Properties

Label 4-429e2-1.1-c1e2-0-9
Degree $4$
Conductor $184041$
Sign $-1$
Analytic cond. $11.7346$
Root an. cond. $1.85083$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4-s + 3·9-s − 2·12-s − 2·13-s − 3·16-s − 10·17-s − 4·23-s + 2·25-s + 4·27-s − 2·29-s − 3·36-s − 4·39-s − 10·43-s − 6·48-s − 10·49-s − 20·51-s + 2·52-s + 8·53-s + 7·64-s + 10·68-s − 8·69-s + 4·75-s − 10·79-s + 5·81-s − 4·87-s + 4·92-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s − 0.554·13-s − 3/4·16-s − 2.42·17-s − 0.834·23-s + 2/5·25-s + 0.769·27-s − 0.371·29-s − 1/2·36-s − 0.640·39-s − 1.52·43-s − 0.866·48-s − 1.42·49-s − 2.80·51-s + 0.277·52-s + 1.09·53-s + 7/8·64-s + 1.21·68-s − 0.963·69-s + 0.461·75-s − 1.12·79-s + 5/9·81-s − 0.428·87-s + 0.417·92-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(184041\)    =    \(3^{2} \cdot 11^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(11.7346\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 184041,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad3$C_1$ \( ( 1 - T )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
17$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.k_cg
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.e_o
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.c_k
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.37.a_ag
41$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.41.a_c
43$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.k_dy
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.47.a_ao
53$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.ai_w
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.59.a_bu
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.67.a_de
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.71.a_c
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.k_gc
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.83.a_be
89$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.89.a_abi
97$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.97.a_de
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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

−8.860494698215530127063807797339, −8.497970162517267172412421665587, −8.207491730113895807631465130580, −7.48257230710667863743307259922, −6.98981697685309159863826431263, −6.66244953999459066797053731978, −6.09636590812008148818778014711, −5.17518429006024096144659161484, −4.69564765327541447085079814085, −4.27406084242480709662783503642, −3.76162170433943117975221951661, −2.93700569376732943734172751748, −2.30085494840846695028256733145, −1.78230359840574640133712286870, 0, 1.78230359840574640133712286870, 2.30085494840846695028256733145, 2.93700569376732943734172751748, 3.76162170433943117975221951661, 4.27406084242480709662783503642, 4.69564765327541447085079814085, 5.17518429006024096144659161484, 6.09636590812008148818778014711, 6.66244953999459066797053731978, 6.98981697685309159863826431263, 7.48257230710667863743307259922, 8.207491730113895807631465130580, 8.497970162517267172412421665587, 8.860494698215530127063807797339

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