Properties

Label 4-182e2-1.1-c1e2-0-11
Degree $4$
Conductor $33124$
Sign $1$
Analytic cond. $2.11201$
Root an. cond. $1.20551$
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  + 2·2-s + 3·4-s + 7-s + 4·8-s + 3·9-s − 4·11-s + 2·14-s + 5·16-s + 6·18-s − 8·22-s − 8·23-s − 9·25-s + 3·28-s + 4·29-s + 6·32-s + 9·36-s + 6·37-s − 10·43-s − 12·44-s − 16·46-s − 6·49-s − 18·50-s + 24·53-s + 4·56-s + 8·58-s + 3·63-s + 7·64-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.377·7-s + 1.41·8-s + 9-s − 1.20·11-s + 0.534·14-s + 5/4·16-s + 1.41·18-s − 1.70·22-s − 1.66·23-s − 9/5·25-s + 0.566·28-s + 0.742·29-s + 1.06·32-s + 3/2·36-s + 0.986·37-s − 1.52·43-s − 1.80·44-s − 2.35·46-s − 6/7·49-s − 2.54·50-s + 3.29·53-s + 0.534·56-s + 1.05·58-s + 0.377·63-s + 7/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33124 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33124 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(33124\)    =    \(2^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(2.11201\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 33124,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.963921441\)
\(L(\frac12)\) \(\approx\) \(2.963921441\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7$C_2$ \( 1 - T + p T^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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

−10.28025762974353066563394220777, −10.09771482430661132709989954261, −9.844456230784246016329367154502, −8.614986352743724302378425623006, −8.182719893822782692587346408124, −7.45985560667588965978165906144, −7.34214085422022263171286152752, −6.38518439150442613685083077370, −5.88908450915949561478136129218, −5.38416318583757143224345400913, −4.61153453491182141362175201622, −4.21013218923093497223883532225, −3.52000805801584107473496848741, −2.50572325173735716694981409632, −1.80655038993973180674017505592, 1.80655038993973180674017505592, 2.50572325173735716694981409632, 3.52000805801584107473496848741, 4.21013218923093497223883532225, 4.61153453491182141362175201622, 5.38416318583757143224345400913, 5.88908450915949561478136129218, 6.38518439150442613685083077370, 7.34214085422022263171286152752, 7.45985560667588965978165906144, 8.182719893822782692587346408124, 8.614986352743724302378425623006, 9.844456230784246016329367154502, 10.09771482430661132709989954261, 10.28025762974353066563394220777

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