Properties

Label 4-1638e2-1.1-c1e2-0-13
Degree $4$
Conductor $2683044$
Sign $1$
Analytic cond. $171.073$
Root an. cond. $3.61655$
Motivic weight $1$
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  − 2-s − 2·5-s − 7-s + 8-s + 2·10-s + 3·11-s − 2·13-s + 14-s − 16-s + 17-s + 3·19-s − 3·22-s + 5·25-s + 2·26-s − 6·29-s + 4·31-s − 34-s + 2·35-s + 2·37-s − 3·38-s − 2·40-s − 4·41-s + 12·43-s − 9·47-s − 6·49-s − 5·50-s + 53-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s + 0.904·11-s − 0.554·13-s + 0.267·14-s − 1/4·16-s + 0.242·17-s + 0.688·19-s − 0.639·22-s + 25-s + 0.392·26-s − 1.11·29-s + 0.718·31-s − 0.171·34-s + 0.338·35-s + 0.328·37-s − 0.486·38-s − 0.316·40-s − 0.624·41-s + 1.82·43-s − 1.31·47-s − 6/7·49-s − 0.707·50-s + 0.137·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2683044\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(171.073\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2683044,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7806040529\)
\(L(\frac12)\) \(\approx\) \(0.7806040529\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 + T + p T^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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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

−9.672481010363890674105407211481, −9.208571340503914392687349137221, −8.893036341207232356492805279194, −8.332366972309569115980189846031, −7.940742001477880350837348973020, −7.81161454955308346082540621441, −7.17232402715223280822933737689, −6.86608899977973603398128103009, −6.66592709979992486070529473756, −5.80425415664890855541541799128, −5.70778858712568040992584885481, −4.89023396517969183226630630722, −4.64741608702335347379096003528, −3.91882111798872013054063487347, −3.84764931436428616548726039874, −2.96933749095443871348859057802, −2.81050914477012288358012806517, −1.78301854277926037327661008589, −1.23258050980158801320788493223, −0.42831090412420055374994169855, 0.42831090412420055374994169855, 1.23258050980158801320788493223, 1.78301854277926037327661008589, 2.81050914477012288358012806517, 2.96933749095443871348859057802, 3.84764931436428616548726039874, 3.91882111798872013054063487347, 4.64741608702335347379096003528, 4.89023396517969183226630630722, 5.70778858712568040992584885481, 5.80425415664890855541541799128, 6.66592709979992486070529473756, 6.86608899977973603398128103009, 7.17232402715223280822933737689, 7.81161454955308346082540621441, 7.940742001477880350837348973020, 8.332366972309569115980189846031, 8.893036341207232356492805279194, 9.208571340503914392687349137221, 9.672481010363890674105407211481

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