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Properties

Label 4-143748-1.1-c1e2-0-6
Degree $4$
Conductor $143748$
Sign $-1$
Analytic cond. $9.16549$
Root an. cond. $1.73995$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 3·4-s − 2·6-s − 4·8-s + 9-s − 11-s + 3·12-s + 5·16-s − 12·17-s − 2·18-s + 2·22-s − 4·24-s − 10·25-s + 27-s + 12·29-s + 16·31-s − 6·32-s − 33-s + 24·34-s + 3·36-s − 20·37-s + 12·41-s − 3·44-s + 5·48-s − 10·49-s + 20·50-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.816·6-s − 1.41·8-s + 1/3·9-s − 0.301·11-s + 0.866·12-s + 5/4·16-s − 2.91·17-s − 0.471·18-s + 0.426·22-s − 0.816·24-s − 2·25-s + 0.192·27-s + 2.22·29-s + 2.87·31-s − 1.06·32-s − 0.174·33-s + 4.11·34-s + 1/2·36-s − 3.28·37-s + 1.87·41-s − 0.452·44-s + 0.721·48-s − 1.42·49-s + 2.82·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143748 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143748 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(143748\)    =    \(2^{2} \cdot 3^{3} \cdot 11^{3}\)
Sign: $-1$
Analytic conductor: \(9.16549\)
Root analytic conductor: \(1.73995\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 143748,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( 1 - T \)
11$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.79.a_abm
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.97.abc_pa
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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

−8.958855021658912251315559727512, −8.672134764185654449392944782672, −8.107075144714634957439631011601, −8.013104010237496671597514230187, −7.11812804222970713232777270208, −6.77506565559004391547494912024, −6.37459703187143981641083529758, −5.86376231731980296819583852288, −4.69333471332706230755932945396, −4.54132076139220643267324954006, −3.59300699702492954515692805621, −2.63915482624572011540809421288, −2.37545091357255938865951638025, −1.45668918986239249375622086896, 0, 1.45668918986239249375622086896, 2.37545091357255938865951638025, 2.63915482624572011540809421288, 3.59300699702492954515692805621, 4.54132076139220643267324954006, 4.69333471332706230755932945396, 5.86376231731980296819583852288, 6.37459703187143981641083529758, 6.77506565559004391547494912024, 7.11812804222970713232777270208, 8.013104010237496671597514230187, 8.107075144714634957439631011601, 8.672134764185654449392944782672, 8.958855021658912251315559727512

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