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Properties

Label 4-912e2-1.1-c1e2-0-47
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 2·7-s + 6·9-s + 8·19-s − 6·21-s − 2·25-s + 9·27-s + 18·29-s − 4·43-s − 11·49-s − 18·53-s + 24·57-s + 6·59-s − 16·61-s − 12·63-s + 24·71-s + 22·73-s − 6·75-s + 9·81-s + 54·87-s + 12·89-s − 6·107-s − 24·113-s + 10·121-s + 127-s − 12·129-s + 131-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.755·7-s + 2·9-s + 1.83·19-s − 1.30·21-s − 2/5·25-s + 1.73·27-s + 3.34·29-s − 0.609·43-s − 1.57·49-s − 2.47·53-s + 3.17·57-s + 0.781·59-s − 2.04·61-s − 1.51·63-s + 2.84·71-s + 2.57·73-s − 0.692·75-s + 81-s + 5.78·87-s + 1.27·89-s − 0.580·107-s − 2.25·113-s + 0.909·121-s + 0.0887·127-s − 1.05·129-s + 0.0873·131-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

\(L(1)\) \(\approx\) \(4.059507167\)
\(L(\frac12)\) \(\approx\) \(4.059507167\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.7.c_p
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.a_ax
17$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \) 2.17.a_abf
23$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \) 2.23.a_at
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.29.as_fj
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.43.e_dm
47$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.47.a_ade
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.53.s_hf
59$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.59.ag_ex
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2^2$ \( 1 - 59 T^{2} + p^{2} T^{4} \) 2.67.a_ach
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.73.aw_kh
79$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \) 2.79.a_aeg
83$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.83.a_acg
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.a_ac
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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

−10.03442972649494056335382487408, −9.602994139561981452123843556916, −9.515191572864907516077214823731, −9.294512334634324008400509642944, −8.386583985166649078361424791657, −8.302578397461485617962229000044, −7.919638778770828429176326313830, −7.54907452133010029137486358438, −6.88067588100079673853203211155, −6.47517911242109154942842088453, −6.35130228305099953158678794439, −5.36289130826234626919889280487, −4.84615344726309464006343551243, −4.56078980769246016118577449845, −3.68828982477715838594221815009, −3.31130025298000745540489865115, −3.02482974477889927597006412686, −2.48557632430000821629214565507, −1.69028020783213549665740772792, −0.923466411521167865382362748965, 0.923466411521167865382362748965, 1.69028020783213549665740772792, 2.48557632430000821629214565507, 3.02482974477889927597006412686, 3.31130025298000745540489865115, 3.68828982477715838594221815009, 4.56078980769246016118577449845, 4.84615344726309464006343551243, 5.36289130826234626919889280487, 6.35130228305099953158678794439, 6.47517911242109154942842088453, 6.88067588100079673853203211155, 7.54907452133010029137486358438, 7.919638778770828429176326313830, 8.302578397461485617962229000044, 8.386583985166649078361424791657, 9.294512334634324008400509642944, 9.515191572864907516077214823731, 9.602994139561981452123843556916, 10.03442972649494056335382487408

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