Properties

Label 10-280e5-1.1-c5e5-0-0
Degree $10$
Conductor $1.721\times 10^{12}$
Sign $1$
Analytic cond. $1.82638\times 10^{8}$
Root an. cond. $6.70130$
Motivic weight $5$
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  − 3·3-s − 125·5-s − 245·7-s − 417·9-s − 263·11-s − 729·13-s + 375·15-s − 1.00e3·17-s − 2.50e3·19-s + 735·21-s − 1.06e3·23-s + 9.37e3·25-s + 1.70e3·27-s + 3.48e3·29-s + 7.88e3·31-s + 789·33-s + 3.06e4·35-s + 1.31e4·37-s + 2.18e3·39-s + 2.39e4·41-s + 3.97e3·43-s + 5.21e4·45-s + 9.05e3·47-s + 3.60e4·49-s + 3.00e3·51-s + 2.12e3·53-s + 3.28e4·55-s + ⋯
L(s)  = 1  − 0.192·3-s − 2.23·5-s − 1.88·7-s − 1.71·9-s − 0.655·11-s − 1.19·13-s + 0.430·15-s − 0.841·17-s − 1.59·19-s + 0.363·21-s − 0.420·23-s + 3·25-s + 0.450·27-s + 0.770·29-s + 1.47·31-s + 0.126·33-s + 4.22·35-s + 1.57·37-s + 0.230·39-s + 2.22·41-s + 0.328·43-s + 3.83·45-s + 0.598·47-s + 15/7·49-s + 0.161·51-s + 0.104·53-s + 1.46·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{5} \cdot 7^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{5} \cdot 7^{5}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{15} \cdot 5^{5} \cdot 7^{5}\)
Sign: $1$
Analytic conductor: \(1.82638\times 10^{8}\)
Root analytic conductor: \(6.70130\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{15} \cdot 5^{5} \cdot 7^{5} ,\ ( \ : 5/2, 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(0.6821258556\)
\(L(\frac12)\) \(\approx\) \(0.6821258556\)
\(L(\frac{7}{2})\) 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)$
bad2 \( 1 \)
5$C_1$ \( ( 1 + p^{2} T )^{5} \)
7$C_1$ \( ( 1 + p^{2} T )^{5} \)
good3$C_2 \wr S_5$ \( 1 + p T + 142 p T^{2} + 821 T^{3} + 12775 p T^{4} + 12008 p^{2} T^{5} + 12775 p^{6} T^{6} + 821 p^{10} T^{7} + 142 p^{16} T^{8} + p^{21} T^{9} + p^{25} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 263 T + 340178 T^{2} + 1195339 p T^{3} + 27905351205 T^{4} - 9263250336808 T^{5} + 27905351205 p^{5} T^{6} + 1195339 p^{11} T^{7} + 340178 p^{15} T^{8} + 263 p^{20} T^{9} + p^{25} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 729 T + 96500 p T^{2} + 745138011 T^{3} + 779098752607 T^{4} + 369174790586904 T^{5} + 779098752607 p^{5} T^{6} + 745138011 p^{10} T^{7} + 96500 p^{16} T^{8} + 729 p^{20} T^{9} + p^{25} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 59 p T + 1555088 T^{2} + 165139337 p T^{3} + 3679973533323 T^{4} + 3277346054906080 T^{5} + 3679973533323 p^{5} T^{6} + 165139337 p^{11} T^{7} + 1555088 p^{15} T^{8} + 59 p^{21} T^{9} + p^{25} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 2506 T + 4256187 T^{2} + 2573144496 T^{3} - 7461797422938 T^{4} - 17819619410062356 T^{5} - 7461797422938 p^{5} T^{6} + 2573144496 p^{10} T^{7} + 4256187 p^{15} T^{8} + 2506 p^{20} T^{9} + p^{25} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 1066 T + 7611455 T^{2} + 4542998992 T^{3} + 45705742349382 T^{4} - 68950154051855156 T^{5} + 45705742349382 p^{5} T^{6} + 4542998992 p^{10} T^{7} + 7611455 p^{15} T^{8} + 1066 p^{20} T^{9} + p^{25} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 3489 T + 63784132 T^{2} - 232779125283 T^{3} + 1878775274724991 T^{4} - 6605265740923215768 T^{5} + 1878775274724991 p^{5} T^{6} - 232779125283 p^{10} T^{7} + 63784132 p^{15} T^{8} - 3489 p^{20} T^{9} + p^{25} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 7880 T + 74785563 T^{2} - 379523483232 T^{3} + 2781696993608586 T^{4} - 12868521079574790576 T^{5} + 2781696993608586 p^{5} T^{6} - 379523483232 p^{10} T^{7} + 74785563 p^{15} T^{8} - 7880 p^{20} T^{9} + p^{25} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 13118 T + 317175441 T^{2} - 3006879972168 T^{3} + 40647453094560546 T^{4} - \)\(29\!\cdots\!84\)\( T^{5} + 40647453094560546 p^{5} T^{6} - 3006879972168 p^{10} T^{7} + 317175441 p^{15} T^{8} - 13118 p^{20} T^{9} + p^{25} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 23972 T + 514192529 T^{2} - 8421338142800 T^{3} + 117433086367025910 T^{4} - \)\(13\!\cdots\!96\)\( T^{5} + 117433086367025910 p^{5} T^{6} - 8421338142800 p^{10} T^{7} + 514192529 p^{15} T^{8} - 23972 p^{20} T^{9} + p^{25} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 3978 T + 89929955 T^{2} + 995358621360 T^{3} + 31200365104356022 T^{4} - \)\(26\!\cdots\!36\)\( T^{5} + 31200365104356022 p^{5} T^{6} + 995358621360 p^{10} T^{7} + 89929955 p^{15} T^{8} - 3978 p^{20} T^{9} + p^{25} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 9057 T + 788168230 T^{2} - 3808702905423 T^{3} + 270675631910429017 T^{4} - \)\(79\!\cdots\!80\)\( T^{5} + 270675631910429017 p^{5} T^{6} - 3808702905423 p^{10} T^{7} + 788168230 p^{15} T^{8} - 9057 p^{20} T^{9} + p^{25} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 2128 T + 494176973 T^{2} - 1532972410672 T^{3} + 341492141219982054 T^{4} - \)\(17\!\cdots\!48\)\( T^{5} + 341492141219982054 p^{5} T^{6} - 1532972410672 p^{10} T^{7} + 494176973 p^{15} T^{8} - 2128 p^{20} T^{9} + p^{25} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 15512 T + 779934647 T^{2} - 10102720080352 T^{3} + 870060510542985018 T^{4} + \)\(71\!\cdots\!80\)\( T^{5} + 870060510542985018 p^{5} T^{6} - 10102720080352 p^{10} T^{7} + 779934647 p^{15} T^{8} + 15512 p^{20} T^{9} + p^{25} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 4560 T + 1307115749 T^{2} - 17145781465584 T^{3} + 786645069913257622 T^{4} - \)\(18\!\cdots\!44\)\( T^{5} + 786645069913257622 p^{5} T^{6} - 17145781465584 p^{10} T^{7} + 1307115749 p^{15} T^{8} - 4560 p^{20} T^{9} + p^{25} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 7780 T + 2643859359 T^{2} - 54894680801328 T^{3} + 4463568945488665146 T^{4} - \)\(12\!\cdots\!32\)\( T^{5} + 4463568945488665146 p^{5} T^{6} - 54894680801328 p^{10} T^{7} + 2643859359 p^{15} T^{8} - 7780 p^{20} T^{9} + p^{25} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 32752 T + 7969049603 T^{2} - 184412145501760 T^{3} + 26270191719722487786 T^{4} - \)\(45\!\cdots\!96\)\( T^{5} + 26270191719722487786 p^{5} T^{6} - 184412145501760 p^{10} T^{7} + 7969049603 p^{15} T^{8} - 32752 p^{20} T^{9} + p^{25} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 189498 T + 21760396853 T^{2} - 1775690065855992 T^{3} + \)\(11\!\cdots\!62\)\( T^{4} - \)\(57\!\cdots\!08\)\( T^{5} + \)\(11\!\cdots\!62\)\( p^{5} T^{6} - 1775690065855992 p^{10} T^{7} + 21760396853 p^{15} T^{8} - 189498 p^{20} T^{9} + p^{25} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 42055 T + 6995657358 T^{2} - 193752235090281 T^{3} + 27010902499411523097 T^{4} - \)\(48\!\cdots\!72\)\( T^{5} + 27010902499411523097 p^{5} T^{6} - 193752235090281 p^{10} T^{7} + 6995657358 p^{15} T^{8} - 42055 p^{20} T^{9} + p^{25} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 58420 T + 13252788191 T^{2} - 411449540580400 T^{3} + 67750483608546119466 T^{4} - \)\(13\!\cdots\!84\)\( T^{5} + 67750483608546119466 p^{5} T^{6} - 411449540580400 p^{10} T^{7} + 13252788191 p^{15} T^{8} - 58420 p^{20} T^{9} + p^{25} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 231324 T + 38173479169 T^{2} - 4773757016381136 T^{3} + \)\(47\!\cdots\!82\)\( T^{4} - \)\(39\!\cdots\!00\)\( T^{5} + \)\(47\!\cdots\!82\)\( p^{5} T^{6} - 4773757016381136 p^{10} T^{7} + 38173479169 p^{15} T^{8} - 231324 p^{20} T^{9} + p^{25} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 247569 T + 50969116136 T^{2} - 7043938986181395 T^{3} + \)\(89\!\cdots\!59\)\( T^{4} - \)\(87\!\cdots\!76\)\( T^{5} + \)\(89\!\cdots\!59\)\( p^{5} T^{6} - 7043938986181395 p^{10} T^{7} + 50969116136 p^{15} T^{8} - 247569 p^{20} T^{9} + p^{25} T^{10} \)
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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

−6.28290161101631193290815822826, −6.27220707141396356483205874696, −6.19848875372754340784910144960, −5.94172859624845606528446434299, −5.77678604503444622287676911262, −5.13654933576731337883504875784, −5.01871032265332143307195155038, −4.82106404147341616184262581478, −4.69218487704418856627163516373, −4.46250194286798660573710805607, −4.19958753551216650697715036816, −3.64406156055255957449885518535, −3.58013481530626237507925495428, −3.56173261002247765137778720409, −3.43080036049146812033461974036, −2.64460244334617040008738670158, −2.49857781181719345416259241690, −2.47280133145965438542753142423, −2.46992631114777564647962591854, −2.07956137909602239816666221932, −1.11769705925842525684704428444, −0.77490516035717051737954373031, −0.55660942989208648484104759188, −0.50530900576386933728930881134, −0.19520177569918713715354960241, 0.19520177569918713715354960241, 0.50530900576386933728930881134, 0.55660942989208648484104759188, 0.77490516035717051737954373031, 1.11769705925842525684704428444, 2.07956137909602239816666221932, 2.46992631114777564647962591854, 2.47280133145965438542753142423, 2.49857781181719345416259241690, 2.64460244334617040008738670158, 3.43080036049146812033461974036, 3.56173261002247765137778720409, 3.58013481530626237507925495428, 3.64406156055255957449885518535, 4.19958753551216650697715036816, 4.46250194286798660573710805607, 4.69218487704418856627163516373, 4.82106404147341616184262581478, 5.01871032265332143307195155038, 5.13654933576731337883504875784, 5.77678604503444622287676911262, 5.94172859624845606528446434299, 6.19848875372754340784910144960, 6.27220707141396356483205874696, 6.28290161101631193290815822826

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