Properties

Label 4-162e2-1.1-c1e2-0-2
Degree $4$
Conductor $26244$
Sign $1$
Analytic cond. $1.67334$
Root an. cond. $1.13735$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 2·7-s + 4·13-s + 16-s + 7·19-s − 25-s − 2·28-s − 2·31-s + 4·37-s + 4·43-s + 7·49-s + 4·52-s − 20·61-s + 64-s + 10·67-s − 14·73-s + 7·76-s − 8·79-s − 8·91-s + 97-s − 100-s + 101-s + 103-s + 107-s + 109-s − 2·112-s + 113-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.755·7-s + 1.10·13-s + 1/4·16-s + 1.60·19-s − 1/5·25-s − 0.377·28-s − 0.359·31-s + 0.657·37-s + 0.609·43-s + 49-s + 0.554·52-s − 2.56·61-s + 1/8·64-s + 1.22·67-s − 1.63·73-s + 0.802·76-s − 0.900·79-s − 0.838·91-s + 0.101·97-s − 0.0999·100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s − 0.188·112-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(26244\)    =    \(2^{2} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.67334\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{26244} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 26244,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.338675918\)
\(L(\frac12)\) \(\approx\) \(1.338675918\)
\(L(\frac{3}{2})\) 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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$C_2^2$ \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \)
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

−15.4738682026, −15.2061796229, −14.3541745719, −14.0955975719, −13.5214137619, −13.1169070671, −12.6552850422, −12.0414265806, −11.6443876848, −11.1681028432, −10.6400066683, −10.1236855964, −9.59662038792, −9.05283650211, −8.63287232751, −7.63374641497, −7.55355051287, −6.73743458349, −6.08281493088, −5.78552929899, −4.94064252938, −3.98416540164, −3.35198302075, −2.65475943668, −1.31523113022, 1.31523113022, 2.65475943668, 3.35198302075, 3.98416540164, 4.94064252938, 5.78552929899, 6.08281493088, 6.73743458349, 7.55355051287, 7.63374641497, 8.63287232751, 9.05283650211, 9.59662038792, 10.1236855964, 10.6400066683, 11.1681028432, 11.6443876848, 12.0414265806, 12.6552850422, 13.1169070671, 13.5214137619, 14.0955975719, 14.3541745719, 15.2061796229, 15.4738682026

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