Properties

Label 10-280e5-1.1-c5e5-0-1
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  + 15·3-s + 125·5-s + 245·7-s + 71·9-s + 281·11-s + 909·13-s + 1.87e3·15-s + 1.49e3·17-s − 422·19-s + 3.67e3·21-s − 62·23-s + 9.37e3·25-s − 2.39e3·27-s − 2.04e3·29-s + 1.63e3·31-s + 4.21e3·33-s + 3.06e4·35-s − 1.03e4·37-s + 1.36e4·39-s + 6.42e3·41-s + 2.83e4·43-s + 8.87e3·45-s + 2.09e4·47-s + 3.60e4·49-s + 2.24e4·51-s + 4.37e4·53-s + 3.51e4·55-s + ⋯
L(s)  = 1  + 0.962·3-s + 2.23·5-s + 1.88·7-s + 0.292·9-s + 0.700·11-s + 1.49·13-s + 2.15·15-s + 1.25·17-s − 0.268·19-s + 1.81·21-s − 0.0244·23-s + 3·25-s − 0.632·27-s − 0.451·29-s + 0.305·31-s + 0.673·33-s + 4.22·35-s − 1.24·37-s + 1.43·39-s + 0.596·41-s + 2.33·43-s + 0.653·45-s + 1.38·47-s + 15/7·49-s + 1.20·51-s + 2.13·53-s + 1.56·55-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: induced by $\chi_{280} (1, \cdot )$
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

\(L(3)\) \(\approx\) \(82.78089068\)
\(L(\frac12)\) \(\approx\) \(82.78089068\)
\(L(\frac{7}{2})\) 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$C_2 \wr S_5$ \( 1 - 5 p T + 154 T^{2} + 1151 T^{3} - 17969 p T^{4} + 63736 p^{2} T^{5} - 17969 p^{6} T^{6} + 1151 p^{10} T^{7} + 154 p^{15} T^{8} - 5 p^{21} T^{9} + p^{25} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 281 T + 221602 T^{2} + 4570105 T^{3} + 40747949317 T^{4} - 4335881998760 T^{5} + 40747949317 p^{5} T^{6} + 4570105 p^{10} T^{7} + 221602 p^{15} T^{8} - 281 p^{20} T^{9} + p^{25} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 909 T + 1436620 T^{2} - 858523871 T^{3} + 839117392375 T^{4} - 396610941659336 T^{5} + 839117392375 p^{5} T^{6} - 858523871 p^{10} T^{7} + 1436620 p^{15} T^{8} - 909 p^{20} T^{9} + p^{25} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 1495 T + 5368176 T^{2} - 6041848165 T^{3} + 13573892472707 T^{4} - 12001168625572880 T^{5} + 13573892472707 p^{5} T^{6} - 6041848165 p^{10} T^{7} + 5368176 p^{15} T^{8} - 1495 p^{20} T^{9} + p^{25} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 422 T + 390361 p T^{2} + 3236021232 T^{3} + 29599412304902 T^{4} + 12005015702197876 T^{5} + 29599412304902 p^{5} T^{6} + 3236021232 p^{10} T^{7} + 390361 p^{16} T^{8} + 422 p^{20} T^{9} + p^{25} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 62 T + 20580223 T^{2} - 2921201488 T^{3} + 222654385879558 T^{4} - 18821069520565660 T^{5} + 222654385879558 p^{5} T^{6} - 2921201488 p^{10} T^{7} + 20580223 p^{15} T^{8} + 62 p^{20} T^{9} + p^{25} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 2047 T + 12605620 T^{2} - 13927401731 T^{3} - 1908393559861 p T^{4} - 2737380069812364536 T^{5} - 1908393559861 p^{6} T^{6} - 13927401731 p^{10} T^{7} + 12605620 p^{15} T^{8} + 2047 p^{20} T^{9} + p^{25} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 1636 T + 109234699 T^{2} - 172511938992 T^{3} + 5563121129882810 T^{4} - 6956514622281569752 T^{5} + 5563121129882810 p^{5} T^{6} - 172511938992 p^{10} T^{7} + 109234699 p^{15} T^{8} - 1636 p^{20} T^{9} + p^{25} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 10358 T + 114652225 T^{2} + 201619793736 T^{3} + 4341551200446386 T^{4} + 446228899272382372 T^{5} + 4341551200446386 p^{5} T^{6} + 201619793736 p^{10} T^{7} + 114652225 p^{15} T^{8} + 10358 p^{20} T^{9} + p^{25} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 6424 T + 395117553 T^{2} - 646274781232 T^{3} + 60937741661540822 T^{4} + 34763395578700777680 T^{5} + 60937741661540822 p^{5} T^{6} - 646274781232 p^{10} T^{7} + 395117553 p^{15} T^{8} - 6424 p^{20} T^{9} + p^{25} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 28306 T + 642480547 T^{2} - 8887632516816 T^{3} + 119006154611192822 T^{4} - \)\(12\!\cdots\!16\)\( T^{5} + 119006154611192822 p^{5} T^{6} - 8887632516816 p^{10} T^{7} + 642480547 p^{15} T^{8} - 28306 p^{20} T^{9} + p^{25} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 20955 T + 1089343902 T^{2} - 18546466912749 T^{3} + 485237221486674017 T^{4} - \)\(63\!\cdots\!64\)\( T^{5} + 485237221486674017 p^{5} T^{6} - 18546466912749 p^{10} T^{7} + 1089343902 p^{15} T^{8} - 20955 p^{20} T^{9} + p^{25} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 43748 T + 838202445 T^{2} - 12691980106256 T^{3} + 562895427828965030 T^{4} - \)\(16\!\cdots\!60\)\( T^{5} + 562895427828965030 p^{5} T^{6} - 12691980106256 p^{10} T^{7} + 838202445 p^{15} T^{8} - 43748 p^{20} T^{9} + p^{25} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 45788 T + 3593077975 T^{2} - 121778928024912 T^{3} + 5136498549843134810 T^{4} - \)\(12\!\cdots\!44\)\( T^{5} + 5136498549843134810 p^{5} T^{6} - 121778928024912 p^{10} T^{7} + 3593077975 p^{15} T^{8} - 45788 p^{20} T^{9} + p^{25} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 50432 T + 4030546853 T^{2} - 136393865631120 T^{3} + 6194450556799903702 T^{4} - \)\(25\!\cdots\!36\)\( p T^{5} + 6194450556799903702 p^{5} T^{6} - 136393865631120 p^{10} T^{7} + 4030546853 p^{15} T^{8} - 50432 p^{20} T^{9} + p^{25} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 40712 T + 5343481519 T^{2} - 173526050513376 T^{3} + 12800459062427075018 T^{4} - \)\(32\!\cdots\!72\)\( T^{5} + 12800459062427075018 p^{5} T^{6} - 173526050513376 p^{10} T^{7} + 5343481519 p^{15} T^{8} - 40712 p^{20} T^{9} + p^{25} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 3096 T + 1680533635 T^{2} - 83824474204512 T^{3} + 6649208184641447530 T^{4} + \)\(25\!\cdots\!84\)\( T^{5} + 6649208184641447530 p^{5} T^{6} - 83824474204512 p^{10} T^{7} + 1680533635 p^{15} T^{8} + 3096 p^{20} T^{9} + p^{25} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 135438 T + 11904560997 T^{2} - 760960528321992 T^{3} + 42362862874340730882 T^{4} - \)\(20\!\cdots\!20\)\( T^{5} + 42362862874340730882 p^{5} T^{6} - 760960528321992 p^{10} T^{7} + 11904560997 p^{15} T^{8} - 135438 p^{20} T^{9} + p^{25} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 13191 T + 6068072158 T^{2} - 343248053940809 T^{3} + 16534271595589609465 T^{4} - \)\(17\!\cdots\!84\)\( T^{5} + 16534271595589609465 p^{5} T^{6} - 343248053940809 p^{10} T^{7} + 6068072158 p^{15} T^{8} - 13191 p^{20} T^{9} + p^{25} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 35108 T + 14240686639 T^{2} - 416149643153968 T^{3} + 97125793235490599674 T^{4} - \)\(23\!\cdots\!68\)\( T^{5} + 97125793235490599674 p^{5} T^{6} - 416149643153968 p^{10} T^{7} + 14240686639 p^{15} T^{8} - 35108 p^{20} T^{9} + p^{25} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 213772 T + 36142472577 T^{2} - 3766577124770512 T^{3} + \)\(36\!\cdots\!74\)\( T^{4} - \)\(26\!\cdots\!88\)\( T^{5} + \)\(36\!\cdots\!74\)\( p^{5} T^{6} - 3766577124770512 p^{10} T^{7} + 36142472577 p^{15} T^{8} - 213772 p^{20} T^{9} + p^{25} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 10659 T + 24686333912 T^{2} - 320233522422945 T^{3} + \)\(33\!\cdots\!39\)\( T^{4} - \)\(48\!\cdots\!36\)\( T^{5} + \)\(33\!\cdots\!39\)\( p^{5} T^{6} - 320233522422945 p^{10} T^{7} + 24686333912 p^{15} T^{8} - 10659 p^{20} T^{9} + p^{25} T^{10} \)
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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

−6.31271445615688204044001848628, −6.05840469128741337504519881562, −5.94806940550914159509243190371, −5.91477804335380785657986465922, −5.65634974429323574879703168117, −5.36099569442125102891702181573, −5.05431980456017445520473720436, −4.93587495361630402307697591659, −4.92965258894169621162018710222, −4.35418940734519868825688185379, −4.08335927893765529112401797346, −3.81042601850168948400750303476, −3.66708966895661366146336163250, −3.47932538402751942718831840385, −3.31567293072768676370663262547, −2.54963616492546237563332231851, −2.41221777884955871814435888195, −2.24429078518721282654680591133, −2.15540302193688309202603114133, −1.92760779923858033828989494429, −1.47507092495482398870471603134, −1.07977122861935902003253480741, −1.02368648561560470155452809102, −0.77675293585368835529628785413, −0.59153553102278329838432045121, 0.59153553102278329838432045121, 0.77675293585368835529628785413, 1.02368648561560470155452809102, 1.07977122861935902003253480741, 1.47507092495482398870471603134, 1.92760779923858033828989494429, 2.15540302193688309202603114133, 2.24429078518721282654680591133, 2.41221777884955871814435888195, 2.54963616492546237563332231851, 3.31567293072768676370663262547, 3.47932538402751942718831840385, 3.66708966895661366146336163250, 3.81042601850168948400750303476, 4.08335927893765529112401797346, 4.35418940734519868825688185379, 4.92965258894169621162018710222, 4.93587495361630402307697591659, 5.05431980456017445520473720436, 5.36099569442125102891702181573, 5.65634974429323574879703168117, 5.91477804335380785657986465922, 5.94806940550914159509243190371, 6.05840469128741337504519881562, 6.31271445615688204044001848628

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