Properties

Degree $10$
Conductor $1419857$
Sign $-1$
Motivic weight $9$
Primitive no
Self-dual yes
Analytic rank $5$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 33·2-s − 236·3-s − 309·4-s + 1.48e3·5-s + 7.78e3·6-s − 1.32e4·7-s + 1.36e4·8-s − 1.58e4·9-s − 4.88e4·10-s − 6.80e4·11-s + 7.29e4·12-s − 1.58e5·13-s + 4.35e5·14-s − 3.49e5·15-s + 2.08e5·16-s − 4.17e5·17-s + 5.23e5·18-s − 3.70e5·19-s − 4.57e5·20-s + 3.11e6·21-s + 2.24e6·22-s + 1.64e6·23-s − 3.22e6·24-s − 2.15e6·25-s + 5.24e6·26-s + 8.67e6·27-s + 4.07e6·28-s + ⋯
L(s)  = 1  − 1.45·2-s − 1.68·3-s − 0.603·4-s + 1.05·5-s + 2.45·6-s − 2.07·7-s + 1.17·8-s − 0.806·9-s − 1.54·10-s − 1.40·11-s + 1.01·12-s − 1.54·13-s + 3.03·14-s − 1.78·15-s + 0.793·16-s − 1.21·17-s + 1.17·18-s − 0.653·19-s − 0.639·20-s + 3.49·21-s + 2.04·22-s + 1.22·23-s − 1.98·24-s − 1.10·25-s + 2.24·26-s + 3.14·27-s + 1.25·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1419857 ^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1419857 ^{s/2} \, \Gamma_{\C}(s+9/2)^{5} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(1419857\)    =    \(17^{5}\)
Sign: $-1$
Motivic weight: \(9\)
Character: induced by $\chi_{17} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 1419857,\ (\ :9/2, 9/2, 9/2, 9/2, 9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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)$
bad17$C_1$ \( ( 1 + p^{4} T )^{5} \)
good2$C_2 \wr S_5$ \( 1 + 33 T + 699 p T^{2} + 5333 p^{3} T^{3} + 9225 p^{7} T^{4} + 26215 p^{10} T^{5} + 9225 p^{16} T^{6} + 5333 p^{21} T^{7} + 699 p^{28} T^{8} + 33 p^{36} T^{9} + p^{45} T^{10} \)
3$C_2 \wr S_5$ \( 1 + 236 T + 23855 p T^{2} + 442924 p^{3} T^{3} + 74219104 p^{3} T^{4} + 3544592944 p^{4} T^{5} + 74219104 p^{12} T^{6} + 442924 p^{21} T^{7} + 23855 p^{28} T^{8} + 236 p^{36} T^{9} + p^{45} T^{10} \)
5$C_2 \wr S_5$ \( 1 - 296 p T + 4342893 T^{2} - 225254208 p^{2} T^{3} + 13283863659206 T^{4} - 3355838628707504 p T^{5} + 13283863659206 p^{9} T^{6} - 225254208 p^{20} T^{7} + 4342893 p^{27} T^{8} - 296 p^{37} T^{9} + p^{45} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 1886 p T + 3957601 p^{2} T^{2} + 35472586200 p^{2} T^{3} + 42464048103472 p^{3} T^{4} + 40407723395812460 p^{4} T^{5} + 42464048103472 p^{12} T^{6} + 35472586200 p^{20} T^{7} + 3957601 p^{29} T^{8} + 1886 p^{37} T^{9} + p^{45} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 68036 T + 11950978605 T^{2} + 590046858288436 T^{3} + 56156610305285905000 T^{4} + \)\(20\!\cdots\!56\)\( T^{5} + 56156610305285905000 p^{9} T^{6} + 590046858288436 p^{18} T^{7} + 11950978605 p^{27} T^{8} + 68036 p^{36} T^{9} + p^{45} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 158862 T + 44696846525 T^{2} + 5023313750605552 T^{3} + \)\(87\!\cdots\!26\)\( T^{4} + \)\(75\!\cdots\!04\)\( T^{5} + \)\(87\!\cdots\!26\)\( p^{9} T^{6} + 5023313750605552 p^{18} T^{7} + 44696846525 p^{27} T^{8} + 158862 p^{36} T^{9} + p^{45} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 370992 T + 1200288343895 T^{2} + 302533541054135968 T^{3} + \)\(63\!\cdots\!06\)\( T^{4} + \)\(11\!\cdots\!84\)\( T^{5} + \)\(63\!\cdots\!06\)\( p^{9} T^{6} + 302533541054135968 p^{18} T^{7} + 1200288343895 p^{27} T^{8} + 370992 p^{36} T^{9} + p^{45} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 1645870 T + 6739301712993 T^{2} - 324362283820467360 p T^{3} + \)\(18\!\cdots\!36\)\( T^{4} - \)\(16\!\cdots\!20\)\( T^{5} + \)\(18\!\cdots\!36\)\( p^{9} T^{6} - 324362283820467360 p^{19} T^{7} + 6739301712993 p^{27} T^{8} - 1645870 p^{36} T^{9} + p^{45} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 126504 p T + 63864470853029 T^{2} - \)\(16\!\cdots\!84\)\( T^{3} + \)\(16\!\cdots\!46\)\( T^{4} - \)\(33\!\cdots\!40\)\( T^{5} + \)\(16\!\cdots\!46\)\( p^{9} T^{6} - \)\(16\!\cdots\!84\)\( p^{18} T^{7} + 63864470853029 p^{27} T^{8} - 126504 p^{37} T^{9} + p^{45} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 7262362 T + 77110044378049 T^{2} + \)\(15\!\cdots\!96\)\( T^{3} + \)\(10\!\cdots\!16\)\( T^{4} - \)\(26\!\cdots\!24\)\( T^{5} + \)\(10\!\cdots\!16\)\( p^{9} T^{6} + \)\(15\!\cdots\!96\)\( p^{18} T^{7} + 77110044378049 p^{27} T^{8} + 7262362 p^{36} T^{9} + p^{45} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 31420708 T + 1002758371700605 T^{2} + \)\(18\!\cdots\!04\)\( T^{3} + \)\(30\!\cdots\!70\)\( T^{4} + \)\(36\!\cdots\!00\)\( T^{5} + \)\(30\!\cdots\!70\)\( p^{9} T^{6} + \)\(18\!\cdots\!04\)\( p^{18} T^{7} + 1002758371700605 p^{27} T^{8} + 31420708 p^{36} T^{9} + p^{45} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 7996938 T + 692313717454277 T^{2} + \)\(70\!\cdots\!40\)\( T^{3} + \)\(30\!\cdots\!18\)\( T^{4} + \)\(37\!\cdots\!36\)\( T^{5} + \)\(30\!\cdots\!18\)\( p^{9} T^{6} + \)\(70\!\cdots\!40\)\( p^{18} T^{7} + 692313717454277 p^{27} T^{8} + 7996938 p^{36} T^{9} + p^{45} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 56908268 T + 60882626535085 p T^{2} + \)\(72\!\cdots\!32\)\( T^{3} + \)\(19\!\cdots\!78\)\( T^{4} + \)\(40\!\cdots\!00\)\( T^{5} + \)\(19\!\cdots\!78\)\( p^{9} T^{6} + \)\(72\!\cdots\!32\)\( p^{18} T^{7} + 60882626535085 p^{28} T^{8} + 56908268 p^{36} T^{9} + p^{45} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 16903336 T + 3512750780124219 T^{2} + \)\(89\!\cdots\!32\)\( T^{3} + \)\(56\!\cdots\!42\)\( T^{4} + \)\(15\!\cdots\!12\)\( T^{5} + \)\(56\!\cdots\!42\)\( p^{9} T^{6} + \)\(89\!\cdots\!32\)\( p^{18} T^{7} + 3512750780124219 p^{27} T^{8} + 16903336 p^{36} T^{9} + p^{45} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 83362982 T + 7119393401614065 T^{2} + \)\(17\!\cdots\!08\)\( T^{3} + \)\(17\!\cdots\!54\)\( T^{4} - \)\(59\!\cdots\!24\)\( T^{5} + \)\(17\!\cdots\!54\)\( p^{9} T^{6} + \)\(17\!\cdots\!08\)\( p^{18} T^{7} + 7119393401614065 p^{27} T^{8} + 83362982 p^{36} T^{9} + p^{45} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 37946604 T + 12306874879678271 T^{2} - \)\(10\!\cdots\!68\)\( T^{3} - \)\(15\!\cdots\!26\)\( T^{4} - \)\(18\!\cdots\!44\)\( T^{5} - \)\(15\!\cdots\!26\)\( p^{9} T^{6} - \)\(10\!\cdots\!68\)\( p^{18} T^{7} + 12306874879678271 p^{27} T^{8} + 37946604 p^{36} T^{9} + p^{45} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 1273532 p T + 25007203254305509 T^{2} + \)\(32\!\cdots\!96\)\( T^{3} + \)\(44\!\cdots\!94\)\( T^{4} + \)\(47\!\cdots\!28\)\( T^{5} + \)\(44\!\cdots\!94\)\( p^{9} T^{6} + \)\(32\!\cdots\!96\)\( p^{18} T^{7} + 25007203254305509 p^{27} T^{8} + 1273532 p^{37} T^{9} + p^{45} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 304503600 T + 143053385462762239 T^{2} + \)\(27\!\cdots\!24\)\( T^{3} + \)\(73\!\cdots\!14\)\( T^{4} + \)\(10\!\cdots\!56\)\( T^{5} + \)\(73\!\cdots\!14\)\( p^{9} T^{6} + \)\(27\!\cdots\!24\)\( p^{18} T^{7} + 143053385462762239 p^{27} T^{8} + 304503600 p^{36} T^{9} + p^{45} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 476602922 T + 238846177016120393 T^{2} + \)\(73\!\cdots\!44\)\( T^{3} + \)\(21\!\cdots\!16\)\( T^{4} + \)\(47\!\cdots\!08\)\( T^{5} + \)\(21\!\cdots\!16\)\( p^{9} T^{6} + \)\(73\!\cdots\!44\)\( p^{18} T^{7} + 238846177016120393 p^{27} T^{8} + 476602922 p^{36} T^{9} + p^{45} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 289980486 T + 185381942449843029 T^{2} + \)\(43\!\cdots\!56\)\( T^{3} + \)\(17\!\cdots\!14\)\( T^{4} + \)\(35\!\cdots\!80\)\( T^{5} + \)\(17\!\cdots\!14\)\( p^{9} T^{6} + \)\(43\!\cdots\!56\)\( p^{18} T^{7} + 185381942449843029 p^{27} T^{8} + 289980486 p^{36} T^{9} + p^{45} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 828240610 T + 703459215778757465 T^{2} + \)\(34\!\cdots\!20\)\( T^{3} + \)\(17\!\cdots\!56\)\( T^{4} + \)\(60\!\cdots\!12\)\( T^{5} + \)\(17\!\cdots\!56\)\( p^{9} T^{6} + \)\(34\!\cdots\!20\)\( p^{18} T^{7} + 703459215778757465 p^{27} T^{8} + 828240610 p^{36} T^{9} + p^{45} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 2345556 p T + 649644875905793847 T^{2} - \)\(51\!\cdots\!68\)\( T^{3} + \)\(18\!\cdots\!38\)\( T^{4} - \)\(62\!\cdots\!60\)\( T^{5} + \)\(18\!\cdots\!38\)\( p^{9} T^{6} - \)\(51\!\cdots\!68\)\( p^{18} T^{7} + 649644875905793847 p^{27} T^{8} - 2345556 p^{37} T^{9} + p^{45} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 376848106 T + 917386117924854825 T^{2} - \)\(24\!\cdots\!92\)\( T^{3} + \)\(37\!\cdots\!98\)\( T^{4} - \)\(82\!\cdots\!88\)\( T^{5} + \)\(37\!\cdots\!98\)\( p^{9} T^{6} - \)\(24\!\cdots\!92\)\( p^{18} T^{7} + 917386117924854825 p^{27} T^{8} - 376848106 p^{36} T^{9} + p^{45} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 692035246 T + 585352914478317997 T^{2} - \)\(16\!\cdots\!84\)\( T^{3} + \)\(14\!\cdots\!18\)\( T^{4} - \)\(63\!\cdots\!40\)\( T^{5} + \)\(14\!\cdots\!18\)\( p^{9} T^{6} - \)\(16\!\cdots\!84\)\( p^{18} T^{7} + 585352914478317997 p^{27} T^{8} - 692035246 p^{36} T^{9} + p^{45} T^{10} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.08745444055847505602446487641, −10.34105374452628142326963725185, −10.17244007977755615560269768856, −10.03847427370142506228505963970, −9.978535697138332836363457657338, −9.293119528811000527869424931187, −9.231449938266936398061284756306, −8.787412359712218941459899605605, −8.601871956367533709247863674306, −8.598136357211658841608598836856, −7.65275779143657110832899147965, −7.57885072522298143691231925353, −6.80615706385180649153702521673, −6.46223949309428521033859739932, −6.26986494640548247187786581752, −6.01247830173674070548198063867, −5.40054711480333106571365523165, −5.24458633008911581840466429301, −4.91100464538083217406147490907, −4.54604993543842503714961192477, −3.41401003838981088558527340527, −3.02354470458563044341929806169, −2.92077445352434282481516541031, −2.07571285542999134835128239610, −1.46726334419730167440098000473, 0, 0, 0, 0, 0, 1.46726334419730167440098000473, 2.07571285542999134835128239610, 2.92077445352434282481516541031, 3.02354470458563044341929806169, 3.41401003838981088558527340527, 4.54604993543842503714961192477, 4.91100464538083217406147490907, 5.24458633008911581840466429301, 5.40054711480333106571365523165, 6.01247830173674070548198063867, 6.26986494640548247187786581752, 6.46223949309428521033859739932, 6.80615706385180649153702521673, 7.57885072522298143691231925353, 7.65275779143657110832899147965, 8.598136357211658841608598836856, 8.601871956367533709247863674306, 8.787412359712218941459899605605, 9.231449938266936398061284756306, 9.293119528811000527869424931187, 9.978535697138332836363457657338, 10.03847427370142506228505963970, 10.17244007977755615560269768856, 10.34105374452628142326963725185, 11.08745444055847505602446487641

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