Properties

Label 4-882e2-1.1-c3e2-0-16
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 + 12·5-s − 8·8-s + 24·10-s + 48·11-s − 112·13-s − 16·16-s + 114·17-s + 2·19-s + 96·22-s − 120·23-s + 125·25-s − 224·26-s + 108·29-s + 236·31-s + 228·34-s − 146·37-s + 4·38-s − 96·40-s + 252·41-s − 752·43-s − 240·46-s + 12·47-s + 250·50-s + 174·53-s + 576·55-s + 216·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.07·5-s − 0.353·8-s + 0.758·10-s + 1.31·11-s − 2.38·13-s − 1/4·16-s + 1.62·17-s + 0.0241·19-s + 0.930·22-s − 1.08·23-s + 25-s − 1.68·26-s + 0.691·29-s + 1.36·31-s + 1.15·34-s − 0.648·37-s + 0.0170·38-s − 0.379·40-s + 0.959·41-s − 2.66·43-s − 0.769·46-s + 0.0372·47-s + 0.707·50-s + 0.450·53-s + 1.41·55-s + 0.489·58-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

\(L(2)\) \(\approx\) \(4.617395229\)
\(L(\frac12)\) \(\approx\) \(4.617395229\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 - 12 T + 19 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 48 T + 973 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 56 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 114 T + 8083 T^{2} - 114 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 2 T - 6855 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 120 T + 2233 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 54 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 236 T + 25905 T^{2} - 236 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 146 T - 29337 T^{2} + 146 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 126 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 376 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 12 T - 103679 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 174 T - 118601 T^{2} - 174 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 138 T - 186335 T^{2} + 138 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 380 T - 82581 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 484 T - 66507 T^{2} - 484 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 576 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 1150 T + 933483 T^{2} + 1150 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 776 T + 109137 T^{2} + 776 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 378 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 390 T - 552869 T^{2} - 390 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 1330 T + p^{3} T^{2} )^{2} \)
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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

−9.935701544149692363438345023187, −9.763792912353710620449209565304, −9.181115071686692938601913878219, −8.825543463473312514983365764538, −8.266613280009776784303157147613, −7.82263217262482659145075055596, −7.19254278864143682365348419750, −7.00780620700644231623354704790, −6.29173850874581406427328007487, −6.11346436661775963350640502755, −5.50607698969204585179748222972, −5.13158060715214338603530428372, −4.52646837043423810501434606529, −4.45486784181315420070080859124, −3.39515941283079181160923057581, −3.20636866839926964328854875056, −2.44698001843132376037635940762, −1.95990059510989396449734175574, −1.27736906756257935114135507374, −0.50340224632015936634852455496, 0.50340224632015936634852455496, 1.27736906756257935114135507374, 1.95990059510989396449734175574, 2.44698001843132376037635940762, 3.20636866839926964328854875056, 3.39515941283079181160923057581, 4.45486784181315420070080859124, 4.52646837043423810501434606529, 5.13158060715214338603530428372, 5.50607698969204585179748222972, 6.11346436661775963350640502755, 6.29173850874581406427328007487, 7.00780620700644231623354704790, 7.19254278864143682365348419750, 7.82263217262482659145075055596, 8.266613280009776784303157147613, 8.825543463473312514983365764538, 9.181115071686692938601913878219, 9.763792912353710620449209565304, 9.935701544149692363438345023187

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