Properties

Label 4-429e2-1.1-c1e2-0-0
Degree $4$
Conductor $184041$
Sign $1$
Analytic cond. $11.7346$
Root an. cond. $1.85083$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·4-s + 4·5-s + 3·9-s + 4·11-s + 6·12-s − 8·15-s + 5·16-s − 12·20-s + 2·25-s − 4·27-s + 8·31-s − 8·33-s − 9·36-s − 4·37-s − 12·44-s + 12·45-s − 10·48-s + 2·49-s + 12·53-s + 16·55-s + 24·59-s + 24·60-s − 3·64-s − 16·67-s − 4·75-s + 20·80-s + ⋯
L(s)  = 1  − 1.15·3-s − 3/2·4-s + 1.78·5-s + 9-s + 1.20·11-s + 1.73·12-s − 2.06·15-s + 5/4·16-s − 2.68·20-s + 2/5·25-s − 0.769·27-s + 1.43·31-s − 1.39·33-s − 3/2·36-s − 0.657·37-s − 1.80·44-s + 1.78·45-s − 1.44·48-s + 2/7·49-s + 1.64·53-s + 2.15·55-s + 3.12·59-s + 3.09·60-s − 3/8·64-s − 1.95·67-s − 0.461·75-s + 2.23·80-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: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 184041,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.139890800\)
\(L(\frac12)\) \(\approx\) \(1.139890800\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 4 T + p T^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.2.a_d
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.5.ae_o
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
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_bm
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.a_acg
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.67.q_hq
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.89.e_ha
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−9.192740554674130435254875131876, −8.834175589947553165436256852057, −8.478952292319463444748938809705, −7.69260660341361319285944067655, −6.98840363121339158549167656149, −6.54556545042783689480045128663, −6.04842165221778667272231121190, −5.65486780279705070512904066158, −5.26961973469360163578902530891, −4.72415369923024439602351421783, −4.11268717107454539243862196325, −3.71205447024015551816570926916, −2.48759719466323003968210266499, −1.63373109287078803691952048226, −0.810173689173891123301166698476, 0.810173689173891123301166698476, 1.63373109287078803691952048226, 2.48759719466323003968210266499, 3.71205447024015551816570926916, 4.11268717107454539243862196325, 4.72415369923024439602351421783, 5.26961973469360163578902530891, 5.65486780279705070512904066158, 6.04842165221778667272231121190, 6.54556545042783689480045128663, 6.98840363121339158549167656149, 7.69260660341361319285944067655, 8.478952292319463444748938809705, 8.834175589947553165436256852057, 9.192740554674130435254875131876

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