Properties

Label 4-1216e2-1.1-c2e2-0-1
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $1097.83$
Root an. cond. $5.75617$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·5-s + 10·7-s + 5·9-s − 20·11-s + 30·17-s − 12·19-s − 70·23-s − 2·25-s − 80·35-s − 40·43-s − 40·45-s − 20·47-s − 23·49-s + 160·55-s + 80·61-s + 50·63-s + 210·73-s − 200·77-s − 56·81-s − 80·83-s − 240·85-s + 96·95-s − 100·99-s + 560·115-s + 300·119-s + 58·121-s + 344·125-s + ⋯
L(s)  = 1  − 8/5·5-s + 10/7·7-s + 5/9·9-s − 1.81·11-s + 1.76·17-s − 0.631·19-s − 3.04·23-s − 0.0799·25-s − 2.28·35-s − 0.930·43-s − 8/9·45-s − 0.425·47-s − 0.469·49-s + 2.90·55-s + 1.31·61-s + 0.793·63-s + 2.87·73-s − 2.59·77-s − 0.691·81-s − 0.963·83-s − 2.82·85-s + 1.01·95-s − 1.01·99-s + 4.86·115-s + 2.52·119-s + 0.479·121-s + 2.75·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1097.83\)
Root analytic conductor: \(5.75617\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3409544982\)
\(L(\frac12)\) \(\approx\) \(0.3409544982\)
\(L(2)\) 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 \( 1 \)
19$C_2$ \( 1 + 12 T + p^{2} T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{4} T^{4} \)
5$C_2$ \( ( 1 + 4 T + p^{2} T^{2} )^{2} \)
7$C_2$ \( ( 1 - 5 T + p^{2} T^{2} )^{2} \)
11$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 p T^{2} + p^{4} T^{4} \)
17$C_2$ \( ( 1 - 15 T + p^{2} T^{2} )^{2} \)
23$C_2$ \( ( 1 + 35 T + p^{2} T^{2} )^{2} \)
29$C_2^2$ \( 1 - 1357 T^{2} + p^{4} T^{4} \)
31$C_2^2$ \( 1 - 622 T^{2} + p^{4} T^{4} \)
37$C_2^2$ \( 1 - 2270 T^{2} + p^{4} T^{4} \)
41$C_2^2$ \( 1 - 2062 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 20 T + p^{2} T^{2} )^{2} \)
47$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{2} \)
53$C_2^2$ \( 1 + 115 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 6637 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 40 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 7405 T^{2} + p^{4} T^{4} \)
71$C_2$ \( ( 1 - 92 T + p^{2} T^{2} )( 1 + 92 T + p^{2} T^{2} ) \)
73$C_2$ \( ( 1 - 105 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 11182 T^{2} + p^{4} T^{4} \)
83$C_2$ \( ( 1 + 40 T + p^{2} T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
97$C_2^2$ \( 1 - 3790 T^{2} + p^{4} T^{4} \)
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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

−9.855705431160310149988817256651, −9.589567326250622356930278152546, −8.471060404125945718378618311284, −8.313943586375826253115996016706, −7.953874850726800222253321761387, −7.935097899905143739865265091968, −7.58252707996188553207210961000, −7.08250472108462689397100210186, −6.45442594173889976787522414484, −5.69627482715020209623018994046, −5.62871307630947176080034633845, −4.94434417578348280169650173334, −4.55960095006345851383012882988, −4.17253266087763365785599155327, −3.47817550771349350694678232705, −3.46198570509865113918864939610, −2.20983681933586124752378825093, −2.08797418286486006571866143206, −1.20854645194228079265580948691, −0.18093373546281496422424141430, 0.18093373546281496422424141430, 1.20854645194228079265580948691, 2.08797418286486006571866143206, 2.20983681933586124752378825093, 3.46198570509865113918864939610, 3.47817550771349350694678232705, 4.17253266087763365785599155327, 4.55960095006345851383012882988, 4.94434417578348280169650173334, 5.62871307630947176080034633845, 5.69627482715020209623018994046, 6.45442594173889976787522414484, 7.08250472108462689397100210186, 7.58252707996188553207210961000, 7.935097899905143739865265091968, 7.953874850726800222253321761387, 8.313943586375826253115996016706, 8.471060404125945718378618311284, 9.589567326250622356930278152546, 9.855705431160310149988817256651

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