Properties

Label 4-484128-1.1-c1e2-0-31
Degree $4$
Conductor $484128$
Sign $-1$
Analytic cond. $30.8684$
Root an. cond. $2.35710$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 2·5-s + 8-s + 9-s + 2·10-s − 14·13-s + 16-s + 14·17-s + 18-s + 2·20-s − 7·25-s − 14·26-s − 16·29-s + 32-s + 14·34-s + 36-s − 20·37-s + 2·40-s − 2·41-s + 2·45-s − 10·49-s − 7·50-s − 14·52-s − 4·53-s − 16·58-s + 12·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 3.88·13-s + 1/4·16-s + 3.39·17-s + 0.235·18-s + 0.447·20-s − 7/5·25-s − 2.74·26-s − 2.97·29-s + 0.176·32-s + 2.40·34-s + 1/6·36-s − 3.28·37-s + 0.316·40-s − 0.312·41-s + 0.298·45-s − 1.42·49-s − 0.989·50-s − 1.94·52-s − 0.549·53-s − 2.10·58-s + 1.53·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484128 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484128 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(484128\)    =    \(2^{5} \cdot 3^{2} \cdot 41^{2}\)
Sign: $-1$
Analytic conductor: \(30.8684\)
Root analytic conductor: \(2.35710\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 484128,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
41$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.5.ac_l
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.13.o_cx
17$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.17.ao_df
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.a_al
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.23.a_bq
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.29.q_es
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.31.a_bl
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.a_da
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.59.a_bl
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.67.a_fd
71$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.71.a_adf
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.73.ac_fr
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.83.a_bt
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.89.ag_hf
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−7.85036087354118043603776124871, −7.75592472796631807124661081148, −7.41213221502769362591996263945, −7.08254152630261167101762047985, −6.44821010025703438356174730880, −5.54857754180669542541462628828, −5.49499909546950468199128317821, −5.17681088119848965933027323367, −4.75649511013744037015141927715, −3.60416049175465027848015448281, −3.59393288410863630933359242281, −2.73461219069641236200963837828, −1.90766854502000788463134943618, −1.80360495700018896488038381445, 0, 1.80360495700018896488038381445, 1.90766854502000788463134943618, 2.73461219069641236200963837828, 3.59393288410863630933359242281, 3.60416049175465027848015448281, 4.75649511013744037015141927715, 5.17681088119848965933027323367, 5.49499909546950468199128317821, 5.54857754180669542541462628828, 6.44821010025703438356174730880, 7.08254152630261167101762047985, 7.41213221502769362591996263945, 7.75592472796631807124661081148, 7.85036087354118043603776124871

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