Properties

Label 4-28432-1.1-c1e2-0-0
Degree $4$
Conductor $28432$
Sign $1$
Analytic cond. $1.81284$
Root an. cond. $1.16035$
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  + 2-s − 4-s + 2·5-s − 3·8-s + 2·9-s + 2·10-s − 16-s + 8·17-s + 2·18-s − 2·20-s − 6·25-s + 2·29-s + 5·32-s + 8·34-s − 2·36-s − 6·40-s + 12·41-s + 4·45-s − 6·49-s − 6·50-s − 4·53-s + 2·58-s − 2·61-s + 7·64-s − 8·68-s − 6·72-s − 20·73-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.06·8-s + 2/3·9-s + 0.632·10-s − 1/4·16-s + 1.94·17-s + 0.471·18-s − 0.447·20-s − 6/5·25-s + 0.371·29-s + 0.883·32-s + 1.37·34-s − 1/3·36-s − 0.948·40-s + 1.87·41-s + 0.596·45-s − 6/7·49-s − 0.848·50-s − 0.549·53-s + 0.262·58-s − 0.256·61-s + 7/8·64-s − 0.970·68-s − 0.707·72-s − 2.34·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28432 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28432 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(28432\)    =    \(2^{4} \cdot 1777\)
Sign: $1$
Analytic conductor: \(1.81284\)
Root analytic conductor: \(1.16035\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 28432,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.772478234\)
\(L(\frac12)\) \(\approx\) \(1.772478234\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
1777$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 18 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
5$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.5.ac_k
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.ai_bu
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.ac_bi
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.a_k
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.43.a_cg
47$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.47.a_ak
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.e_dq
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.59.a_ak
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.c_bq
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.67.a_abi
71$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.71.a_ao
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.79.a_eg
83$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.83.a_acs
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.e_eo
97$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.ai_hy
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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

−10.33759283520705414330246409182, −9.925861680934046289344392881081, −9.684513040578502301331101523791, −9.132719232092148282025976190498, −8.483835829017907123698621486479, −7.72350325726488011846424008434, −7.45778784527001726286877639627, −6.41271233170927062310320545104, −5.92151120535948119665413671999, −5.57169388265821507841035523243, −4.84112876509543059977411580296, −4.17472410450190054935759446532, −3.48688867316083525298390165538, −2.66465623767694407982305826694, −1.40813581034372894689705112493, 1.40813581034372894689705112493, 2.66465623767694407982305826694, 3.48688867316083525298390165538, 4.17472410450190054935759446532, 4.84112876509543059977411580296, 5.57169388265821507841035523243, 5.92151120535948119665413671999, 6.41271233170927062310320545104, 7.45778784527001726286877639627, 7.72350325726488011846424008434, 8.483835829017907123698621486479, 9.132719232092148282025976190498, 9.684513040578502301331101523791, 9.925861680934046289344392881081, 10.33759283520705414330246409182

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