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Properties

Label 4-972e2-1.1-c1e2-0-10
Degree $4$
Conductor $944784$
Sign $1$
Analytic cond. $60.2402$
Root an. cond. $2.78593$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Related objects

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·7-s + 10·13-s − 14·19-s − 10·25-s − 14·31-s − 20·37-s − 26·43-s + 34·49-s − 26·61-s + 22·67-s + 34·73-s + 34·79-s − 80·91-s − 38·97-s − 14·103-s + 34·109-s − 22·121-s + 127-s + 131-s + 112·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 3.02·7-s + 2.77·13-s − 3.21·19-s − 2·25-s − 2.51·31-s − 3.28·37-s − 3.96·43-s + 34/7·49-s − 3.32·61-s + 2.68·67-s + 3.97·73-s + 3.82·79-s − 8.38·91-s − 3.85·97-s − 1.37·103-s + 3.25·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 9.71·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 944784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 944784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(944784\)    =    \(2^{4} \cdot 3^{10}\)
Sign: $1$
Analytic conductor: \(60.2402\)
Root analytic conductor: \(2.78593\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 944784,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.7.i_be
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.13.ak_bz
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.19.o_dj
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.31.o_eh
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.43.ba_jv
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.61.ba_lf
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.67.aw_jv
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \) 2.73.abi_qt
79$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \) 2.79.abi_rf
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 + 19 T + p T^{2} )^{2} \) 2.97.bm_vj
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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

−12.82058976104446547776475724049, −12.12289525585334004101347249137, −12.12289525585334004101347249137, −11.03001156038449382906652178216, −11.03001156038449382906652178216, −10.44556393191177276025815146812, −10.44556393191177276025815146812, −9.560406478930196694623363606584, −9.560406478930196694623363606584, −8.836919150676161187346044032312, −8.836919150676161187346044032312, −8.048301102429563301256457101837, −8.048301102429563301256457101837, −6.72127363228842394258218909510, −6.72127363228842394258218909510, −6.37963360432194233363479609739, −6.37963360432194233363479609739, −5.37914816586411840428812893207, −5.37914816586411840428812893207, −3.91733393921845546677555767878, −3.91733393921845546677555767878, −3.37170556741266467753859084934, −3.37170556741266467753859084934, −1.91268222393441456759898930404, −1.91268222393441456759898930404, 0, 0, 1.91268222393441456759898930404, 1.91268222393441456759898930404, 3.37170556741266467753859084934, 3.37170556741266467753859084934, 3.91733393921845546677555767878, 3.91733393921845546677555767878, 5.37914816586411840428812893207, 5.37914816586411840428812893207, 6.37963360432194233363479609739, 6.37963360432194233363479609739, 6.72127363228842394258218909510, 6.72127363228842394258218909510, 8.048301102429563301256457101837, 8.048301102429563301256457101837, 8.836919150676161187346044032312, 8.836919150676161187346044032312, 9.560406478930196694623363606584, 9.560406478930196694623363606584, 10.44556393191177276025815146812, 10.44556393191177276025815146812, 11.03001156038449382906652178216, 11.03001156038449382906652178216, 12.12289525585334004101347249137, 12.12289525585334004101347249137, 12.82058976104446547776475724049

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