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Properties

Label 4-912e2-1.1-c1e2-0-33
Degree $4$
Conductor $831744$
Sign $1$
Analytic cond. $53.0327$
Root an. cond. $2.69858$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·7-s + 4·11-s − 5·13-s + 4·17-s + 8·19-s + 2·21-s − 4·23-s + 5·25-s + 27-s + 8·29-s + 6·31-s − 4·33-s + 6·37-s + 5·39-s + 12·41-s − 43-s − 6·47-s − 11·49-s − 4·51-s − 4·53-s − 8·57-s + 10·59-s + 13·61-s + 11·67-s + 4·69-s + 6·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1.20·11-s − 1.38·13-s + 0.970·17-s + 1.83·19-s + 0.436·21-s − 0.834·23-s + 25-s + 0.192·27-s + 1.48·29-s + 1.07·31-s − 0.696·33-s + 0.986·37-s + 0.800·39-s + 1.87·41-s − 0.152·43-s − 0.875·47-s − 1.57·49-s − 0.560·51-s − 0.549·53-s − 1.05·57-s + 1.30·59-s + 1.66·61-s + 1.34·67-s + 0.481·69-s + 0.712·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(831744\)    =    \(2^{8} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(53.0327\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 831744,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.721569065\)
\(L(\frac12)\) \(\approx\) \(1.721569065\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + T + T^{2} \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.7.c_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.ae_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.ai_bj
31$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.31.ag_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.am_dz
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) 2.43.b_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.e_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$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.al_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.ab_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.ag_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

−9.976310125551603916714078857945, −9.846940528791508155438682464043, −9.614120158777879377862355068818, −9.402993109267397406533377407274, −8.465389779110257953042058928766, −8.309023106357371923203651429449, −7.63332701472433553381900100158, −7.39410359462783531046833040830, −6.64925044299159055433327378284, −6.53926519184705695833700468458, −6.12592983552435378604554141944, −5.46339732494784042743075950482, −4.91374114911062976195849874026, −4.83845410283345096510038428529, −3.87269598346932198121200767351, −3.58752547743725086819724628841, −2.75316935781477094691283772688, −2.54014659762008265712800764059, −1.23184700208324681817707627677, −0.76300889156741984406867492217, 0.76300889156741984406867492217, 1.23184700208324681817707627677, 2.54014659762008265712800764059, 2.75316935781477094691283772688, 3.58752547743725086819724628841, 3.87269598346932198121200767351, 4.83845410283345096510038428529, 4.91374114911062976195849874026, 5.46339732494784042743075950482, 6.12592983552435378604554141944, 6.53926519184705695833700468458, 6.64925044299159055433327378284, 7.39410359462783531046833040830, 7.63332701472433553381900100158, 8.309023106357371923203651429449, 8.465389779110257953042058928766, 9.402993109267397406533377407274, 9.614120158777879377862355068818, 9.846940528791508155438682464043, 9.976310125551603916714078857945

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