Properties

Label 4-882e2-1.1-c3e2-0-4
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 + 12·11-s − 76·13-s − 16·16-s + 126·17-s + 20·19-s − 24·22-s + 168·23-s + 125·25-s + 152·26-s − 60·29-s − 88·31-s − 252·34-s − 254·37-s − 40·38-s − 48·40-s + 84·41-s − 104·43-s − 336·46-s + 96·47-s − 250·50-s + 198·53-s − 72·55-s + 120·58-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.536·5-s + 0.353·8-s + 0.379·10-s + 0.328·11-s − 1.62·13-s − 1/4·16-s + 1.79·17-s + 0.241·19-s − 0.232·22-s + 1.52·23-s + 25-s + 1.14·26-s − 0.384·29-s − 0.509·31-s − 1.27·34-s − 1.12·37-s − 0.170·38-s − 0.189·40-s + 0.319·41-s − 0.368·43-s − 1.07·46-s + 0.297·47-s − 0.707·50-s + 0.513·53-s − 0.176·55-s + 0.271·58-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

\(L(2)\) \(\approx\) \(0.4362901348\)
\(L(\frac12)\) \(\approx\) \(0.4362901348\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 - 12 T - 1187 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 38 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 126 T + 10963 T^{2} - 126 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 20 T - 6459 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 168 T + 16057 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 30 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 88 T - 22047 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 254 T + 13863 T^{2} + 254 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 42 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 52 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 96 T - 94607 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 198 T - 109673 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 660 T + 230221 T^{2} - 660 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 538 T + 62463 T^{2} + 538 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 884 T + 480693 T^{2} + 884 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 792 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 218 T - 341493 T^{2} - 218 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 520 T - 222639 T^{2} - 520 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 492 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 810 T - 48869 T^{2} + 810 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 1154 T + p^{3} T^{2} )^{2} \)
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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

−10.27472298059006282094981156693, −9.403893733397744327525409419475, −9.254120021796929534445468051654, −8.638352818429540955154229683483, −8.453354736127816002320170989756, −7.62606402421002978639344468309, −7.57412715223556482736072848398, −7.05495832581985940437560931493, −6.92265702912014084475065900190, −6.03210777588729638093086971357, −5.51438297287800821489360250648, −5.03780803396016091230541940197, −4.82284836812762754502982698673, −4.00857799430211390603494479550, −3.61273600239515311305668593921, −2.78712185451989871271143488283, −2.69915411882516241300638816146, −1.34499351151781540492126025199, −1.30054712577225999602830800987, −0.21358927988003252036095267343, 0.21358927988003252036095267343, 1.30054712577225999602830800987, 1.34499351151781540492126025199, 2.69915411882516241300638816146, 2.78712185451989871271143488283, 3.61273600239515311305668593921, 4.00857799430211390603494479550, 4.82284836812762754502982698673, 5.03780803396016091230541940197, 5.51438297287800821489360250648, 6.03210777588729638093086971357, 6.92265702912014084475065900190, 7.05495832581985940437560931493, 7.57412715223556482736072848398, 7.62606402421002978639344468309, 8.453354736127816002320170989756, 8.638352818429540955154229683483, 9.254120021796929534445468051654, 9.403893733397744327525409419475, 10.27472298059006282094981156693

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