Properties

Label 4-882e2-1.1-c3e2-0-18
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 + 6·5-s + 8·8-s − 12·10-s − 30·11-s − 106·13-s − 16·16-s + 84·17-s − 97·19-s + 60·22-s + 84·23-s + 125·25-s + 212·26-s + 360·29-s + 179·31-s − 168·34-s + 145·37-s + 194·38-s + 48·40-s + 252·41-s − 650·43-s − 168·46-s + 366·47-s − 250·50-s − 768·53-s − 180·55-s − 720·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.536·5-s + 0.353·8-s − 0.379·10-s − 0.822·11-s − 2.26·13-s − 1/4·16-s + 1.19·17-s − 1.17·19-s + 0.581·22-s + 0.761·23-s + 25-s + 1.59·26-s + 2.30·29-s + 1.03·31-s − 0.847·34-s + 0.644·37-s + 0.828·38-s + 0.189·40-s + 0.959·41-s − 2.30·43-s − 0.538·46-s + 1.13·47-s − 0.707·50-s − 1.99·53-s − 0.441·55-s − 1.63·58-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

\(L(2)\) \(\approx\) \(2.015328069\)
\(L(\frac12)\) \(\approx\) \(2.015328069\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 - 6 T - 89 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 30 T - 431 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 53 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 84 T + 2143 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 97 T + 2550 T^{2} + 97 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 84 T - 5111 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 180 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 179 T + 2250 T^{2} - 179 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 145 T - 29628 T^{2} - 145 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 126 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 325 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 366 T + 30133 T^{2} - 366 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 768 T + 440947 T^{2} + 768 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 264 T - 135683 T^{2} - 264 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 818 T + 442143 T^{2} - 818 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 523 T - 27234 T^{2} - 523 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 342 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 43 T - 387168 T^{2} + 43 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 1171 T + 878202 T^{2} - 1171 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 810 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 600 T - 344969 T^{2} - 600 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 386 T + p^{3} T^{2} )^{2} \)
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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

−9.823579701990239421355464197369, −9.811059772996028032671376095765, −9.250281953240399211949457615188, −8.500020955045673934973615890006, −8.354584522892701799115657533375, −8.060511558356104849808430341502, −7.31363511003180748540107930928, −7.19014298159028862334067531642, −6.52075572437335769202827385101, −6.28542715394234372574205124823, −5.42596306802656806337613276836, −5.11042506835334813545510859284, −4.65856801786842666626219859070, −4.44208240083751175276253115551, −3.31987311241712133345987925301, −2.85477868235839737366656932100, −2.41969051922330492129602437876, −1.88762298925123600881300149651, −0.817551161758007230210496891808, −0.57501820450557357591991923647, 0.57501820450557357591991923647, 0.817551161758007230210496891808, 1.88762298925123600881300149651, 2.41969051922330492129602437876, 2.85477868235839737366656932100, 3.31987311241712133345987925301, 4.44208240083751175276253115551, 4.65856801786842666626219859070, 5.11042506835334813545510859284, 5.42596306802656806337613276836, 6.28542715394234372574205124823, 6.52075572437335769202827385101, 7.19014298159028862334067531642, 7.31363511003180748540107930928, 8.060511558356104849808430341502, 8.354584522892701799115657533375, 8.500020955045673934973615890006, 9.250281953240399211949457615188, 9.811059772996028032671376095765, 9.823579701990239421355464197369

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