Properties

Label 4-136e2-1.1-c1e2-0-0
Degree $4$
Conductor $18496$
Sign $1$
Analytic cond. $1.17932$
Root an. cond. $1.04209$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 6·5-s − 6·7-s + 2·9-s + 2·11-s + 4·13-s + 12·15-s + 2·17-s + 12·21-s + 2·23-s + 18·25-s − 6·27-s + 2·29-s − 2·31-s − 4·33-s + 36·35-s − 6·37-s − 8·39-s + 2·41-s − 12·45-s + 18·49-s − 4·51-s − 12·55-s + 2·61-s − 12·63-s − 24·65-s − 8·67-s + ⋯
L(s)  = 1  − 1.15·3-s − 2.68·5-s − 2.26·7-s + 2/3·9-s + 0.603·11-s + 1.10·13-s + 3.09·15-s + 0.485·17-s + 2.61·21-s + 0.417·23-s + 18/5·25-s − 1.15·27-s + 0.371·29-s − 0.359·31-s − 0.696·33-s + 6.08·35-s − 0.986·37-s − 1.28·39-s + 0.312·41-s − 1.78·45-s + 18/7·49-s − 0.560·51-s − 1.61·55-s + 0.256·61-s − 1.51·63-s − 2.97·65-s − 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(18496\)    =    \(2^{6} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1.17932\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 18496,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2394068819\)
\(L(\frac12)\) \(\approx\) \(0.2394068819\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
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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

−13.71249579319047191607397152491, −12.59556677850495166800387194345, −12.41116369979000795244311433793, −11.97051129764903560079559465174, −11.34998283212830154083112140063, −11.31494624200146311970892108287, −10.48398298727560462919490705387, −10.09143045403590978449096306444, −9.202507382653078951283222717035, −8.854843738420638136300929156248, −8.080179679553920263428034571344, −7.36939679385969926518453818079, −7.11515201242943145173795287105, −6.19492569190470344880461988066, −6.15322450586680848039117927547, −4.98280709685367234778156530488, −4.10703016265840933503824180951, −3.48480366989158766057115751419, −3.38791501966806882203789319927, −0.54099548438271655965028962282, 0.54099548438271655965028962282, 3.38791501966806882203789319927, 3.48480366989158766057115751419, 4.10703016265840933503824180951, 4.98280709685367234778156530488, 6.15322450586680848039117927547, 6.19492569190470344880461988066, 7.11515201242943145173795287105, 7.36939679385969926518453818079, 8.080179679553920263428034571344, 8.854843738420638136300929156248, 9.202507382653078951283222717035, 10.09143045403590978449096306444, 10.48398298727560462919490705387, 11.31494624200146311970892108287, 11.34998283212830154083112140063, 11.97051129764903560079559465174, 12.41116369979000795244311433793, 12.59556677850495166800387194345, 13.71249579319047191607397152491

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