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Properties

Label 4-18432-1.1-c1e2-0-11
Degree $4$
Conductor $18432$
Sign $-1$
Analytic cond. $1.17524$
Root an. cond. $1.04119$
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  − 8·7-s + 9-s − 12·17-s − 6·25-s + 8·31-s + 4·41-s + 16·47-s + 34·49-s − 8·63-s − 32·71-s − 12·73-s + 8·79-s + 81-s + 20·89-s − 28·97-s − 24·103-s + 4·113-s + 96·119-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + 157-s + ⋯
L(s)  = 1  − 3.02·7-s + 1/3·9-s − 2.91·17-s − 6/5·25-s + 1.43·31-s + 0.624·41-s + 2.33·47-s + 34/7·49-s − 1.00·63-s − 3.79·71-s − 1.40·73-s + 0.900·79-s + 1/9·81-s + 2.11·89-s − 2.84·97-s − 2.36·103-s + 0.376·113-s + 8.80·119-s − 0.545·121-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 − 0.970·153-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18432 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18432 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(18432\)    =    \(2^{11} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(1.17524\)
Root analytic conductor: \(1.04119\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 18432,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.7.i_be
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
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$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.a_g
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.a_di
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.71.bg_pi
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.79.ai_gs
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.97.bc_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

−10.44913304188171401892835914004, −10.20586918892848667331432436971, −9.488699237661810053920246470193, −9.133901360741508328144538474134, −8.839886292016704738441339481672, −7.83182394757322498939219938639, −6.91469103626909406223255043804, −6.80946897903754876349072202165, −6.15267013015603292112906972931, −5.77092473843782707280190480124, −4.31158789218116792910899233548, −4.10625798014363516984369255368, −3.00571163091752467289605693825, −2.44821729992400328434125228202, 0, 2.44821729992400328434125228202, 3.00571163091752467289605693825, 4.10625798014363516984369255368, 4.31158789218116792910899233548, 5.77092473843782707280190480124, 6.15267013015603292112906972931, 6.80946897903754876349072202165, 6.91469103626909406223255043804, 7.83182394757322498939219938639, 8.839886292016704738441339481672, 9.133901360741508328144538474134, 9.488699237661810053920246470193, 10.20586918892848667331432436971, 10.44913304188171401892835914004

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