Properties

Label 4-882e2-1.1-c3e2-0-11
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 − 7·5-s + 8·8-s + 14·10-s + 35·11-s − 132·13-s − 16·16-s − 59·17-s + 137·19-s − 70·22-s − 7·23-s + 125·25-s + 264·26-s − 212·29-s + 75·31-s + 118·34-s − 11·37-s − 274·38-s − 56·40-s − 996·41-s + 520·43-s + 14·46-s + 171·47-s − 250·50-s − 417·53-s − 245·55-s + 424·58-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.626·5-s + 0.353·8-s + 0.442·10-s + 0.959·11-s − 2.81·13-s − 1/4·16-s − 0.841·17-s + 1.65·19-s − 0.678·22-s − 0.0634·23-s + 25-s + 1.99·26-s − 1.35·29-s + 0.434·31-s + 0.595·34-s − 0.0488·37-s − 1.16·38-s − 0.221·40-s − 3.79·41-s + 1.84·43-s + 0.0448·46-s + 0.530·47-s − 0.707·50-s − 1.08·53-s − 0.600·55-s + 0.959·58-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

\(L(2)\) \(\approx\) \(1.046325656\)
\(L(\frac12)\) \(\approx\) \(1.046325656\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 + 7 T - 76 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 35 T - 106 T^{2} - 35 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 66 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 59 T - 1432 T^{2} + 59 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 137 T + 11910 T^{2} - 137 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 7 T - 12118 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 106 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 75 T - 24166 T^{2} - 75 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 11 T - 50532 T^{2} + 11 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 498 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 260 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 171 T - 74582 T^{2} - 171 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 417 T + 25012 T^{2} + 417 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 17 T - 205090 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 51 T - 224380 T^{2} - 51 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 439 T - 108042 T^{2} + 439 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 784 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 295 T - 301992 T^{2} - 295 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 495 T - 248014 T^{2} - 495 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 932 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 873 T + 57160 T^{2} - 873 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 290 T + p^{3} T^{2} )^{2} \)
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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

−9.783844137183922726823207024269, −9.520398906676197926588163413478, −9.076260019416269867917586082603, −8.997042314847306825319242943007, −8.187452947395834703355436028831, −7.78204159044455625372466350830, −7.51927821940859444470299721761, −7.07178880730073154365133204922, −6.75390374056931334616334687400, −6.30000082153689471637851609023, −5.23451480780146545295883837705, −5.21205112577994915283732849663, −4.67988562440756188989188847818, −4.21354458573971250352090719392, −3.32198324117737817605549476503, −3.25449409154516855377114838133, −2.07252921039306952139782110772, −2.03693154474428314659159210349, −0.832043311305833913523232234021, −0.40338328084342301524641908582, 0.40338328084342301524641908582, 0.832043311305833913523232234021, 2.03693154474428314659159210349, 2.07252921039306952139782110772, 3.25449409154516855377114838133, 3.32198324117737817605549476503, 4.21354458573971250352090719392, 4.67988562440756188989188847818, 5.21205112577994915283732849663, 5.23451480780146545295883837705, 6.30000082153689471637851609023, 6.75390374056931334616334687400, 7.07178880730073154365133204922, 7.51927821940859444470299721761, 7.78204159044455625372466350830, 8.187452947395834703355436028831, 8.997042314847306825319242943007, 9.076260019416269867917586082603, 9.520398906676197926588163413478, 9.783844137183922726823207024269

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