↑

Properties

Label 4-8788-1.1-c1e2-0-1
Degree $4$
Conductor $8788$
Sign $1$
Analytic cond. $0.560330$
Root an. cond. $0.865189$
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 + 13-s + 16-s − 6·17-s − 25-s − 14·27-s + 12·29-s − 3·36-s + 2·39-s − 2·43-s + 2·48-s − 13·49-s − 12·51-s + 52-s + 16·61-s + 64-s − 6·68-s − 2·75-s + 16·79-s − 4·81-s + 24·87-s − 100-s − 24·101-s − 8·103-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s − 9-s + 0.577·12-s + 0.277·13-s + 1/4·16-s − 1.45·17-s − 1/5·25-s − 2.69·27-s + 2.22·29-s − 1/2·36-s + 0.320·39-s − 0.304·43-s + 0.288·48-s − 1.85·49-s − 1.68·51-s + 0.138·52-s + 2.04·61-s + 1/8·64-s − 0.727·68-s − 0.230·75-s + 1.80·79-s − 4/9·81-s + 2.57·87-s − 0.0999·100-s − 2.38·101-s − 0.788·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8788 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8788 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8788\)    =    \(2^{2} \cdot 13^{3}\)
Sign: $1$
Analytic conductor: \(0.560330\)
Root analytic conductor: \(0.865189\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8788,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.327050077\)
\(L(\frac12)\) \(\approx\) \(1.327050077\)
\(L(\frac{3}{2})\) 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$ \( ( 1 - T )( 1 + T ) \)
13$C_1$ \( 1 - T \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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

−11.52736358384083380205994337321, −11.21486999262721520146961750481, −10.59579744741587591987377046637, −9.848425754886298251834442498391, −9.252086119982469784390910809299, −8.631851122063147597031259612932, −8.289125051595985253760732717123, −7.85100333312502477176899037562, −6.76423132327857768444846951818, −6.43741307804194710670707219872, −5.56442463767257268744193982616, −4.68671862775252250000074194944, −3.64028761626013442697591583469, −2.86630826934752796258245605486, −2.19431989234797768682378556675, 2.19431989234797768682378556675, 2.86630826934752796258245605486, 3.64028761626013442697591583469, 4.68671862775252250000074194944, 5.56442463767257268744193982616, 6.43741307804194710670707219872, 6.76423132327857768444846951818, 7.85100333312502477176899037562, 8.289125051595985253760732717123, 8.631851122063147597031259612932, 9.252086119982469784390910809299, 9.848425754886298251834442498391, 10.59579744741587591987377046637, 11.21486999262721520146961750481, 11.52736358384083380205994337321

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