Properties

Label 4-882e2-1.1-c3e2-0-29
Degree $4$
Conductor $777924$
Sign $1$
Analytic cond. $2708.12$
Root an. cond. $7.21385$
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  − 2·2-s + 15·5-s + 8·8-s − 30·10-s − 9·11-s + 176·13-s − 16·16-s + 84·17-s + 104·19-s + 18·22-s − 84·23-s + 125·25-s − 352·26-s − 102·29-s + 185·31-s − 168·34-s − 44·37-s − 208·38-s + 120·40-s − 336·41-s + 652·43-s + 168·46-s + 138·47-s − 250·50-s + 639·53-s − 135·55-s + 204·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.34·5-s + 0.353·8-s − 0.948·10-s − 0.246·11-s + 3.75·13-s − 1/4·16-s + 1.19·17-s + 1.25·19-s + 0.174·22-s − 0.761·23-s + 25-s − 2.65·26-s − 0.653·29-s + 1.07·31-s − 0.847·34-s − 0.195·37-s − 0.887·38-s + 0.474·40-s − 1.27·41-s + 2.31·43-s + 0.538·46-s + 0.428·47-s − 0.707·50-s + 1.65·53-s − 0.330·55-s + 0.461·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(777924\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2708.12\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 777924,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.847933696\)
\(L(\frac12)\) \(\approx\) \(4.847933696\)
\(L(\frac{5}{2})\) 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)$
bad2$C_2$ \( 1 + p T + p^{2} T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 3 p T + 4 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 9 T - 1250 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 88 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 84 T + 2143 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 104 T + 3957 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 84 T - 5111 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 51 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 185 T + 4434 T^{2} - 185 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 44 T - 48717 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 168 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 326 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 138 T - 84779 T^{2} - 138 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 639 T + 259444 T^{2} - 639 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 159 T - 180098 T^{2} + 159 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 722 T + 294303 T^{2} - 722 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 166 T - 273207 T^{2} - 166 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 1086 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 218 T - 341493 T^{2} - 218 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 583 T - 153150 T^{2} - 583 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 597 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 1038 T + 372475 T^{2} - 1038 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 169 T + p^{3} T^{2} )^{2} \)
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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

−10.03245559958017064118179415095, −9.487197656409406714988870145216, −9.108405651408045119036910910646, −8.726937585838005903889681790942, −8.352408264812863941792285192478, −8.152505378630798299159528353568, −7.28174100178960210009641039443, −7.21621616189477991160094011090, −6.28576223143553926093655555174, −6.02214964050695360950919815442, −5.64530164833488729003380350719, −5.57183141114174089979452010651, −4.59365991708199108903706476370, −4.01945838171820216718624890850, −3.43342528532717521611183859716, −3.16705677521830510990103354373, −2.20253531835108513668716009281, −1.61099360543715266666972509201, −0.989890241946753010844531822044, −0.871371088139521781582173889999, 0.871371088139521781582173889999, 0.989890241946753010844531822044, 1.61099360543715266666972509201, 2.20253531835108513668716009281, 3.16705677521830510990103354373, 3.43342528532717521611183859716, 4.01945838171820216718624890850, 4.59365991708199108903706476370, 5.57183141114174089979452010651, 5.64530164833488729003380350719, 6.02214964050695360950919815442, 6.28576223143553926093655555174, 7.21621616189477991160094011090, 7.28174100178960210009641039443, 8.152505378630798299159528353568, 8.352408264812863941792285192478, 8.726937585838005903889681790942, 9.108405651408045119036910910646, 9.487197656409406714988870145216, 10.03245559958017064118179415095

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