Properties

Label 4-882e2-1.1-c1e2-0-8
Degree $4$
Conductor $777924$
Sign $1$
Analytic cond. $49.6011$
Root an. cond. $2.65382$
Motivic weight $1$
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-s + 2·5-s − 8-s + 2·10-s − 4·11-s − 12·13-s − 16-s − 2·17-s − 4·19-s − 4·22-s + 8·23-s + 5·25-s − 12·26-s + 4·29-s − 2·34-s + 10·37-s − 4·38-s − 2·40-s − 12·41-s − 8·43-s + 8·46-s + 5·50-s + 6·53-s − 8·55-s + 4·58-s − 4·59-s + 6·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.894·5-s − 0.353·8-s + 0.632·10-s − 1.20·11-s − 3.32·13-s − 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.852·22-s + 1.66·23-s + 25-s − 2.35·26-s + 0.742·29-s − 0.342·34-s + 1.64·37-s − 0.648·38-s − 0.316·40-s − 1.87·41-s − 1.21·43-s + 1.17·46-s + 0.707·50-s + 0.824·53-s − 1.07·55-s + 0.525·58-s − 0.520·59-s + 0.768·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(777924\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(49.6011\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 777924,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.725533575\)
\(L(\frac12)\) \(\approx\) \(1.725533575\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−10.22625386472067229882020250663, −9.863661167547984375619952395857, −9.859893420753569555582262348779, −9.005953684881011955540837994079, −8.801598430319248119453100341416, −8.304920417701320089062950205098, −7.63976104653277913101100762676, −7.16120562476150645619837818524, −7.09289247807004359454370261014, −6.37224329538730761845383534777, −5.99546887150647875495752095784, −5.14340923638633090293219468768, −5.13827866328819252008261178183, −4.60837865356506512641534448293, −4.51100270128946100487323118536, −3.08259396129458407694553315513, −3.08043975772074178389062491122, −2.19951601080366493466920784925, −2.13514168951973436735378160616, −0.52611535156862844656460187403, 0.52611535156862844656460187403, 2.13514168951973436735378160616, 2.19951601080366493466920784925, 3.08043975772074178389062491122, 3.08259396129458407694553315513, 4.51100270128946100487323118536, 4.60837865356506512641534448293, 5.13827866328819252008261178183, 5.14340923638633090293219468768, 5.99546887150647875495752095784, 6.37224329538730761845383534777, 7.09289247807004359454370261014, 7.16120562476150645619837818524, 7.63976104653277913101100762676, 8.304920417701320089062950205098, 8.801598430319248119453100341416, 9.005953684881011955540837994079, 9.859893420753569555582262348779, 9.863661167547984375619952395857, 10.22625386472067229882020250663

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