Properties

Label 4-882e2-1.1-c3e2-0-19
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 − 8·5-s + 8·8-s + 16·10-s + 40·11-s − 8·13-s − 16·16-s + 84·17-s + 148·19-s − 80·22-s + 84·23-s + 125·25-s + 16·26-s − 116·29-s − 136·31-s − 168·34-s + 222·37-s − 296·38-s − 64·40-s + 840·41-s − 328·43-s − 168·46-s − 488·47-s − 250·50-s + 478·53-s − 320·55-s + 232·58-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.715·5-s + 0.353·8-s + 0.505·10-s + 1.09·11-s − 0.170·13-s − 1/4·16-s + 1.19·17-s + 1.78·19-s − 0.775·22-s + 0.761·23-s + 25-s + 0.120·26-s − 0.742·29-s − 0.787·31-s − 0.847·34-s + 0.986·37-s − 1.26·38-s − 0.252·40-s + 3.19·41-s − 1.16·43-s − 0.538·46-s − 1.51·47-s − 0.707·50-s + 1.23·53-s − 0.784·55-s + 0.525·58-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

\(L(2)\) \(\approx\) \(2.265071366\)
\(L(\frac12)\) \(\approx\) \(2.265071366\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 + 8 T - 61 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 40 T + 269 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 84 T + 2143 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 148 T + 15045 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 84 T - 5111 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 2 p T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 136 T - 11295 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 6 p T - p^{2} T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 420 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 164 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 488 T + 134321 T^{2} + 488 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 478 T + 79607 T^{2} - 478 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 548 T + 94925 T^{2} + 548 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 692 T + 251883 T^{2} - 692 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 908 T + 523701 T^{2} - 908 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 524 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 440 T - 195417 T^{2} - 440 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 1216 T + 985617 T^{2} + 1216 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 684 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 604 T - 340153 T^{2} + 604 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 832 T + p^{3} T^{2} )^{2} \)
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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

−9.714879919780118228268093871157, −9.528140775371864134018027101228, −9.306199235491868336321694692864, −8.712882560227662252369905312791, −8.179525701153965263108620433752, −8.005443474786458595116267818284, −7.32867786215593760669409647592, −7.16423219140379823302192824912, −6.84522350296149077361290414139, −5.85287937243293545413355382586, −5.82198346397583932902458569575, −4.98498115317466576903732732524, −4.75992817625116901059784829602, −3.79315530065622872314601504341, −3.77550680248969001489138332926, −3.06494867622510365688247579511, −2.47656822013302420966938165988, −1.47543416799411403831628700372, −1.00421005397588998533956616212, −0.57707210975312479493743956858, 0.57707210975312479493743956858, 1.00421005397588998533956616212, 1.47543416799411403831628700372, 2.47656822013302420966938165988, 3.06494867622510365688247579511, 3.77550680248969001489138332926, 3.79315530065622872314601504341, 4.75992817625116901059784829602, 4.98498115317466576903732732524, 5.82198346397583932902458569575, 5.85287937243293545413355382586, 6.84522350296149077361290414139, 7.16423219140379823302192824912, 7.32867786215593760669409647592, 8.005443474786458595116267818284, 8.179525701153965263108620433752, 8.712882560227662252369905312791, 9.306199235491868336321694692864, 9.528140775371864134018027101228, 9.714879919780118228268093871157

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