Properties

Label 4-21e4-1.1-c3e2-0-11
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $677.032$
Root an. cond. $5.10096$
Motivic weight $3$
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  + 4·2-s + 8·4-s + 18·5-s + 32·8-s + 72·10-s − 50·11-s − 72·13-s + 128·16-s + 126·17-s + 72·19-s + 144·20-s − 200·22-s + 14·23-s + 125·25-s − 288·26-s − 316·29-s + 36·31-s + 256·32-s + 504·34-s + 162·37-s + 288·38-s + 576·40-s + 540·41-s − 648·43-s − 400·44-s + 56·46-s − 72·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.60·5-s + 1.41·8-s + 2.27·10-s − 1.37·11-s − 1.53·13-s + 2·16-s + 1.79·17-s + 0.869·19-s + 1.60·20-s − 1.93·22-s + 0.126·23-s + 25-s − 2.17·26-s − 2.02·29-s + 0.208·31-s + 1.41·32-s + 2.54·34-s + 0.719·37-s + 1.22·38-s + 2.27·40-s + 2.05·41-s − 2.29·43-s − 1.37·44-s + 0.179·46-s − 0.223·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(677.032\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(8.924582409\)
\(L(\frac12)\) \(\approx\) \(8.924582409\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( 1 - p^{2} T + p^{3} T^{2} - p^{5} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 - 18 T + 199 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 50 T + 1169 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 36 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 126 T + 10963 T^{2} - 126 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 72 T - 1675 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 14 T - 11971 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 158 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 36 T - 28495 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 162 T - 24409 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 270 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 324 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 72 T - 98639 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 22 T - 148393 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 468 T + 13645 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 792 T + 400283 T^{2} + 792 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 232 T - 246939 T^{2} + 232 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 734 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 180 T - 356617 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 236 T - 437343 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 36 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 234 T - 650213 T^{2} - 234 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 468 T + p^{3} T^{2} )^{2} \)
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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

−11.11750125645526920489569439214, −10.37959236046794352065600998233, −9.973782429067275128168820471918, −9.707839772485549190304386758801, −9.590909811573335505431432400739, −8.645623040322338854191749242535, −7.73862628648548741588356976419, −7.67563324164051724819918980550, −7.35113664707516585753102203571, −6.55473114019585918765795369822, −5.81647534471014863245159835337, −5.54182344074111998080655328658, −5.28670544784905077182741335878, −4.84811686510826080903172194078, −4.25789809106925798375841256425, −3.28761552118328841932336690979, −3.03973785342368663940085066393, −2.10985879352052494052591334413, −1.82981204546499058838735934758, −0.75128843312636456064363221741, 0.75128843312636456064363221741, 1.82981204546499058838735934758, 2.10985879352052494052591334413, 3.03973785342368663940085066393, 3.28761552118328841932336690979, 4.25789809106925798375841256425, 4.84811686510826080903172194078, 5.28670544784905077182741335878, 5.54182344074111998080655328658, 5.81647534471014863245159835337, 6.55473114019585918765795369822, 7.35113664707516585753102203571, 7.67563324164051724819918980550, 7.73862628648548741588356976419, 8.645623040322338854191749242535, 9.590909811573335505431432400739, 9.707839772485549190304386758801, 9.973782429067275128168820471918, 10.37959236046794352065600998233, 11.11750125645526920489569439214

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