Properties

Label 4-21e4-1.1-c3e2-0-4
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $677.032$
Root an. cond. $5.10096$
Motivic weight $3$
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  + 8·4-s − 140·13-s − 56·19-s + 125·25-s − 308·31-s − 110·37-s − 1.04e3·43-s − 1.12e3·52-s − 182·61-s − 512·64-s + 880·67-s − 1.19e3·73-s − 448·76-s − 884·79-s − 2.66e3·97-s + 1.00e3·100-s − 1.82e3·103-s + 646·109-s + 1.33e3·121-s − 2.46e3·124-s + 127-s + 131-s + 137-s + 139-s − 880·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 4-s − 2.98·13-s − 0.676·19-s + 25-s − 1.78·31-s − 0.488·37-s − 3.68·43-s − 2.98·52-s − 0.382·61-s − 64-s + 1.60·67-s − 1.90·73-s − 0.676·76-s − 1.25·79-s − 2.78·97-s + 100-s − 1.74·103-s + 0.567·109-s + 121-s − 1.78·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 0.488·148-s + 0.000549·149-s + 0.000538·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(677.032\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5455554896\)
\(L(\frac12)\) \(\approx\) \(0.5455554896\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
5$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 70 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 - 107 T + p^{3} T^{2} )( 1 + 163 T + p^{3} T^{2} ) \)
23$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 19 T + p^{3} T^{2} )( 1 + 289 T + p^{3} T^{2} ) \)
37$C_2$ \( ( 1 - 323 T + p^{3} T^{2} )( 1 + 433 T + p^{3} T^{2} ) \)
41$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 520 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
61$C_2$ \( ( 1 - 719 T + p^{3} T^{2} )( 1 + 901 T + p^{3} T^{2} ) \)
67$C_2$ \( ( 1 - 1007 T + p^{3} T^{2} )( 1 + 127 T + p^{3} T^{2} ) \)
71$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 271 T + p^{3} T^{2} )( 1 + 919 T + p^{3} T^{2} ) \)
79$C_2$ \( ( 1 - 503 T + p^{3} T^{2} )( 1 + 1387 T + p^{3} T^{2} ) \)
83$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 1330 T + p^{3} T^{2} )^{2} \)
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

−11.30453194257185347936891286881, −10.27956083209803175970728823408, −10.25081169568286480137101051612, −9.567272421880884948901266038398, −9.332757193244271912365143230436, −8.529049672316125847966618328334, −8.251388036442077242208349159031, −7.45779498421284055779773259075, −7.22313989641582163104713892595, −6.70967973899054809091711450815, −6.64668630759271982649604512118, −5.41828436421701897590208947581, −5.39489699364415509355371136752, −4.65829605843546070449210196188, −4.20949255296742531605535240181, −3.05964471367929791532014632990, −2.92776921553246232766972661783, −1.94842047405757395210193268107, −1.80493315745799743006436770142, −0.21407224702063321874449893990, 0.21407224702063321874449893990, 1.80493315745799743006436770142, 1.94842047405757395210193268107, 2.92776921553246232766972661783, 3.05964471367929791532014632990, 4.20949255296742531605535240181, 4.65829605843546070449210196188, 5.39489699364415509355371136752, 5.41828436421701897590208947581, 6.64668630759271982649604512118, 6.70967973899054809091711450815, 7.22313989641582163104713892595, 7.45779498421284055779773259075, 8.251388036442077242208349159031, 8.529049672316125847966618328334, 9.332757193244271912365143230436, 9.567272421880884948901266038398, 10.25081169568286480137101051612, 10.27956083209803175970728823408, 11.30453194257185347936891286881

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