Properties

Label 4-637e2-1.1-c5e2-0-0
Degree $4$
Conductor $405769$
Sign $1$
Analytic cond. $10437.5$
Root an. cond. $10.1076$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s + 28·3-s − 41·4-s + 42·5-s − 140·6-s + 375·8-s + 255·9-s − 210·10-s − 376·11-s − 1.14e3·12-s + 338·13-s + 1.17e3·15-s + 607·16-s + 2.63e3·17-s − 1.27e3·18-s + 312·19-s − 1.72e3·20-s + 1.88e3·22-s − 2.62e3·23-s + 1.05e4·24-s + 1.87e3·25-s − 1.69e3·26-s − 868·27-s − 812·29-s − 5.88e3·30-s − 7.72e3·31-s − 1.61e4·32-s + ⋯
L(s)  = 1  − 0.883·2-s + 1.79·3-s − 1.28·4-s + 0.751·5-s − 1.58·6-s + 2.07·8-s + 1.04·9-s − 0.664·10-s − 0.936·11-s − 2.30·12-s + 0.554·13-s + 1.34·15-s + 0.592·16-s + 2.20·17-s − 0.927·18-s + 0.198·19-s − 0.962·20-s + 0.828·22-s − 1.03·23-s + 3.72·24-s + 0.599·25-s − 0.490·26-s − 0.229·27-s − 0.179·29-s − 1.19·30-s − 1.44·31-s − 2.78·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(405769\)    =    \(7^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(10437.5\)
Root analytic conductor: \(10.1076\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 405769,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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)$
bad7 \( 1 \)
13$C_1$ \( ( 1 - p^{2} T )^{2} \)
good2$D_{4}$ \( 1 + 5 T + 33 p T^{2} + 5 p^{5} T^{3} + p^{10} T^{4} \)
3$D_{4}$ \( 1 - 28 T + 529 T^{2} - 28 p^{5} T^{3} + p^{10} T^{4} \)
5$D_{4}$ \( 1 - 42 T - 109 T^{2} - 42 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 + 376 T + 327458 T^{2} + 376 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 2630 T + 4499307 T^{2} - 2630 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 - 312 T + 4973202 T^{2} - 312 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 2624 T + 6072542 T^{2} + 2624 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 28 p T + 41177342 T^{2} + 28 p^{6} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 7720 T + 71304910 T^{2} + 7720 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 16858 T + 195347155 T^{2} + 16858 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 7840 T + 230885010 T^{2} + 7840 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 2420 T + 200878009 T^{2} - 2420 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 9972 T + 86137385 T^{2} + 9972 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 + 43720 T + 1116804698 T^{2} + 43720 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 - 38936 T + 1489531170 T^{2} - 38936 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 1984 T + 1322910298 T^{2} + 1984 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 69928 T + 3922568242 T^{2} + 69928 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 67396 T + 4686895793 T^{2} - 67396 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 + 74412 T + 4066416822 T^{2} + 74412 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 55296 T + 6515668494 T^{2} + 55296 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 75712 T + 7946370822 T^{2} - 75712 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 116508 T + 12213785846 T^{2} + 116508 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 34756 T + 2822512198 T^{2} - 34756 p^{5} T^{3} + p^{10} T^{4} \)
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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

−9.474844443797862105883349465573, −9.139752428640512558875019916449, −8.649985246785161414705484367631, −8.473327231309248507456592759139, −7.967679615778552638650845728834, −7.78865741391671480032546302054, −7.36632872221057329113670884557, −6.63770170683152492902145019605, −5.69377371335231716021401628620, −5.48077874112184312682676486589, −5.11202613037993555590117288000, −4.38086849779077522154023347756, −3.56806117171018219690518309750, −3.51181334336415981220763235608, −2.93818325536681287795418861814, −2.17233930710598054491751766750, −1.37381246337704936611807263317, −1.36875283227590074929825351736, 0, 0, 1.36875283227590074929825351736, 1.37381246337704936611807263317, 2.17233930710598054491751766750, 2.93818325536681287795418861814, 3.51181334336415981220763235608, 3.56806117171018219690518309750, 4.38086849779077522154023347756, 5.11202613037993555590117288000, 5.48077874112184312682676486589, 5.69377371335231716021401628620, 6.63770170683152492902145019605, 7.36632872221057329113670884557, 7.78865741391671480032546302054, 7.967679615778552638650845728834, 8.473327231309248507456592759139, 8.649985246785161414705484367631, 9.139752428640512558875019916449, 9.474844443797862105883349465573

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