Properties

Label 4-18e4-1.1-c5e2-0-2
Degree $4$
Conductor $104976$
Sign $1$
Analytic cond. $2700.29$
Root an. cond. $7.20863$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 54·5-s + 88·7-s − 540·11-s + 418·13-s + 1.18e3·17-s + 1.67e3·19-s + 4.10e3·23-s + 3.12e3·25-s + 594·29-s − 4.25e3·31-s − 4.75e3·35-s − 596·37-s − 1.72e4·41-s + 1.21e4·43-s + 1.29e3·47-s + 1.68e4·49-s + 3.89e4·53-s + 2.91e4·55-s + 7.66e3·59-s + 3.47e4·61-s − 2.25e4·65-s − 2.18e4·67-s − 9.37e4·71-s + 1.35e5·73-s − 4.75e4·77-s + 7.69e4·79-s − 6.77e4·83-s + ⋯
L(s)  = 1  − 0.965·5-s + 0.678·7-s − 1.34·11-s + 0.685·13-s + 0.996·17-s + 1.06·19-s + 1.61·23-s + 25-s + 0.131·29-s − 0.795·31-s − 0.655·35-s − 0.0715·37-s − 1.60·41-s + 0.997·43-s + 0.0855·47-s + 49-s + 1.90·53-s + 1.29·55-s + 0.286·59-s + 1.19·61-s − 0.662·65-s − 0.593·67-s − 2.20·71-s + 2.96·73-s − 0.913·77-s + 1.38·79-s − 1.07·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(104976\)    =    \(2^{4} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(2700.29\)
Root analytic conductor: \(7.20863\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 104976,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.423104591\)
\(L(\frac12)\) \(\approx\) \(3.423104591\)
\(L(\frac{7}{2})\) 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$C_2^2$ \( 1 + 54 T - 209 T^{2} + 54 p^{5} T^{3} + p^{10} T^{4} \)
7$C_2^2$ \( 1 - 88 T - 9063 T^{2} - 88 p^{5} T^{3} + p^{10} T^{4} \)
11$C_2^2$ \( 1 + 540 T + 130549 T^{2} + 540 p^{5} T^{3} + p^{10} T^{4} \)
13$C_2^2$ \( 1 - 418 T - 196569 T^{2} - 418 p^{5} T^{3} + p^{10} T^{4} \)
17$C_2$ \( ( 1 - 594 T + p^{5} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 44 p T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 4104 T + 10406473 T^{2} - 4104 p^{5} T^{3} + p^{10} T^{4} \)
29$C_2^2$ \( 1 - 594 T - 20158313 T^{2} - 594 p^{5} T^{3} + p^{10} T^{4} \)
31$C_2^2$ \( 1 + 4256 T - 10515615 T^{2} + 4256 p^{5} T^{3} + p^{10} T^{4} \)
37$C_2$ \( ( 1 + 298 T + p^{5} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 17226 T + 180878875 T^{2} + 17226 p^{5} T^{3} + p^{10} T^{4} \)
43$C_2^2$ \( 1 - 12100 T - 598443 T^{2} - 12100 p^{5} T^{3} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 1296 T - 227665391 T^{2} - 1296 p^{5} T^{3} + p^{10} T^{4} \)
53$C_2$ \( ( 1 - 19494 T + p^{5} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 7668 T - 656126075 T^{2} - 7668 p^{5} T^{3} + p^{10} T^{4} \)
61$C_2^2$ \( 1 - 34738 T + 362132343 T^{2} - 34738 p^{5} T^{3} + p^{10} T^{4} \)
67$C_2^2$ \( 1 + 21812 T - 874361763 T^{2} + 21812 p^{5} T^{3} + p^{10} T^{4} \)
71$C_2$ \( ( 1 + 46872 T + p^{5} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 67562 T + p^{5} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 76912 T + 2838399345 T^{2} - 76912 p^{5} T^{3} + p^{10} T^{4} \)
83$C_2^2$ \( 1 + 67716 T + 646416013 T^{2} + 67716 p^{5} T^{3} + p^{10} T^{4} \)
89$C_2$ \( ( 1 - 29754 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 122398 T + 6393930147 T^{2} - 122398 p^{5} T^{3} + p^{10} T^{4} \)
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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

−10.93924823538706199944332151788, −10.68498397130898562629171487870, −10.20111016237332780534751718337, −9.703343829035993967420227198512, −8.804654938864817023428976233836, −8.775207778350747690106384689866, −8.119913496441408337851387342849, −7.68745361364208909805983655063, −7.14277190123082386922962872499, −7.08466225466417310760178082753, −5.95615042300228144968883086750, −5.48829332235196035976712933288, −5.02217756792865302365399463699, −4.64249939845563387571794624803, −3.65161833350762234541314170447, −3.40253923728506648921637001545, −2.71039620040613701431055701534, −1.90256510124934022263725276195, −0.858445915920504867879388613228, −0.68481102638740035370880259847, 0.68481102638740035370880259847, 0.858445915920504867879388613228, 1.90256510124934022263725276195, 2.71039620040613701431055701534, 3.40253923728506648921637001545, 3.65161833350762234541314170447, 4.64249939845563387571794624803, 5.02217756792865302365399463699, 5.48829332235196035976712933288, 5.95615042300228144968883086750, 7.08466225466417310760178082753, 7.14277190123082386922962872499, 7.68745361364208909805983655063, 8.119913496441408337851387342849, 8.775207778350747690106384689866, 8.804654938864817023428976233836, 9.703343829035993967420227198512, 10.20111016237332780534751718337, 10.68498397130898562629171487870, 10.93924823538706199944332151788

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