Properties

Label 4-882e2-1.1-c3e2-0-5
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

\(L(2)\) \(\approx\) \(0.9359736227\)
\(L(\frac12)\) \(\approx\) \(0.9359736227\)
\(L(\frac{5}{2})\) 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

−10.04424337270821070038079942943, −9.430103555112888622989388064035, −8.922553411953916553859947489529, −8.606168293349679099328712125878, −8.414768520957178967010800674330, −8.062528080053267827281816832176, −7.21201218608357151135395158566, −6.73063353873411338409650450755, −6.61565577414794532100974835964, −6.14910841885089154079067712168, −5.56275776426569389861747546061, −5.05181550809757590497152944516, −4.37761479730857268629575623636, −4.06879366356212266112056526597, −3.66008697534077786369446962037, −3.44784384529873341657743074169, −2.63490221577512492950750090011, −1.54290879248312543222384624022, −1.44557325087370246360990278162, −0.22635467242426747302159979827, 0.22635467242426747302159979827, 1.44557325087370246360990278162, 1.54290879248312543222384624022, 2.63490221577512492950750090011, 3.44784384529873341657743074169, 3.66008697534077786369446962037, 4.06879366356212266112056526597, 4.37761479730857268629575623636, 5.05181550809757590497152944516, 5.56275776426569389861747546061, 6.14910841885089154079067712168, 6.61565577414794532100974835964, 6.73063353873411338409650450755, 7.21201218608357151135395158566, 8.062528080053267827281816832176, 8.414768520957178967010800674330, 8.606168293349679099328712125878, 8.922553411953916553859947489529, 9.430103555112888622989388064035, 10.04424337270821070038079942943

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