Properties

Label 4-78e2-1.1-c1e2-0-2
Degree $4$
Conductor $6084$
Sign $1$
Analytic cond. $0.387921$
Root an. cond. $0.789197$
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  + 2·3-s − 4-s + 3·9-s − 2·12-s − 6·13-s + 16-s − 4·17-s + 8·23-s + 6·25-s + 4·27-s − 20·29-s − 3·36-s − 12·39-s + 8·43-s + 2·48-s + 10·49-s − 8·51-s + 6·52-s − 12·53-s + 4·61-s − 64-s + 4·68-s + 16·69-s + 12·75-s + 5·81-s − 40·87-s − 8·92-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s − 1.66·13-s + 1/4·16-s − 0.970·17-s + 1.66·23-s + 6/5·25-s + 0.769·27-s − 3.71·29-s − 1/2·36-s − 1.92·39-s + 1.21·43-s + 0.288·48-s + 10/7·49-s − 1.12·51-s + 0.832·52-s − 1.64·53-s + 0.512·61-s − 1/8·64-s + 0.485·68-s + 1.92·69-s + 1.38·75-s + 5/9·81-s − 4.28·87-s − 0.834·92-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6084\)    =    \(2^{2} \cdot 3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.387921\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{78} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6084,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.069277895\)
\(L(\frac12)\) \(\approx\) \(1.069277895\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
13$C_2$ \( 1 + 6 T + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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

−14.81502284481490964825941282866, −14.21173017908092884179812633889, −13.74217914458887684779145517637, −12.91154339683631396817715304873, −12.89700338987594582710171365238, −12.35987154513940471821027925503, −11.12830875965079958923609098612, −11.10594759747666845413619050874, −10.08531224531131389879503994586, −9.511202414981599377293774012324, −8.984026452504948289969072761145, −8.851745622938911644148999350723, −7.66415074492954259569590107456, −7.41831476835950984450226175229, −6.79786971902108167486082243354, −5.54761891959354108169770988651, −4.85249707345187774716120514420, −4.09590559657322186514860263713, −3.11282312486137849791066275163, −2.16401043466288981441025505835, 2.16401043466288981441025505835, 3.11282312486137849791066275163, 4.09590559657322186514860263713, 4.85249707345187774716120514420, 5.54761891959354108169770988651, 6.79786971902108167486082243354, 7.41831476835950984450226175229, 7.66415074492954259569590107456, 8.851745622938911644148999350723, 8.984026452504948289969072761145, 9.511202414981599377293774012324, 10.08531224531131389879503994586, 11.10594759747666845413619050874, 11.12830875965079958923609098612, 12.35987154513940471821027925503, 12.89700338987594582710171365238, 12.91154339683631396817715304873, 13.74217914458887684779145517637, 14.21173017908092884179812633889, 14.81502284481490964825941282866

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