Properties

Label 10-272e5-1.1-c9e5-0-0
Degree $10$
Conductor $1.489\times 10^{12}$
Sign $1$
Analytic cond. $5.39550\times 10^{10}$
Root an. cond. $11.8359$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 236·3-s + 1.48e3·5-s + 1.32e4·7-s − 1.58e4·9-s + 6.80e4·11-s − 1.58e5·13-s + 3.49e5·15-s − 4.17e5·17-s + 3.70e5·19-s + 3.11e6·21-s − 1.64e6·23-s − 2.15e6·25-s − 8.67e6·27-s + 3.66e6·29-s + 7.26e6·31-s + 1.60e7·33-s + 1.95e7·35-s − 3.14e7·37-s − 3.74e7·39-s − 7.99e6·41-s + 5.69e7·43-s − 2.34e7·45-s + 1.69e7·47-s − 1.96e7·49-s − 9.85e7·51-s − 8.33e7·53-s + 1.00e8·55-s + ⋯
L(s)  = 1  + 1.68·3-s + 1.05·5-s + 2.07·7-s − 0.806·9-s + 1.40·11-s − 1.54·13-s + 1.78·15-s − 1.21·17-s + 0.653·19-s + 3.49·21-s − 1.22·23-s − 1.10·25-s − 3.14·27-s + 0.963·29-s + 1.41·31-s + 2.35·33-s + 2.20·35-s − 2.75·37-s − 2.59·39-s − 0.441·41-s + 2.53·43-s − 0.853·45-s + 0.505·47-s − 0.486·49-s − 2.03·51-s − 1.45·53-s + 1.48·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 17^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 17^{5}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{20} \cdot 17^{5}\)
Sign: $1$
Analytic conductor: \(5.39550\times 10^{10}\)
Root analytic conductor: \(11.8359\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{20} \cdot 17^{5} ,\ ( \ : 9/2, 9/2, 9/2, 9/2, 9/2 ),\ 1 )\)

Particular Values

\(L(5)\) \(\approx\) \(13.75349815\)
\(L(\frac12)\) \(\approx\) \(13.75349815\)
\(L(\frac{11}{2})\) 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)$
bad2 \( 1 \)
17$C_1$ \( ( 1 + p^{4} T )^{5} \)
good3$C_2 \wr S_5$ \( 1 - 236 T + 23855 p T^{2} - 442924 p^{3} T^{3} + 74219104 p^{3} T^{4} - 3544592944 p^{4} T^{5} + 74219104 p^{12} T^{6} - 442924 p^{21} T^{7} + 23855 p^{28} T^{8} - 236 p^{36} T^{9} + p^{45} T^{10} \)
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

−5.57097021949361362042577053302, −5.31820528819888926120048203586, −5.27952004939618832514928050248, −5.20756077777799967817079104107, −5.18003978440908109615766464440, −4.51529846814718957394177994208, −4.49344933081747920737152060026, −4.16654509275371244323638478729, −4.02077057490679885294377798184, −3.94240940537509023381989184067, −3.29696398692107505753022360022, −3.27961879988995199474718342078, −3.00514429989698296046252399683, −2.87519935451421313596493456276, −2.61290134381911024279078685006, −2.06467809055782479132797575832, −2.04317097962619882521512685955, −2.02186411785145499094443803818, −2.00564195903360612601227489408, −1.60904669376382880009188016657, −1.23014855692568482565874080478, −1.02815163578928402115276311143, −0.53349071579340325735062639807, −0.52856701714122070409894936477, −0.19785623225894636513856731954, 0.19785623225894636513856731954, 0.52856701714122070409894936477, 0.53349071579340325735062639807, 1.02815163578928402115276311143, 1.23014855692568482565874080478, 1.60904669376382880009188016657, 2.00564195903360612601227489408, 2.02186411785145499094443803818, 2.04317097962619882521512685955, 2.06467809055782479132797575832, 2.61290134381911024279078685006, 2.87519935451421313596493456276, 3.00514429989698296046252399683, 3.27961879988995199474718342078, 3.29696398692107505753022360022, 3.94240940537509023381989184067, 4.02077057490679885294377798184, 4.16654509275371244323638478729, 4.49344933081747920737152060026, 4.51529846814718957394177994208, 5.18003978440908109615766464440, 5.20756077777799967817079104107, 5.27952004939618832514928050248, 5.31820528819888926120048203586, 5.57097021949361362042577053302

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