Properties

Degree $4$
Conductor $777924$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s + 3·5-s − 3·6-s − 8-s + 6·9-s + 3·10-s + 3·11-s − 13-s − 9·15-s − 16-s − 6·17-s + 6·18-s + 14·19-s + 3·22-s + 9·23-s + 3·24-s + 5·25-s − 26-s − 9·27-s − 3·29-s − 9·30-s + 8·31-s − 9·33-s − 6·34-s − 2·37-s + 14·38-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1.34·5-s − 1.22·6-s − 0.353·8-s + 2·9-s + 0.948·10-s + 0.904·11-s − 0.277·13-s − 2.32·15-s − 1/4·16-s − 1.45·17-s + 1.41·18-s + 3.21·19-s + 0.639·22-s + 1.87·23-s + 0.612·24-s + 25-s − 0.196·26-s − 1.73·27-s − 0.557·29-s − 1.64·30-s + 1.43·31-s − 1.56·33-s − 1.02·34-s − 0.328·37-s + 2.27·38-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: $1$
Motivic weight: \(1\)
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

\(L(1)\) \(\approx\) \(2.47799\)
\(L(\frac12)\) \(\approx\) \(2.47799\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−10.31890396833786160270954866036, −9.877447480455617376993334873529, −9.673463905284701070644229080629, −9.255505984517883432505675079881, −8.936261980336884443152273871707, −8.301183231056055855344608894092, −7.29117996637768372246160041753, −7.28306231394502376893488186712, −6.57704481571504673144800439881, −6.54754633643403299961289665079, −5.68671321128062997522276327147, −5.67476906995538082895750696529, −5.01123085682411587890281074804, −4.91636534811541873833526297319, −4.30918883849380669987224604026, −3.59513487770469630493419161079, −2.96324445485016578225124427418, −2.28667728761921153767464113895, −1.27136214522902862082356429554, −0.908358593200701456434204711347, 0.908358593200701456434204711347, 1.27136214522902862082356429554, 2.28667728761921153767464113895, 2.96324445485016578225124427418, 3.59513487770469630493419161079, 4.30918883849380669987224604026, 4.91636534811541873833526297319, 5.01123085682411587890281074804, 5.67476906995538082895750696529, 5.68671321128062997522276327147, 6.54754633643403299961289665079, 6.57704481571504673144800439881, 7.28306231394502376893488186712, 7.29117996637768372246160041753, 8.301183231056055855344608894092, 8.936261980336884443152273871707, 9.255505984517883432505675079881, 9.673463905284701070644229080629, 9.877447480455617376993334873529, 10.31890396833786160270954866036

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