Properties

Label 4-168e2-1.1-c7e2-0-0
Degree $4$
Conductor $28224$
Sign $1$
Analytic cond. $2754.22$
Root an. cond. $7.24435$
Motivic weight $7$
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  + 54·3-s + 348·5-s − 686·7-s + 2.18e3·9-s + 424·11-s − 1.81e3·13-s + 1.87e4·15-s + 1.19e4·17-s + 6.54e4·19-s − 3.70e4·21-s + 1.88e4·23-s − 1.42e4·25-s + 7.87e4·27-s + 8.08e4·29-s + 3.40e5·31-s + 2.28e4·33-s − 2.38e5·35-s + 3.43e5·37-s − 9.78e4·39-s − 4.06e5·41-s − 2.89e5·43-s + 7.61e5·45-s + 2.26e5·47-s + 3.52e5·49-s + 6.44e5·51-s + 7.31e5·53-s + 1.47e5·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.24·5-s − 0.755·7-s + 9-s + 0.0960·11-s − 0.228·13-s + 1.43·15-s + 0.589·17-s + 2.18·19-s − 0.872·21-s + 0.323·23-s − 0.182·25-s + 0.769·27-s + 0.615·29-s + 2.05·31-s + 0.110·33-s − 0.941·35-s + 1.11·37-s − 0.264·39-s − 0.920·41-s − 0.555·43-s + 1.24·45-s + 0.317·47-s + 3/7·49-s + 0.680·51-s + 0.675·53-s + 0.119·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(28224\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2754.22\)
Root analytic conductor: \(7.24435\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 28224,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(8.594324147\)
\(L(\frac12)\) \(\approx\) \(8.594324147\)
\(L(\frac{9}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - p^{3} T )^{2} \)
7$C_1$ \( ( 1 + p^{3} T )^{2} \)
good5$D_{4}$ \( 1 - 348 T + 27078 p T^{2} - 348 p^{7} T^{3} + p^{14} T^{4} \)
11$D_{4}$ \( 1 - 424 T + 37740886 T^{2} - 424 p^{7} T^{3} + p^{14} T^{4} \)
13$D_{4}$ \( 1 + 1812 T + 105863470 T^{2} + 1812 p^{7} T^{3} + p^{14} T^{4} \)
17$D_{4}$ \( 1 - 11940 T + 752767846 T^{2} - 11940 p^{7} T^{3} + p^{14} T^{4} \)
19$D_{4}$ \( 1 - 65464 T + 2636378886 T^{2} - 65464 p^{7} T^{3} + p^{14} T^{4} \)
23$D_{4}$ \( 1 - 18896 T - 368992946 T^{2} - 18896 p^{7} T^{3} + p^{14} T^{4} \)
29$D_{4}$ \( 1 - 80844 T + 4542074446 T^{2} - 80844 p^{7} T^{3} + p^{14} T^{4} \)
31$D_{4}$ \( 1 - 340704 T + 74054898622 T^{2} - 340704 p^{7} T^{3} + p^{14} T^{4} \)
37$D_{4}$ \( 1 - 343644 T + 156154639966 T^{2} - 343644 p^{7} T^{3} + p^{14} T^{4} \)
41$D_{4}$ \( 1 + 406284 T + 428140232982 T^{2} + 406284 p^{7} T^{3} + p^{14} T^{4} \)
43$D_{4}$ \( 1 + 289624 T - 22359088426 T^{2} + 289624 p^{7} T^{3} + p^{14} T^{4} \)
47$D_{4}$ \( 1 - 226080 T + 83156419230 T^{2} - 226080 p^{7} T^{3} + p^{14} T^{4} \)
53$D_{4}$ \( 1 - 731868 T + 2464177290046 T^{2} - 731868 p^{7} T^{3} + p^{14} T^{4} \)
59$D_{4}$ \( 1 + 590360 T - 519744837962 T^{2} + 590360 p^{7} T^{3} + p^{14} T^{4} \)
61$D_{4}$ \( 1 - 2916236 T + 7445751528590 T^{2} - 2916236 p^{7} T^{3} + p^{14} T^{4} \)
67$D_{4}$ \( 1 - 3306680 T + 8066320800422 T^{2} - 3306680 p^{7} T^{3} + p^{14} T^{4} \)
71$D_{4}$ \( 1 - 2514480 T + 7439454465006 T^{2} - 2514480 p^{7} T^{3} + p^{14} T^{4} \)
73$D_{4}$ \( 1 - 2854644 T + 23117603230102 T^{2} - 2854644 p^{7} T^{3} + p^{14} T^{4} \)
79$D_{4}$ \( 1 - 169152 T + 15842512838494 T^{2} - 169152 p^{7} T^{3} + p^{14} T^{4} \)
83$D_{4}$ \( 1 - 9037112 T + 51541810456454 T^{2} - 9037112 p^{7} T^{3} + p^{14} T^{4} \)
89$D_{4}$ \( 1 - 6141972 T + 97780858417654 T^{2} - 6141972 p^{7} T^{3} + p^{14} T^{4} \)
97$D_{4}$ \( 1 + 6506108 T + 160224145545542 T^{2} + 6506108 p^{7} T^{3} + p^{14} T^{4} \)
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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

−11.75160843771417980998915508327, −11.38585824379926537285934090921, −10.23088607558507534297478473635, −10.07411747165324528023953931942, −9.716221956377585248330760993209, −9.418600961445818660134879608115, −8.724496971303160538883365833847, −8.240352382279186262595062075019, −7.55384495090783080445210502028, −7.20614590105802663954361117248, −6.24331874093064330561874876149, −6.20276031533803563156502179710, −5.10730036492092715915663524107, −4.88893045648378033240569326341, −3.54592220118887460387568623430, −3.47549710604148067683908469667, −2.47102494784375780309201349023, −2.28989335327504677010851668476, −1.11712992927830761906462832814, −0.837036120672870089143735722268, 0.837036120672870089143735722268, 1.11712992927830761906462832814, 2.28989335327504677010851668476, 2.47102494784375780309201349023, 3.47549710604148067683908469667, 3.54592220118887460387568623430, 4.88893045648378033240569326341, 5.10730036492092715915663524107, 6.20276031533803563156502179710, 6.24331874093064330561874876149, 7.20614590105802663954361117248, 7.55384495090783080445210502028, 8.240352382279186262595062075019, 8.724496971303160538883365833847, 9.418600961445818660134879608115, 9.716221956377585248330760993209, 10.07411747165324528023953931942, 10.23088607558507534297478473635, 11.38585824379926537285934090921, 11.75160843771417980998915508327

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