Properties

Label 4-112e2-1.1-c1e2-0-4
Degree $4$
Conductor $12544$
Sign $1$
Analytic cond. $0.799816$
Root an. cond. $0.945687$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·5-s + 4·7-s + 3·9-s − 3·11-s + 4·13-s − 3·15-s − 3·17-s − 19-s + 4·21-s + 3·23-s + 5·25-s + 8·27-s − 12·29-s − 7·31-s − 3·33-s − 12·35-s + 37-s + 4·39-s + 12·41-s + 8·43-s − 9·45-s − 9·47-s + 9·49-s − 3·51-s − 3·53-s + 9·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s + 1.51·7-s + 9-s − 0.904·11-s + 1.10·13-s − 0.774·15-s − 0.727·17-s − 0.229·19-s + 0.872·21-s + 0.625·23-s + 25-s + 1.53·27-s − 2.22·29-s − 1.25·31-s − 0.522·33-s − 2.02·35-s + 0.164·37-s + 0.640·39-s + 1.87·41-s + 1.21·43-s − 1.34·45-s − 1.31·47-s + 9/7·49-s − 0.420·51-s − 0.412·53-s + 1.21·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12544\)    =    \(2^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.799816\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{112} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12544,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.188600650\)
\(L(\frac12)\) \(\approx\) \(1.188600650\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.10393178619044678202324047216, −13.15731248317590097915692395944, −12.77709897150667300725432770036, −12.69847235360425222607566044537, −11.46234776756091867399631572804, −11.29704054042343866233602528609, −10.96936263850379735102289178171, −10.44293406708577394241929796142, −9.492989260528380339639210897307, −8.872568670958147305227252780654, −8.392723117255656987425958037339, −7.974416743167004649698970511426, −7.25119079791951144259261746664, −7.19914933579514161714021419198, −5.86494321785812432165727224010, −5.14048749834935377839706087154, −4.14091713354568277008552963907, −4.13834254377419881126812925505, −2.86536388145239218181672587187, −1.61351067204266654033921989241, 1.61351067204266654033921989241, 2.86536388145239218181672587187, 4.13834254377419881126812925505, 4.14091713354568277008552963907, 5.14048749834935377839706087154, 5.86494321785812432165727224010, 7.19914933579514161714021419198, 7.25119079791951144259261746664, 7.974416743167004649698970511426, 8.392723117255656987425958037339, 8.872568670958147305227252780654, 9.492989260528380339639210897307, 10.44293406708577394241929796142, 10.96936263850379735102289178171, 11.29704054042343866233602528609, 11.46234776756091867399631572804, 12.69847235360425222607566044537, 12.77709897150667300725432770036, 13.15731248317590097915692395944, 14.10393178619044678202324047216

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