Properties

Label 4-882e2-1.1-c3e2-0-26
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 + 22·5-s − 8·8-s + 44·10-s − 26·11-s + 108·13-s − 16·16-s + 74·17-s + 116·19-s − 52·22-s + 58·23-s + 125·25-s + 216·26-s + 416·29-s − 252·31-s + 148·34-s − 50·37-s + 232·38-s − 176·40-s − 252·41-s + 328·43-s + 116·46-s − 444·47-s + 250·50-s − 12·53-s − 572·55-s + 832·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.96·5-s − 0.353·8-s + 1.39·10-s − 0.712·11-s + 2.30·13-s − 1/4·16-s + 1.05·17-s + 1.40·19-s − 0.503·22-s + 0.525·23-s + 25-s + 1.62·26-s + 2.66·29-s − 1.46·31-s + 0.746·34-s − 0.222·37-s + 0.990·38-s − 0.695·40-s − 0.959·41-s + 1.16·43-s + 0.371·46-s − 1.37·47-s + 0.707·50-s − 0.0311·53-s − 1.40·55-s + 1.88·58-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

\(L(2)\) \(\approx\) \(8.944124845\)
\(L(\frac12)\) \(\approx\) \(8.944124845\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 - 22 T + 359 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 26 T - 655 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 54 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 74 T + 563 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 116 T + 6597 T^{2} - 116 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 58 T - 8803 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 208 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 252 T + 33713 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 50 T - 48153 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 126 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 164 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 444 T + 93313 T^{2} + 444 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 12 T - 148733 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 124 T - 190003 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 162 T - 200737 T^{2} + 162 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 860 T + 438837 T^{2} - 860 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 238 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 2 p T - 69 p^{2} T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 984 T + 475217 T^{2} - 984 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 656 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 954 T + 205147 T^{2} + 954 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 526 T + p^{3} T^{2} )^{2} \)
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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

−10.10324970872910118563993969350, −9.472611466745523857975637391098, −9.337004952024934762886630898959, −8.719982553543262981811542874834, −8.246969507104555018800315631606, −8.056738954645379092987383435277, −7.25548772435266672522130566870, −6.75584249299222292181317890609, −6.33608442100946324051894453340, −5.85836316998607796306946506728, −5.67300117162461797580523395188, −5.18868273799001947003019907772, −4.92954993586069015402247425827, −4.00994707850092303655913556912, −3.58646644756029227980934631957, −2.96086020034435439386371356470, −2.68802811088496267993818996845, −1.65304744305588622253064255264, −1.40803933143798587166975529332, −0.72073454422683459054382020477, 0.72073454422683459054382020477, 1.40803933143798587166975529332, 1.65304744305588622253064255264, 2.68802811088496267993818996845, 2.96086020034435439386371356470, 3.58646644756029227980934631957, 4.00994707850092303655913556912, 4.92954993586069015402247425827, 5.18868273799001947003019907772, 5.67300117162461797580523395188, 5.85836316998607796306946506728, 6.33608442100946324051894453340, 6.75584249299222292181317890609, 7.25548772435266672522130566870, 8.056738954645379092987383435277, 8.246969507104555018800315631606, 8.719982553543262981811542874834, 9.337004952024934762886630898959, 9.472611466745523857975637391098, 10.10324970872910118563993969350

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