Properties

Label 4-2592e2-1.1-c2e2-0-3
Degree $4$
Conductor $6718464$
Sign $1$
Analytic cond. $4988.15$
Root an. cond. $8.40398$
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  + 14·11-s − 2·17-s − 34·19-s + 50·25-s + 46·41-s + 14·43-s + 98·49-s − 82·59-s + 62·67-s + 142·73-s − 316·83-s + 292·89-s + 94·97-s − 178·107-s + 196·113-s + 121·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 338·169-s + 173-s + ⋯
L(s)  = 1  + 1.27·11-s − 0.117·17-s − 1.78·19-s + 2·25-s + 1.12·41-s + 0.325·43-s + 2·49-s − 1.38·59-s + 0.925·67-s + 1.94·73-s − 3.80·83-s + 3.28·89-s + 0.969·97-s − 1.66·107-s + 1.73·113-s + 121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2·169-s + 0.00578·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6718464\)    =    \(2^{10} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(4988.15\)
Root analytic conductor: \(8.40398\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6718464,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.901852610\)
\(L(\frac12)\) \(\approx\) \(3.901852610\)
\(L(2)\) 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 \( 1 \)
3 \( 1 \)
good5$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
11$C_2^2$ \( 1 - 14 T + 75 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
17$C_2^2$ \( 1 + 2 T - 285 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \)
19$C_2^2$ \( 1 + 34 T + 795 T^{2} + 34 p^{2} T^{3} + p^{4} T^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
41$C_2^2$ \( 1 - 46 T + 435 T^{2} - 46 p^{2} T^{3} + p^{4} T^{4} \)
43$C_2^2$ \( 1 - 14 T - 1653 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
59$C_2^2$ \( 1 + 82 T + 3243 T^{2} + 82 p^{2} T^{3} + p^{4} T^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
67$C_2^2$ \( 1 - 62 T - 645 T^{2} - 62 p^{2} T^{3} + p^{4} T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2^2$ \( 1 - 142 T + 14835 T^{2} - 142 p^{2} T^{3} + p^{4} T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
83$C_2$ \( ( 1 + 158 T + p^{2} T^{2} )^{2} \)
89$C_2$ \( ( 1 - 146 T + p^{2} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T - 573 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} \)
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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

−8.857182153747146816385881363608, −8.729406144549132514594794571611, −8.022252042629259576671393388416, −7.973387100860133123746028805804, −7.14953962163162773312225419964, −7.02760255156079760739128507717, −6.51635561570790054788188018049, −6.41296118084383562494311376575, −5.68103403440282372830941220883, −5.63564896001323633048724772797, −4.74830322761990273054316301531, −4.56290271023946992281934092488, −4.14359761278418028037056789355, −3.77354038379930965977104934141, −3.16177518169495614150526879708, −2.71727117213999009817114460463, −2.13507806369967156799116807989, −1.71497533667526269050134707403, −0.897807411443900502707627615893, −0.58471480027150311601828499744, 0.58471480027150311601828499744, 0.897807411443900502707627615893, 1.71497533667526269050134707403, 2.13507806369967156799116807989, 2.71727117213999009817114460463, 3.16177518169495614150526879708, 3.77354038379930965977104934141, 4.14359761278418028037056789355, 4.56290271023946992281934092488, 4.74830322761990273054316301531, 5.63564896001323633048724772797, 5.68103403440282372830941220883, 6.41296118084383562494311376575, 6.51635561570790054788188018049, 7.02760255156079760739128507717, 7.14953962163162773312225419964, 7.973387100860133123746028805804, 8.022252042629259576671393388416, 8.729406144549132514594794571611, 8.857182153747146816385881363608

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