Properties

Label 4-429e2-1.1-c1e2-0-7
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 $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·4-s − 4·7-s − 2·9-s + 4·12-s − 2·13-s + 12·16-s + 4·19-s + 4·21-s − 9·25-s + 5·27-s + 16·28-s − 6·31-s + 8·36-s − 22·37-s + 2·39-s − 8·43-s − 12·48-s − 2·49-s + 8·52-s − 4·57-s − 4·61-s + 8·63-s − 32·64-s − 2·67-s − 32·73-s + 9·75-s + ⋯
L(s)  = 1  − 0.577·3-s − 2·4-s − 1.51·7-s − 2/3·9-s + 1.15·12-s − 0.554·13-s + 3·16-s + 0.917·19-s + 0.872·21-s − 9/5·25-s + 0.962·27-s + 3.02·28-s − 1.07·31-s + 4/3·36-s − 3.61·37-s + 0.320·39-s − 1.21·43-s − 1.73·48-s − 2/7·49-s + 1.10·52-s − 0.529·57-s − 0.512·61-s + 1.00·63-s − 4·64-s − 0.244·67-s − 3.74·73-s + 1.03·75-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: \(2\)
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$C_2$ \( 1 + T + p T^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.2.a_e
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.a_j
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.23.a_ad
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.31.g_ct
37$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.37.w_hn
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.a_da
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.a_dy
59$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.59.a_en
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.67.c_ff
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.71.a_cj
73$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.73.bg_pm
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.89.a_ez
97$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.97.ba_nz
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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

−8.585957272776804427928347593039, −8.528601897910997907766991611046, −7.76116009742030896542487448970, −7.21661224338768342466420756206, −6.70861289370917133535141316633, −5.82135849339387357739621546932, −5.76971326436299567396516934204, −5.13164450675750106555447729059, −4.75768041533452306802121891710, −3.93044483272471932820766011024, −3.29304594675951575785139026233, −3.23573160090042234483516934690, −1.66131996401817111295419273715, 0, 0, 1.66131996401817111295419273715, 3.23573160090042234483516934690, 3.29304594675951575785139026233, 3.93044483272471932820766011024, 4.75768041533452306802121891710, 5.13164450675750106555447729059, 5.76971326436299567396516934204, 5.82135849339387357739621546932, 6.70861289370917133535141316633, 7.21661224338768342466420756206, 7.76116009742030896542487448970, 8.528601897910997907766991611046, 8.585957272776804427928347593039

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