Properties

Label 4-1638e2-1.1-c1e2-0-10
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 − 5·7-s − 8-s − 5·11-s − 2·13-s − 5·14-s − 16-s + 7·17-s − 7·19-s − 5·22-s + 2·23-s + 5·25-s − 2·26-s + 18·29-s + 7·34-s − 4·37-s − 7·38-s − 8·41-s + 4·43-s + 2·46-s − 3·47-s + 18·49-s + 5·50-s + 53-s + 5·56-s + 18·58-s + 7·59-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.88·7-s − 0.353·8-s − 1.50·11-s − 0.554·13-s − 1.33·14-s − 1/4·16-s + 1.69·17-s − 1.60·19-s − 1.06·22-s + 0.417·23-s + 25-s − 0.392·26-s + 3.34·29-s + 1.20·34-s − 0.657·37-s − 1.13·38-s − 1.24·41-s + 0.609·43-s + 0.294·46-s − 0.437·47-s + 18/7·49-s + 0.707·50-s + 0.137·53-s + 0.668·56-s + 2.36·58-s + 0.911·59-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

\(L(1)\) \(\approx\) \(1.041320306\)
\(L(\frac12)\) \(\approx\) \(1.041320306\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 5 T + p T^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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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

−9.764717050405730391949446399441, −9.295491744358619932618581458772, −8.481213636942509685412672167072, −8.387192099684624494284264448779, −8.362730255356323489181195412855, −7.25946836630580690033140738911, −7.07025445252110664646804122261, −6.92656342783635747472832860635, −6.14654595655446289822978818093, −5.91383208341202130680338410728, −5.64903030720755033003214975341, −4.90140638607968982471553293470, −4.64080503531638618810366248364, −4.30431630219931507322813451246, −3.34940118302461705559251281734, −3.21109064944812983287736315743, −2.75902094013391765351294794265, −2.47461252438090593952243785767, −1.29561055762111638574518192723, −0.36097349834419984743586074329, 0.36097349834419984743586074329, 1.29561055762111638574518192723, 2.47461252438090593952243785767, 2.75902094013391765351294794265, 3.21109064944812983287736315743, 3.34940118302461705559251281734, 4.30431630219931507322813451246, 4.64080503531638618810366248364, 4.90140638607968982471553293470, 5.64903030720755033003214975341, 5.91383208341202130680338410728, 6.14654595655446289822978818093, 6.92656342783635747472832860635, 7.07025445252110664646804122261, 7.25946836630580690033140738911, 8.362730255356323489181195412855, 8.387192099684624494284264448779, 8.481213636942509685412672167072, 9.295491744358619932618581458772, 9.764717050405730391949446399441

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