Properties

Label 10-1856e5-1.1-c3e5-0-0
Degree $10$
Conductor $2.202\times 10^{16}$
Sign $1$
Analytic cond. $1.57478\times 10^{10}$
Root an. cond. $10.4645$
Motivic weight $3$
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  − 4·3-s − 10·5-s + 32·7-s − 45·9-s − 36·11-s − 26·13-s + 40·15-s + 82·17-s − 156·19-s − 128·21-s + 336·23-s − 187·25-s + 120·27-s − 145·29-s + 432·31-s + 144·33-s − 320·35-s + 18·37-s + 104·39-s + 82·41-s − 340·43-s + 450·45-s + 680·47-s − 403·49-s − 328·51-s + 102·53-s + 360·55-s + ⋯
L(s)  = 1  − 0.769·3-s − 0.894·5-s + 1.72·7-s − 5/3·9-s − 0.986·11-s − 0.554·13-s + 0.688·15-s + 1.16·17-s − 1.88·19-s − 1.33·21-s + 3.04·23-s − 1.49·25-s + 0.855·27-s − 0.928·29-s + 2.50·31-s + 0.759·33-s − 1.54·35-s + 0.0799·37-s + 0.427·39-s + 0.312·41-s − 1.20·43-s + 1.49·45-s + 2.11·47-s − 1.17·49-s − 0.900·51-s + 0.264·53-s + 0.882·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{30} \cdot 29^{5}\)
Sign: $1$
Analytic conductor: \(1.57478\times 10^{10}\)
Root analytic conductor: \(10.4645\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1856} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{30} \cdot 29^{5} ,\ ( \ : 3/2, 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.8480202717\)
\(L(\frac12)\) \(\approx\) \(0.8480202717\)
\(L(\frac{5}{2})\) 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 \( 1 \)
29$C_1$ \( ( 1 + p T )^{5} \)
good3$C_2 \wr S_5$ \( 1 + 4 T + 61 T^{2} + 304 T^{3} + 2821 T^{4} + 9244 T^{5} + 2821 p^{3} T^{6} + 304 p^{6} T^{7} + 61 p^{9} T^{8} + 4 p^{12} T^{9} + p^{15} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 2 p T + 287 T^{2} + 2696 T^{3} + 11009 p T^{4} + 371994 T^{5} + 11009 p^{4} T^{6} + 2696 p^{6} T^{7} + 287 p^{9} T^{8} + 2 p^{13} T^{9} + p^{15} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 32 T + 1427 T^{2} - 36032 T^{3} + 18282 p^{2} T^{4} - 17455168 T^{5} + 18282 p^{5} T^{6} - 36032 p^{6} T^{7} + 1427 p^{9} T^{8} - 32 p^{12} T^{9} + p^{15} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 36 T + 5309 T^{2} + 177032 T^{3} + 12820837 T^{4} + 341964932 T^{5} + 12820837 p^{3} T^{6} + 177032 p^{6} T^{7} + 5309 p^{9} T^{8} + 36 p^{12} T^{9} + p^{15} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 2 p T + 5839 T^{2} + 200496 T^{3} + 22116069 T^{4} + 524580946 T^{5} + 22116069 p^{3} T^{6} + 200496 p^{6} T^{7} + 5839 p^{9} T^{8} + 2 p^{13} T^{9} + p^{15} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 82 T + 21789 T^{2} - 1579928 T^{3} + 201344338 T^{4} - 11565860396 T^{5} + 201344338 p^{3} T^{6} - 1579928 p^{6} T^{7} + 21789 p^{9} T^{8} - 82 p^{12} T^{9} + p^{15} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 156 T + 38511 T^{2} + 4147088 T^{3} + 557756866 T^{4} + 42213259048 T^{5} + 557756866 p^{3} T^{6} + 4147088 p^{6} T^{7} + 38511 p^{9} T^{8} + 156 p^{12} T^{9} + p^{15} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 336 T + 68587 T^{2} - 9632544 T^{3} + 1169621794 T^{4} - 5540381280 p T^{5} + 1169621794 p^{3} T^{6} - 9632544 p^{6} T^{7} + 68587 p^{9} T^{8} - 336 p^{12} T^{9} + p^{15} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 432 T + 201929 T^{2} - 52526032 T^{3} + 13543164109 T^{4} - 2362357787192 T^{5} + 13543164109 p^{3} T^{6} - 52526032 p^{6} T^{7} + 201929 p^{9} T^{8} - 432 p^{12} T^{9} + p^{15} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 18 T + 164945 T^{2} + 1995160 T^{3} + 13003010010 T^{4} + 309782516404 T^{5} + 13003010010 p^{3} T^{6} + 1995160 p^{6} T^{7} + 164945 p^{9} T^{8} - 18 p^{12} T^{9} + p^{15} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 2 p T + 134797 T^{2} + 4859064 T^{3} + 11312527994 T^{4} + 717309134036 T^{5} + 11312527994 p^{3} T^{6} + 4859064 p^{6} T^{7} + 134797 p^{9} T^{8} - 2 p^{13} T^{9} + p^{15} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 340 T + 78557 T^{2} + 13668248 T^{3} + 883246085 T^{4} - 1553774172796 T^{5} + 883246085 p^{3} T^{6} + 13668248 p^{6} T^{7} + 78557 p^{9} T^{8} + 340 p^{12} T^{9} + p^{15} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 680 T + 384593 T^{2} - 98072184 T^{3} + 24706007645 T^{4} - 3145150453528 T^{5} + 24706007645 p^{3} T^{6} - 98072184 p^{6} T^{7} + 384593 p^{9} T^{8} - 680 p^{12} T^{9} + p^{15} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 102 T + 654439 T^{2} - 51999000 T^{3} + 183063886741 T^{4} - 11065291160166 T^{5} + 183063886741 p^{3} T^{6} - 51999000 p^{6} T^{7} + 654439 p^{9} T^{8} - 102 p^{12} T^{9} + p^{15} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 924 T + 866103 T^{2} + 337060304 T^{3} + 171323060562 T^{4} + 43701048197672 T^{5} + 171323060562 p^{3} T^{6} + 337060304 p^{6} T^{7} + 866103 p^{9} T^{8} + 924 p^{12} T^{9} + p^{15} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 618 T + 750561 T^{2} - 351831128 T^{3} + 243423855586 T^{4} - 98404847105884 T^{5} + 243423855586 p^{3} T^{6} - 351831128 p^{6} T^{7} + 750561 p^{9} T^{8} - 618 p^{12} T^{9} + p^{15} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 44 T + 597703 T^{2} + 81163152 T^{3} + 249970117114 T^{4} + 11056591462856 T^{5} + 249970117114 p^{3} T^{6} + 81163152 p^{6} T^{7} + 597703 p^{9} T^{8} + 44 p^{12} T^{9} + p^{15} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 1032 T + 1465835 T^{2} - 983439264 T^{3} + 931340637650 T^{4} - 489572777520432 T^{5} + 931340637650 p^{3} T^{6} - 983439264 p^{6} T^{7} + 1465835 p^{9} T^{8} - 1032 p^{12} T^{9} + p^{15} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 1078 T + 1544061 T^{2} + 895316056 T^{3} + 816827199978 T^{4} + 365553950129028 T^{5} + 816827199978 p^{3} T^{6} + 895316056 p^{6} T^{7} + 1544061 p^{9} T^{8} + 1078 p^{12} T^{9} + p^{15} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 200 T + 2450913 T^{2} - 390614776 T^{3} + 2409756860989 T^{4} - 287955059167144 T^{5} + 2409756860989 p^{3} T^{6} - 390614776 p^{6} T^{7} + 2450913 p^{9} T^{8} - 200 p^{12} T^{9} + p^{15} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 452 T + 1144543 T^{2} + 431114608 T^{3} + 1056135128754 T^{4} + 379282762835224 T^{5} + 1056135128754 p^{3} T^{6} + 431114608 p^{6} T^{7} + 1144543 p^{9} T^{8} + 452 p^{12} T^{9} + p^{15} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 1790 T + 3080861 T^{2} + 2722201848 T^{3} + 2858190637178 T^{4} + 1978995927638068 T^{5} + 2858190637178 p^{3} T^{6} + 2722201848 p^{6} T^{7} + 3080861 p^{9} T^{8} + 1790 p^{12} T^{9} + p^{15} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 2518 T + 6513717 T^{2} + 9477719064 T^{3} + 13480346210298 T^{4} + 13064195578094340 T^{5} + 13480346210298 p^{3} T^{6} + 9477719064 p^{6} T^{7} + 6513717 p^{9} T^{8} + 2518 p^{12} T^{9} + p^{15} T^{10} \)
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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

−5.09072890947972885906481939495, −4.94281166645378493514087334247, −4.81612947147568494393861384767, −4.77339061003286556745061782914, −4.57080040298765080274333379983, −4.26511442615022648066635861166, −4.12868478991709858899137038348, −3.95198572847476000915725640952, −3.78217404160499395309832210868, −3.60461008094319068463718622955, −3.13306760235205853456441211547, −3.00103025955831425961186159401, −2.87990264557674828068989226878, −2.80769808140121024190190184892, −2.73506504140253755132772445379, −2.24711227627512677144299460483, −2.00525198336622792101733123007, −1.90068112808130427246010438757, −1.54499332472573280125183218416, −1.53512241343622635734437426453, −1.03509136178878113026406792672, −0.76241098611340975065378953799, −0.69688779632008960473218383317, −0.25501605763080462599350759825, −0.16612842489638316736938986530, 0.16612842489638316736938986530, 0.25501605763080462599350759825, 0.69688779632008960473218383317, 0.76241098611340975065378953799, 1.03509136178878113026406792672, 1.53512241343622635734437426453, 1.54499332472573280125183218416, 1.90068112808130427246010438757, 2.00525198336622792101733123007, 2.24711227627512677144299460483, 2.73506504140253755132772445379, 2.80769808140121024190190184892, 2.87990264557674828068989226878, 3.00103025955831425961186159401, 3.13306760235205853456441211547, 3.60461008094319068463718622955, 3.78217404160499395309832210868, 3.95198572847476000915725640952, 4.12868478991709858899137038348, 4.26511442615022648066635861166, 4.57080040298765080274333379983, 4.77339061003286556745061782914, 4.81612947147568494393861384767, 4.94281166645378493514087334247, 5.09072890947972885906481939495

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