Properties

Label 4-882e2-1.1-c3e2-0-12
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 + 8·5-s + 8·8-s − 16·10-s + 40·11-s + 8·13-s − 16·16-s − 84·17-s − 148·19-s − 80·22-s + 84·23-s + 125·25-s − 16·26-s − 116·29-s + 136·31-s + 168·34-s + 222·37-s + 296·38-s + 64·40-s − 840·41-s − 328·43-s − 168·46-s + 488·47-s − 250·50-s + 478·53-s + 320·55-s + 232·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.715·5-s + 0.353·8-s − 0.505·10-s + 1.09·11-s + 0.170·13-s − 1/4·16-s − 1.19·17-s − 1.78·19-s − 0.775·22-s + 0.761·23-s + 25-s − 0.120·26-s − 0.742·29-s + 0.787·31-s + 0.847·34-s + 0.986·37-s + 1.26·38-s + 0.252·40-s − 3.19·41-s − 1.16·43-s − 0.538·46-s + 1.51·47-s − 0.707·50-s + 1.23·53-s + 0.784·55-s + 0.525·58-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

\(L(2)\) \(\approx\) \(1.593285748\)
\(L(\frac12)\) \(\approx\) \(1.593285748\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 - 8 T - 61 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 40 T + 269 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 84 T + 2143 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 148 T + 15045 T^{2} + 148 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 84 T - 5111 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 2 p T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 136 T - 11295 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 6 p T - p^{2} T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 420 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 164 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 488 T + 134321 T^{2} - 488 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 478 T + 79607 T^{2} - 478 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 548 T + 94925 T^{2} - 548 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 692 T + 251883 T^{2} + 692 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 908 T + 523701 T^{2} - 908 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 524 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 440 T - 195417 T^{2} + 440 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 1216 T + 985617 T^{2} + 1216 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 684 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 604 T - 340153 T^{2} - 604 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 832 T + p^{3} T^{2} )^{2} \)
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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

−9.910075997239317237751294079883, −9.520824818266280354133375905641, −8.965411909394660092936402786834, −8.746484285154261989225518921380, −8.552416923603640979856184080166, −8.069712964044986680870957581835, −7.34338592388613631723495004630, −6.72590039828169247260887310317, −6.72557698150347431004615973683, −6.28986634520893858265906278529, −5.69226922168415090878288138083, −4.90822406959509140830949012500, −4.84168424903329939716650601645, −3.94561779717954434556179207480, −3.79843187422710614307505161963, −2.85382632787840991472545488910, −2.22051265698941903484376616446, −1.80638605263437487070091983098, −1.12280294265039273695632535460, −0.40372768027252831420413483416, 0.40372768027252831420413483416, 1.12280294265039273695632535460, 1.80638605263437487070091983098, 2.22051265698941903484376616446, 2.85382632787840991472545488910, 3.79843187422710614307505161963, 3.94561779717954434556179207480, 4.84168424903329939716650601645, 4.90822406959509140830949012500, 5.69226922168415090878288138083, 6.28986634520893858265906278529, 6.72557698150347431004615973683, 6.72590039828169247260887310317, 7.34338592388613631723495004630, 8.069712964044986680870957581835, 8.552416923603640979856184080166, 8.746484285154261989225518921380, 8.965411909394660092936402786834, 9.520824818266280354133375905641, 9.910075997239317237751294079883

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