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Properties

Label 4-171e2-1.1-c1e2-0-5
Degree $4$
Conductor $29241$
Sign $1$
Analytic cond. $1.86443$
Root an. cond. $1.16852$
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-s + 2·4-s + 2·7-s + 5·8-s + 4·11-s − 5·13-s + 2·14-s + 5·16-s − 4·17-s − 8·19-s + 4·22-s − 4·23-s + 5·25-s − 5·26-s + 4·28-s − 8·29-s − 6·31-s + 10·32-s − 4·34-s + 6·37-s − 8·38-s − 12·41-s + 43-s + 8·44-s − 4·46-s − 6·47-s − 11·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 4-s + 0.755·7-s + 1.76·8-s + 1.20·11-s − 1.38·13-s + 0.534·14-s + 5/4·16-s − 0.970·17-s − 1.83·19-s + 0.852·22-s − 0.834·23-s + 25-s − 0.980·26-s + 0.755·28-s − 1.48·29-s − 1.07·31-s + 1.76·32-s − 0.685·34-s + 0.986·37-s − 1.29·38-s − 1.87·41-s + 0.152·43-s + 1.20·44-s − 0.589·46-s − 0.875·47-s − 1.57·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(29241\)    =    \(3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1.86443\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 29241,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.295425420\)
\(L(\frac12)\) \(\approx\) \(2.295425420\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 + 8 T + p T^{2} \)
good2$C_2^2$ \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) 2.2.ab_ab
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.7.ac_p
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.f_m
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_ab
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_ah
29$C_2^2$ \( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_bj
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.31.g_ct
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.37.ag_df
41$C_2^2$ \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_dz
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) 2.43.ab_abq
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_al
53$C_2^2$ \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_abl
59$C_2^2$ \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.59.ak_bp
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.an_ee
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.l_cc
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_abj
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.73.al_bw
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) 2.79.b_ada
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_acb
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_adp
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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

−13.00358723600296357930201328921, −12.72990820154975166673138353569, −11.73795319515122549705207836049, −11.72597041430198767534386264957, −10.99314825236889749223271038992, −10.89319498300486686465672884818, −9.991703005421745185780464941918, −9.770082112684664126323073126712, −8.739122200387657435331285421480, −8.465345632336524695421519797279, −7.63756970877524069298361787844, −7.22863667257789865628661188720, −6.62899897096283456553033020916, −6.30003152929570013923168302982, −5.25237231027046152650155683618, −4.73389840297896153033691600548, −4.26218036196311968434327440702, −3.55767824986701375675868417535, −2.05930157279150413425232928117, −1.98588800280289030643560997342, 1.98588800280289030643560997342, 2.05930157279150413425232928117, 3.55767824986701375675868417535, 4.26218036196311968434327440702, 4.73389840297896153033691600548, 5.25237231027046152650155683618, 6.30003152929570013923168302982, 6.62899897096283456553033020916, 7.22863667257789865628661188720, 7.63756970877524069298361787844, 8.465345632336524695421519797279, 8.739122200387657435331285421480, 9.770082112684664126323073126712, 9.991703005421745185780464941918, 10.89319498300486686465672884818, 10.99314825236889749223271038992, 11.72597041430198767534386264957, 11.73795319515122549705207836049, 12.72990820154975166673138353569, 13.00358723600296357930201328921

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