Properties

Label 4-882e2-1.1-c3e2-0-23
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  − 4·2-s + 12·4-s + 7·5-s − 32·8-s − 28·10-s − 25·11-s − 49·13-s + 80·16-s + 98·17-s − 119·19-s + 84·20-s + 100·22-s + 122·23-s − 165·25-s + 196·26-s + 73·29-s + 98·31-s − 192·32-s − 392·34-s + 289·37-s + 476·38-s − 224·40-s + 336·41-s + 307·43-s − 300·44-s − 488·46-s + 672·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.626·5-s − 1.41·8-s − 0.885·10-s − 0.685·11-s − 1.04·13-s + 5/4·16-s + 1.39·17-s − 1.43·19-s + 0.939·20-s + 0.969·22-s + 1.10·23-s − 1.31·25-s + 1.47·26-s + 0.467·29-s + 0.567·31-s − 1.06·32-s − 1.97·34-s + 1.28·37-s + 2.03·38-s − 0.885·40-s + 1.27·41-s + 1.08·43-s − 1.02·44-s − 1.56·46-s + 2.08·47-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 777924,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.857400948\)
\(L(\frac12)\) \(\approx\) \(1.857400948\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$D_{4}$ \( 1 - 7 T + 214 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 25 T + 454 T^{2} + 25 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 49 T + 3788 T^{2} + 49 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 98 T + 7402 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 119 T + 16824 T^{2} + 119 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 122 T + 18598 T^{2} - 122 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 73 T + 47746 T^{2} - 73 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 98 T + 24155 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 289 T + 100908 T^{2} - 289 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 168 T + p^{3} T^{2} )^{2} \)
43$D_{4}$ \( 1 - 307 T + 161298 T^{2} - 307 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 672 T + 292750 T^{2} - 672 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 375 T + 141406 T^{2} + 375 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 763 T + 376762 T^{2} - 763 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 406 T + 439394 T^{2} + 406 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 1041 T + 813340 T^{2} + 1041 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 1652 T + 1360270 T^{2} - 1652 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 189 T + 332980 T^{2} + 189 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 524 T + 591329 T^{2} - 524 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 287 T + 781978 T^{2} - 287 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 2394 T + 2702050 T^{2} - 2394 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 63 T + 667132 T^{2} + 63 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.886347350369213047179066291986, −9.583811678896067826311590396230, −9.057390614250648813669155547659, −8.985443460335952095566491266943, −8.102706811264275455903456275468, −7.965182330107184570630775432286, −7.47801299305153993560941962342, −7.31946933983935022873738461937, −6.38885327937957580439052825754, −6.35006950483747500716571680690, −5.61690552857443810764579156232, −5.43267854165545265076556557550, −4.60737212994496195560113593216, −4.17110124449224150711930530080, −3.30970839536116700272523664417, −2.69614471607526039311167843510, −2.30049847294585263309300547453, −1.87238271952139931688339478832, −0.74936051136377557813929769102, −0.66426432561399122764242441785, 0.66426432561399122764242441785, 0.74936051136377557813929769102, 1.87238271952139931688339478832, 2.30049847294585263309300547453, 2.69614471607526039311167843510, 3.30970839536116700272523664417, 4.17110124449224150711930530080, 4.60737212994496195560113593216, 5.43267854165545265076556557550, 5.61690552857443810764579156232, 6.35006950483747500716571680690, 6.38885327937957580439052825754, 7.31946933983935022873738461937, 7.47801299305153993560941962342, 7.965182330107184570630775432286, 8.102706811264275455903456275468, 8.985443460335952095566491266943, 9.057390614250648813669155547659, 9.583811678896067826311590396230, 9.886347350369213047179066291986

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