Properties

Label 4-1014e2-1.1-c1e2-0-5
Degree $4$
Conductor $1028196$
Sign $1$
Analytic cond. $65.5586$
Root an. cond. $2.84549$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4-s + 2·7-s − 2·9-s + 12-s + 16-s − 4·19-s + 2·21-s − 25-s − 5·27-s + 2·28-s + 8·31-s − 2·36-s + 14·37-s − 2·43-s + 48-s − 11·49-s − 4·57-s + 16·61-s − 4·63-s + 64-s − 28·67-s − 4·73-s − 75-s − 4·76-s + 16·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s + 0.755·7-s − 2/3·9-s + 0.288·12-s + 1/4·16-s − 0.917·19-s + 0.436·21-s − 1/5·25-s − 0.962·27-s + 0.377·28-s + 1.43·31-s − 1/3·36-s + 2.30·37-s − 0.304·43-s + 0.144·48-s − 1.57·49-s − 0.529·57-s + 2.04·61-s − 0.503·63-s + 1/8·64-s − 3.42·67-s − 0.468·73-s − 0.115·75-s − 0.458·76-s + 1.80·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1028196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028196 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1028196\)    =    \(2^{2} \cdot 3^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(65.5586\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1028196,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.868065512\)
\(L(\frac12)\) \(\approx\) \(2.868065512\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
13 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.934997178452817067223303679813, −7.85100333312502477176899037562, −7.46023947905323814388126308639, −6.63349412690092297262196593118, −6.43741307804194710670707219872, −5.90864573169190576818907822580, −5.56442463767257268744193982616, −4.68671862775252250000074194944, −4.60999990936908806684732327864, −3.94348155625524089488702113109, −3.19596255080634681974520626766, −2.86630826934752796258245605486, −2.19431989234797768682378556675, −1.76973285518057250387253289317, −0.74226904844910325205307127476, 0.74226904844910325205307127476, 1.76973285518057250387253289317, 2.19431989234797768682378556675, 2.86630826934752796258245605486, 3.19596255080634681974520626766, 3.94348155625524089488702113109, 4.60999990936908806684732327864, 4.68671862775252250000074194944, 5.56442463767257268744193982616, 5.90864573169190576818907822580, 6.43741307804194710670707219872, 6.63349412690092297262196593118, 7.46023947905323814388126308639, 7.85100333312502477176899037562, 7.934997178452817067223303679813

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