Properties

Label 4-112e2-1.1-c0e2-0-0
Degree $4$
Conductor $12544$
Sign $1$
Analytic cond. $0.00312428$
Root an. cond. $0.236421$
Motivic weight $0$
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  − 4-s − 2·11-s + 16-s − 2·29-s − 2·37-s + 2·43-s + 2·44-s − 49-s + 2·53-s − 64-s + 2·67-s − 81-s + 2·107-s + 2·109-s + 2·116-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·172-s + ⋯
L(s)  = 1  − 4-s − 2·11-s + 16-s − 2·29-s − 2·37-s + 2·43-s + 2·44-s − 49-s + 2·53-s − 64-s + 2·67-s − 81-s + 2·107-s + 2·109-s + 2·116-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·172-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(12544\)    =    \(2^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.00312428\)
Root analytic conductor: \(0.236421\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{112} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12544,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2824099001\)
\(L(\frac12)\) \(\approx\) \(0.2824099001\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−14.30069270320800848660558005175, −13.45286471755867347856734524811, −13.13005281311943114418937395804, −12.75795018895155456044755509988, −12.31230408122088277357194352773, −11.49301810311495345341512613038, −10.90122155092304599712070645171, −10.42149304749246711383091061266, −9.968337038803622567918569441583, −9.372527297341524736564459728269, −8.712784745011879189339577700650, −8.319056652516069500791959709910, −7.48250950853313495465445459749, −7.34148884483076194341677592755, −6.09862088320331059470531847771, −5.28799620715987171801146708145, −5.22631067709461204170447629166, −4.13114150743211819775629176162, −3.39688143770569238837513800590, −2.27803974715281569194064287612, 2.27803974715281569194064287612, 3.39688143770569238837513800590, 4.13114150743211819775629176162, 5.22631067709461204170447629166, 5.28799620715987171801146708145, 6.09862088320331059470531847771, 7.34148884483076194341677592755, 7.48250950853313495465445459749, 8.319056652516069500791959709910, 8.712784745011879189339577700650, 9.372527297341524736564459728269, 9.968337038803622567918569441583, 10.42149304749246711383091061266, 10.90122155092304599712070645171, 11.49301810311495345341512613038, 12.31230408122088277357194352773, 12.75795018895155456044755509988, 13.13005281311943114418937395804, 13.45286471755867347856734524811, 14.30069270320800848660558005175

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