Properties

Degree $14$
Conductor $410338673$
Sign $1$
Motivic weight $9$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 88·3-s − 597·4-s + 1.36e3·5-s − 88·6-s + 9.38e3·7-s + 6.37e3·8-s − 2.43e4·9-s − 1.36e3·10-s + 1.35e5·11-s − 5.25e4·12-s + 1.66e5·13-s − 9.38e3·14-s + 1.19e5·15-s + 2.32e5·16-s + 5.84e5·17-s + 2.43e4·18-s + 7.77e5·19-s − 8.13e5·20-s + 8.26e5·21-s − 1.35e5·22-s + 1.35e6·23-s + 5.61e5·24-s − 5.37e6·25-s − 1.66e5·26-s − 4.45e6·27-s − 5.60e6·28-s + ⋯
L(s)  = 1  − 0.0441·2-s + 0.627·3-s − 1.16·4-s + 0.974·5-s − 0.0277·6-s + 1.47·7-s + 0.550·8-s − 1.23·9-s − 0.0430·10-s + 2.79·11-s − 0.731·12-s + 1.61·13-s − 0.0653·14-s + 0.611·15-s + 0.887·16-s + 1.69·17-s + 0.0545·18-s + 1.36·19-s − 1.13·20-s + 0.926·21-s − 0.123·22-s + 1.01·23-s + 0.345·24-s − 2.75·25-s − 0.0712·26-s − 1.61·27-s − 1.72·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{7}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(17^{7}\)
Sign: $1$
Motivic weight: \(9\)
Character: induced by $\chi_{17} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((14,\ 17^{7} ,\ ( \ : [9/2]^{7} ),\ 1 )\)

Particular Values

\(L(5)\) \(\approx\) \(23.0228\)
\(L(\frac12)\) \(\approx\) \(23.0228\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( ( 1 - p^{4} T )^{7} \)
good2 \( 1 + T + 299 p T^{2} - 1295 p^{2} T^{3} + 14115 p^{3} T^{4} - 80599 p^{5} T^{5} + 668745 p^{7} T^{6} + 1990581 p^{9} T^{7} + 668745 p^{16} T^{8} - 80599 p^{23} T^{9} + 14115 p^{30} T^{10} - 1295 p^{38} T^{11} + 299 p^{46} T^{12} + p^{54} T^{13} + p^{63} T^{14} \)
3 \( 1 - 88 T + 32053 T^{2} - 169928 p T^{3} + 10118477 p^{4} T^{4} - 1162506328 p^{3} T^{5} + 2243718737 p^{8} T^{6} + 1135094102608 p^{5} T^{7} + 2243718737 p^{17} T^{8} - 1162506328 p^{21} T^{9} + 10118477 p^{31} T^{10} - 169928 p^{37} T^{11} + 32053 p^{45} T^{12} - 88 p^{54} T^{13} + p^{63} T^{14} \)
5 \( 1 - 1362 T + 7230567 T^{2} - 180823684 p^{2} T^{3} + 818396294953 p^{2} T^{4} - 125636115174 p^{5} T^{5} + 60356607903632927 p^{4} T^{6} + 3958396497804070312 p^{5} T^{7} + 60356607903632927 p^{13} T^{8} - 125636115174 p^{23} T^{9} + 818396294953 p^{29} T^{10} - 180823684 p^{38} T^{11} + 7230567 p^{45} T^{12} - 1362 p^{54} T^{13} + p^{63} T^{14} \)
7 \( 1 - 9388 T + 171799721 T^{2} - 1171723923712 T^{3} + 1894379515664715 p T^{4} - 1498272325406275700 p^{2} T^{5} + \)\(20\!\cdots\!95\)\( p^{3} T^{6} - \)\(13\!\cdots\!88\)\( p^{4} T^{7} + \)\(20\!\cdots\!95\)\( p^{12} T^{8} - 1498272325406275700 p^{20} T^{9} + 1894379515664715 p^{28} T^{10} - 1171723923712 p^{36} T^{11} + 171799721 p^{45} T^{12} - 9388 p^{54} T^{13} + p^{63} T^{14} \)
11 \( 1 - 135536 T + 18715983613 T^{2} - 1497302316100632 T^{3} + \)\(12\!\cdots\!25\)\( T^{4} - \)\(72\!\cdots\!92\)\( T^{5} + \)\(44\!\cdots\!37\)\( T^{6} - \)\(21\!\cdots\!72\)\( T^{7} + \)\(44\!\cdots\!37\)\( p^{9} T^{8} - \)\(72\!\cdots\!92\)\( p^{18} T^{9} + \)\(12\!\cdots\!25\)\( p^{27} T^{10} - 1497302316100632 p^{36} T^{11} + 18715983613 p^{45} T^{12} - 135536 p^{54} T^{13} + p^{63} T^{14} \)
13 \( 1 - 166122 T + 50414906991 T^{2} - 5203646785769876 T^{3} + \)\(98\!\cdots\!45\)\( T^{4} - \)\(71\!\cdots\!26\)\( T^{5} + \)\(11\!\cdots\!31\)\( T^{6} - \)\(73\!\cdots\!04\)\( T^{7} + \)\(11\!\cdots\!31\)\( p^{9} T^{8} - \)\(71\!\cdots\!26\)\( p^{18} T^{9} + \)\(98\!\cdots\!45\)\( p^{27} T^{10} - 5203646785769876 p^{36} T^{11} + 50414906991 p^{45} T^{12} - 166122 p^{54} T^{13} + p^{63} T^{14} \)
19 \( 1 - 777172 T + 374495139317 T^{2} + 68955098143291288 T^{3} - \)\(30\!\cdots\!75\)\( T^{4} - \)\(79\!\cdots\!32\)\( T^{5} + \)\(64\!\cdots\!53\)\( T^{6} - \)\(36\!\cdots\!88\)\( T^{7} + \)\(64\!\cdots\!53\)\( p^{9} T^{8} - \)\(79\!\cdots\!32\)\( p^{18} T^{9} - \)\(30\!\cdots\!75\)\( p^{27} T^{10} + 68955098143291288 p^{36} T^{11} + 374495139317 p^{45} T^{12} - 777172 p^{54} T^{13} + p^{63} T^{14} \)
23 \( 1 - 1357764 T + 6514524023049 T^{2} - 6721075762811064976 T^{3} + \)\(22\!\cdots\!21\)\( T^{4} - \)\(19\!\cdots\!72\)\( T^{5} + \)\(52\!\cdots\!17\)\( T^{6} - \)\(40\!\cdots\!28\)\( T^{7} + \)\(52\!\cdots\!17\)\( p^{9} T^{8} - \)\(19\!\cdots\!72\)\( p^{18} T^{9} + \)\(22\!\cdots\!21\)\( p^{27} T^{10} - 6721075762811064976 p^{36} T^{11} + 6514524023049 p^{45} T^{12} - 1357764 p^{54} T^{13} + p^{63} T^{14} \)
29 \( 1 - 967002 T + 75160403436591 T^{2} - 93226043731707880788 T^{3} + \)\(26\!\cdots\!41\)\( T^{4} - \)\(34\!\cdots\!58\)\( T^{5} + \)\(58\!\cdots\!31\)\( T^{6} - \)\(65\!\cdots\!44\)\( T^{7} + \)\(58\!\cdots\!31\)\( p^{9} T^{8} - \)\(34\!\cdots\!58\)\( p^{18} T^{9} + \)\(26\!\cdots\!41\)\( p^{27} T^{10} - 93226043731707880788 p^{36} T^{11} + 75160403436591 p^{45} T^{12} - 967002 p^{54} T^{13} + p^{63} T^{14} \)
31 \( 1 - 3546740 T + 97757254742513 T^{2} - \)\(29\!\cdots\!20\)\( T^{3} + \)\(46\!\cdots\!69\)\( T^{4} - \)\(14\!\cdots\!04\)\( T^{5} + \)\(16\!\cdots\!85\)\( T^{6} - \)\(49\!\cdots\!28\)\( p^{2} T^{7} + \)\(16\!\cdots\!85\)\( p^{9} T^{8} - \)\(14\!\cdots\!04\)\( p^{18} T^{9} + \)\(46\!\cdots\!69\)\( p^{27} T^{10} - \)\(29\!\cdots\!20\)\( p^{36} T^{11} + 97757254742513 p^{45} T^{12} - 3546740 p^{54} T^{13} + p^{63} T^{14} \)
37 \( 1 - 18296498 T + 664882288842215 T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!29\)\( T^{4} - \)\(31\!\cdots\!54\)\( T^{5} + \)\(45\!\cdots\!95\)\( T^{6} - \)\(52\!\cdots\!80\)\( T^{7} + \)\(45\!\cdots\!95\)\( p^{9} T^{8} - \)\(31\!\cdots\!54\)\( p^{18} T^{9} + \)\(22\!\cdots\!29\)\( p^{27} T^{10} - \)\(11\!\cdots\!00\)\( p^{36} T^{11} + 664882288842215 p^{45} T^{12} - 18296498 p^{54} T^{13} + p^{63} T^{14} \)
41 \( 1 - 10285686 T + 1382895753635475 T^{2} - \)\(42\!\cdots\!64\)\( T^{3} + \)\(77\!\cdots\!05\)\( T^{4} + \)\(34\!\cdots\!78\)\( T^{5} + \)\(26\!\cdots\!15\)\( T^{6} + \)\(24\!\cdots\!12\)\( T^{7} + \)\(26\!\cdots\!15\)\( p^{9} T^{8} + \)\(34\!\cdots\!78\)\( p^{18} T^{9} + \)\(77\!\cdots\!05\)\( p^{27} T^{10} - \)\(42\!\cdots\!64\)\( p^{36} T^{11} + 1382895753635475 p^{45} T^{12} - 10285686 p^{54} T^{13} + p^{63} T^{14} \)
43 \( 1 - 21913204 T + 2627801175592349 T^{2} - \)\(53\!\cdots\!76\)\( T^{3} + \)\(32\!\cdots\!13\)\( T^{4} - \)\(59\!\cdots\!16\)\( T^{5} + \)\(24\!\cdots\!41\)\( T^{6} - \)\(37\!\cdots\!20\)\( T^{7} + \)\(24\!\cdots\!41\)\( p^{9} T^{8} - \)\(59\!\cdots\!16\)\( p^{18} T^{9} + \)\(32\!\cdots\!13\)\( p^{27} T^{10} - \)\(53\!\cdots\!76\)\( p^{36} T^{11} + 2627801175592349 p^{45} T^{12} - 21913204 p^{54} T^{13} + p^{63} T^{14} \)
47 \( 1 - 56639800 T + 5464864808109577 T^{2} - \)\(17\!\cdots\!24\)\( T^{3} + \)\(10\!\cdots\!61\)\( T^{4} - \)\(20\!\cdots\!96\)\( T^{5} + \)\(11\!\cdots\!17\)\( T^{6} - \)\(17\!\cdots\!92\)\( T^{7} + \)\(11\!\cdots\!17\)\( p^{9} T^{8} - \)\(20\!\cdots\!96\)\( p^{18} T^{9} + \)\(10\!\cdots\!61\)\( p^{27} T^{10} - \)\(17\!\cdots\!24\)\( p^{36} T^{11} + 5464864808109577 p^{45} T^{12} - 56639800 p^{54} T^{13} + p^{63} T^{14} \)
53 \( 1 - 121813562 T + 15960381948040711 T^{2} - \)\(15\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!49\)\( T^{4} - \)\(10\!\cdots\!22\)\( T^{5} + \)\(72\!\cdots\!39\)\( T^{6} - \)\(43\!\cdots\!60\)\( T^{7} + \)\(72\!\cdots\!39\)\( p^{9} T^{8} - \)\(10\!\cdots\!22\)\( p^{18} T^{9} + \)\(14\!\cdots\!49\)\( p^{27} T^{10} - \)\(15\!\cdots\!60\)\( p^{36} T^{11} + 15960381948040711 p^{45} T^{12} - 121813562 p^{54} T^{13} + p^{63} T^{14} \)
59 \( 1 - 29222388 T + 38743784663931213 T^{2} - \)\(10\!\cdots\!32\)\( T^{3} + \)\(78\!\cdots\!81\)\( T^{4} - \)\(19\!\cdots\!68\)\( T^{5} + \)\(99\!\cdots\!09\)\( T^{6} - \)\(20\!\cdots\!44\)\( T^{7} + \)\(99\!\cdots\!09\)\( p^{9} T^{8} - \)\(19\!\cdots\!68\)\( p^{18} T^{9} + \)\(78\!\cdots\!81\)\( p^{27} T^{10} - \)\(10\!\cdots\!32\)\( p^{36} T^{11} + 38743784663931213 p^{45} T^{12} - 29222388 p^{54} T^{13} + p^{63} T^{14} \)
61 \( 1 + 49915846 T + 51036539781826415 T^{2} + \)\(40\!\cdots\!16\)\( T^{3} + \)\(12\!\cdots\!69\)\( T^{4} + \)\(11\!\cdots\!62\)\( T^{5} + \)\(21\!\cdots\!19\)\( T^{6} + \)\(17\!\cdots\!24\)\( T^{7} + \)\(21\!\cdots\!19\)\( p^{9} T^{8} + \)\(11\!\cdots\!62\)\( p^{18} T^{9} + \)\(12\!\cdots\!69\)\( p^{27} T^{10} + \)\(40\!\cdots\!16\)\( p^{36} T^{11} + 51036539781826415 p^{45} T^{12} + 49915846 p^{54} T^{13} + p^{63} T^{14} \)
67 \( 1 - 301863420 T + 174315165377438021 T^{2} - \)\(42\!\cdots\!72\)\( T^{3} + \)\(13\!\cdots\!45\)\( T^{4} - \)\(26\!\cdots\!88\)\( T^{5} + \)\(58\!\cdots\!49\)\( T^{6} - \)\(91\!\cdots\!80\)\( T^{7} + \)\(58\!\cdots\!49\)\( p^{9} T^{8} - \)\(26\!\cdots\!88\)\( p^{18} T^{9} + \)\(13\!\cdots\!45\)\( p^{27} T^{10} - \)\(42\!\cdots\!72\)\( p^{36} T^{11} + 174315165377438021 p^{45} T^{12} - 301863420 p^{54} T^{13} + p^{63} T^{14} \)
71 \( 1 - 652473940 T + 395785453180899961 T^{2} - \)\(16\!\cdots\!56\)\( T^{3} + \)\(59\!\cdots\!37\)\( T^{4} - \)\(17\!\cdots\!96\)\( T^{5} + \)\(46\!\cdots\!01\)\( T^{6} - \)\(10\!\cdots\!16\)\( T^{7} + \)\(46\!\cdots\!01\)\( p^{9} T^{8} - \)\(17\!\cdots\!96\)\( p^{18} T^{9} + \)\(59\!\cdots\!37\)\( p^{27} T^{10} - \)\(16\!\cdots\!56\)\( p^{36} T^{11} + 395785453180899961 p^{45} T^{12} - 652473940 p^{54} T^{13} + p^{63} T^{14} \)
73 \( 1 - 306656342 T + 368816681103768371 T^{2} - \)\(94\!\cdots\!16\)\( T^{3} + \)\(60\!\cdots\!53\)\( T^{4} - \)\(12\!\cdots\!82\)\( T^{5} + \)\(56\!\cdots\!31\)\( T^{6} - \)\(97\!\cdots\!68\)\( T^{7} + \)\(56\!\cdots\!31\)\( p^{9} T^{8} - \)\(12\!\cdots\!82\)\( p^{18} T^{9} + \)\(60\!\cdots\!53\)\( p^{27} T^{10} - \)\(94\!\cdots\!16\)\( p^{36} T^{11} + 368816681103768371 p^{45} T^{12} - 306656342 p^{54} T^{13} + p^{63} T^{14} \)
79 \( 1 - 959147884 T + 1105722859010207057 T^{2} - \)\(69\!\cdots\!28\)\( T^{3} + \)\(45\!\cdots\!45\)\( T^{4} - \)\(20\!\cdots\!44\)\( T^{5} + \)\(95\!\cdots\!77\)\( T^{6} - \)\(33\!\cdots\!68\)\( T^{7} + \)\(95\!\cdots\!77\)\( p^{9} T^{8} - \)\(20\!\cdots\!44\)\( p^{18} T^{9} + \)\(45\!\cdots\!45\)\( p^{27} T^{10} - \)\(69\!\cdots\!28\)\( p^{36} T^{11} + 1105722859010207057 p^{45} T^{12} - 959147884 p^{54} T^{13} + p^{63} T^{14} \)
83 \( 1 + 1512945268 T + 1750810807114580341 T^{2} + \)\(13\!\cdots\!36\)\( T^{3} + \)\(90\!\cdots\!41\)\( T^{4} + \)\(48\!\cdots\!56\)\( T^{5} + \)\(24\!\cdots\!65\)\( T^{6} + \)\(10\!\cdots\!48\)\( T^{7} + \)\(24\!\cdots\!65\)\( p^{9} T^{8} + \)\(48\!\cdots\!56\)\( p^{18} T^{9} + \)\(90\!\cdots\!41\)\( p^{27} T^{10} + \)\(13\!\cdots\!36\)\( p^{36} T^{11} + 1750810807114580341 p^{45} T^{12} + 1512945268 p^{54} T^{13} + p^{63} T^{14} \)
89 \( 1 + 1971327114 T + 3491107103618599779 T^{2} + \)\(40\!\cdots\!96\)\( T^{3} + \)\(42\!\cdots\!65\)\( T^{4} + \)\(34\!\cdots\!70\)\( T^{5} + \)\(25\!\cdots\!63\)\( T^{6} + \)\(16\!\cdots\!20\)\( T^{7} + \)\(25\!\cdots\!63\)\( p^{9} T^{8} + \)\(34\!\cdots\!70\)\( p^{18} T^{9} + \)\(42\!\cdots\!65\)\( p^{27} T^{10} + \)\(40\!\cdots\!96\)\( p^{36} T^{11} + 3491107103618599779 p^{45} T^{12} + 1971327114 p^{54} T^{13} + p^{63} T^{14} \)
97 \( 1 - 2006526254 T + 4610054277620485499 T^{2} - \)\(60\!\cdots\!92\)\( T^{3} + \)\(88\!\cdots\!93\)\( T^{4} - \)\(91\!\cdots\!22\)\( T^{5} + \)\(10\!\cdots\!79\)\( T^{6} - \)\(85\!\cdots\!76\)\( T^{7} + \)\(10\!\cdots\!79\)\( p^{9} T^{8} - \)\(91\!\cdots\!22\)\( p^{18} T^{9} + \)\(88\!\cdots\!93\)\( p^{27} T^{10} - \)\(60\!\cdots\!92\)\( p^{36} T^{11} + 4610054277620485499 p^{45} T^{12} - 2006526254 p^{54} T^{13} + p^{63} T^{14} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.099615789853963887761836891481, −7.70653694709031935913070747997, −7.70572490911238671485960578545, −7.21342054371703754541035211976, −6.74289187633109225193852640621, −6.71926124848699301595070906182, −6.19883438744100674546912363288, −5.78859085187022506234082272096, −5.71504062958116055510467538005, −5.64638718591517569820758078715, −5.41543927693687106766573064045, −4.82801047484950600224407818281, −4.55855265695563168355632802467, −4.05334358887555833762083259030, −3.98718451997191803367848058337, −3.79306451037682429053245026325, −3.24490865549226043494539667760, −3.19151948276565238362434478355, −2.48781855586923147381985174343, −2.05266453409891427308902586184, −1.64154761238917240061481378123, −1.50121256120600685429285899933, −0.915894036795500598442491894205, −0.849171790121337493242783066568, −0.62618691271467735594292864210, 0.62618691271467735594292864210, 0.849171790121337493242783066568, 0.915894036795500598442491894205, 1.50121256120600685429285899933, 1.64154761238917240061481378123, 2.05266453409891427308902586184, 2.48781855586923147381985174343, 3.19151948276565238362434478355, 3.24490865549226043494539667760, 3.79306451037682429053245026325, 3.98718451997191803367848058337, 4.05334358887555833762083259030, 4.55855265695563168355632802467, 4.82801047484950600224407818281, 5.41543927693687106766573064045, 5.64638718591517569820758078715, 5.71504062958116055510467538005, 5.78859085187022506234082272096, 6.19883438744100674546912363288, 6.71926124848699301595070906182, 6.74289187633109225193852640621, 7.21342054371703754541035211976, 7.70572490911238671485960578545, 7.70653694709031935913070747997, 8.099615789853963887761836891481

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

Plot not available for L-functions of degree greater than 10.