367241330174869746, 8.248493668774160775968796430180, 8.627403054786748208048664375201, 9.329791819739600874960477252523, 9.463178162563749165658667546576, 10.50948119523858406985507295777

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