Properties

Label 16-14e8-1.1-c6e8-0-0
Degree $16$
Conductor $1475789056$
Sign $1$
Analytic cond. $11578.8$
Root an. cond. $1.79464$
Motivic weight $6$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 64·4-s − 336·5-s + 652·7-s − 1.08e3·9-s − 1.35e3·11-s + 1.02e3·16-s − 1.73e4·17-s − 3.20e4·19-s + 2.15e4·20-s − 4.12e3·23-s + 2.75e4·25-s − 4.17e4·28-s − 3.03e4·29-s − 3.10e3·31-s − 2.19e5·35-s + 6.91e4·36-s − 6.12e3·37-s − 2.97e5·43-s + 8.67e4·44-s + 3.62e5·45-s + 3.13e5·47-s + 2.28e5·49-s + 2.78e5·53-s + 4.55e5·55-s − 8.35e5·59-s − 9.95e5·61-s − 7.04e5·63-s + ⋯
L(s)  = 1  − 4-s − 2.68·5-s + 1.90·7-s − 1.48·9-s − 1.01·11-s + 1/4·16-s − 3.52·17-s − 4.66·19-s + 2.68·20-s − 0.339·23-s + 1.76·25-s − 1.90·28-s − 1.24·29-s − 0.104·31-s − 5.10·35-s + 1.48·36-s − 0.120·37-s − 3.74·43-s + 1.01·44-s + 3.98·45-s + 3.02·47-s + 1.94·49-s + 1.87·53-s + 2.73·55-s − 4.06·59-s − 4.38·61-s − 2.81·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(11578.8\)
Root analytic conductor: \(1.79464\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{14} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 7^{8} ,\ ( \ : [3]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0006378165735\)
\(L(\frac12)\) \(\approx\) \(0.0006378165735\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p^{5} T^{2} + p^{10} T^{4} )^{2} \)
7 \( 1 - 652 T + 28048 p T^{2} + 63236 p^{3} T^{3} - 1332974 p^{5} T^{4} + 63236 p^{9} T^{5} + 28048 p^{13} T^{6} - 652 p^{18} T^{7} + p^{24} T^{8} \)
good3 \( 1 + 40 p^{3} T^{2} + 16771 p^{2} T^{4} + 25844 p^{5} T^{5} + 2055304 p^{3} T^{6} + 224304052 p^{4} T^{7} + 2551219432 p^{4} T^{8} + 224304052 p^{10} T^{9} + 2055304 p^{15} T^{10} + 25844 p^{23} T^{11} + 16771 p^{26} T^{12} + 40 p^{39} T^{14} + p^{48} T^{16} \)
5 \( 1 + 336 T + 85366 T^{2} + 16038624 T^{3} + 2456362441 T^{4} + 65605622544 p T^{5} + 1589301396406 p^{2} T^{6} + 37401029748096 p^{3} T^{7} + 919144374631636 p^{4} T^{8} + 37401029748096 p^{9} T^{9} + 1589301396406 p^{14} T^{10} + 65605622544 p^{19} T^{11} + 2456362441 p^{24} T^{12} + 16038624 p^{30} T^{13} + 85366 p^{36} T^{14} + 336 p^{42} T^{15} + p^{48} T^{16} \)
11 \( 1 + 1356 T - 122600 p T^{2} - 5127563208 T^{3} - 3659896680029 T^{4} + 3327928404358428 T^{5} + 7354942466938909256 T^{6} + \)\(27\!\cdots\!04\)\( T^{7} - \)\(58\!\cdots\!00\)\( T^{8} + \)\(27\!\cdots\!04\)\( p^{6} T^{9} + 7354942466938909256 p^{12} T^{10} + 3327928404358428 p^{18} T^{11} - 3659896680029 p^{24} T^{12} - 5127563208 p^{30} T^{13} - 122600 p^{37} T^{14} + 1356 p^{42} T^{15} + p^{48} T^{16} \)
13 \( 1 - 26799680 T^{2} + 345775979967292 T^{4} - \)\(28\!\cdots\!12\)\( T^{6} + \)\(16\!\cdots\!94\)\( T^{8} - \)\(28\!\cdots\!12\)\( p^{12} T^{10} + 345775979967292 p^{24} T^{12} - 26799680 p^{36} T^{14} + p^{48} T^{16} \)
17 \( 1 + 17304 T + 206140798 T^{2} + 1839957265104 T^{3} + 14302857981101617 T^{4} + 99192647737413753264 T^{5} + \)\(61\!\cdots\!82\)\( T^{6} + \)\(34\!\cdots\!60\)\( T^{7} + \)\(17\!\cdots\!64\)\( T^{8} + \)\(34\!\cdots\!60\)\( p^{6} T^{9} + \)\(61\!\cdots\!82\)\( p^{12} T^{10} + 99192647737413753264 p^{18} T^{11} + 14302857981101617 p^{24} T^{12} + 1839957265104 p^{30} T^{13} + 206140798 p^{36} T^{14} + 17304 p^{42} T^{15} + p^{48} T^{16} \)
19 \( 1 + 32004 T + 616491088 T^{2} + 8803417601664 T^{3} + 103939442799115627 T^{4} + \)\(10\!\cdots\!20\)\( T^{5} + \)\(95\!\cdots\!20\)\( T^{6} + \)\(77\!\cdots\!72\)\( T^{7} + \)\(55\!\cdots\!72\)\( T^{8} + \)\(77\!\cdots\!72\)\( p^{6} T^{9} + \)\(95\!\cdots\!20\)\( p^{12} T^{10} + \)\(10\!\cdots\!20\)\( p^{18} T^{11} + 103939442799115627 p^{24} T^{12} + 8803417601664 p^{30} T^{13} + 616491088 p^{36} T^{14} + 32004 p^{42} T^{15} + p^{48} T^{16} \)
23 \( 1 + 4128 T - 386373856 T^{2} + 223897068600 T^{3} + 83079437636729539 T^{4} - \)\(23\!\cdots\!96\)\( T^{5} - \)\(13\!\cdots\!80\)\( T^{6} + \)\(19\!\cdots\!12\)\( T^{7} + \)\(18\!\cdots\!12\)\( T^{8} + \)\(19\!\cdots\!12\)\( p^{6} T^{9} - \)\(13\!\cdots\!80\)\( p^{12} T^{10} - \)\(23\!\cdots\!96\)\( p^{18} T^{11} + 83079437636729539 p^{24} T^{12} + 223897068600 p^{30} T^{13} - 386373856 p^{36} T^{14} + 4128 p^{42} T^{15} + p^{48} T^{16} \)
29 \( ( 1 + 15156 T + 1003136368 T^{2} - 3639995041860 T^{3} + 336439222276101918 T^{4} - 3639995041860 p^{6} T^{5} + 1003136368 p^{12} T^{6} + 15156 p^{18} T^{7} + p^{24} T^{8} )^{2} \)
31 \( 1 + 3108 T + 791635696 T^{2} + 79045042944 p T^{3} + 26109511358578261 p T^{4} - \)\(14\!\cdots\!36\)\( p T^{5} - \)\(69\!\cdots\!80\)\( T^{6} - \)\(10\!\cdots\!12\)\( T^{7} - \)\(47\!\cdots\!88\)\( T^{8} - \)\(10\!\cdots\!12\)\( p^{6} T^{9} - \)\(69\!\cdots\!80\)\( p^{12} T^{10} - \)\(14\!\cdots\!36\)\( p^{19} T^{11} + 26109511358578261 p^{25} T^{12} + 79045042944 p^{31} T^{13} + 791635696 p^{36} T^{14} + 3108 p^{42} T^{15} + p^{48} T^{16} \)
37 \( 1 + 6124 T - 2410851594 T^{2} + 105511123683224 T^{3} + 5352778640202062561 T^{4} - \)\(23\!\cdots\!04\)\( T^{5} + \)\(32\!\cdots\!86\)\( T^{6} + \)\(72\!\cdots\!72\)\( T^{7} - \)\(85\!\cdots\!20\)\( T^{8} + \)\(72\!\cdots\!72\)\( p^{6} T^{9} + \)\(32\!\cdots\!86\)\( p^{12} T^{10} - \)\(23\!\cdots\!04\)\( p^{18} T^{11} + 5352778640202062561 p^{24} T^{12} + 105511123683224 p^{30} T^{13} - 2410851594 p^{36} T^{14} + 6124 p^{42} T^{15} + p^{48} T^{16} \)
41 \( 1 - 12858425408 T^{2} + \)\(12\!\cdots\!52\)\( T^{4} - \)\(77\!\cdots\!04\)\( T^{6} + \)\(42\!\cdots\!02\)\( T^{8} - \)\(77\!\cdots\!04\)\( p^{12} T^{10} + \)\(12\!\cdots\!52\)\( p^{24} T^{12} - 12858425408 p^{36} T^{14} + p^{48} T^{16} \)
43 \( ( 1 + 148688 T + 16174962100 T^{2} + 944168320869968 T^{3} + 67784308274406325750 T^{4} + 944168320869968 p^{6} T^{5} + 16174962100 p^{12} T^{6} + 148688 p^{18} T^{7} + p^{24} T^{8} )^{2} \)
47 \( 1 - 313908 T + 61402539880 T^{2} - 8964101996547936 T^{3} + 20135717581678287629 p T^{4} - \)\(68\!\cdots\!56\)\( T^{5} + \)\(17\!\cdots\!12\)\( T^{6} + \)\(38\!\cdots\!92\)\( T^{7} - \)\(62\!\cdots\!88\)\( T^{8} + \)\(38\!\cdots\!92\)\( p^{6} T^{9} + \)\(17\!\cdots\!12\)\( p^{12} T^{10} - \)\(68\!\cdots\!56\)\( p^{18} T^{11} + 20135717581678287629 p^{25} T^{12} - 8964101996547936 p^{30} T^{13} + 61402539880 p^{36} T^{14} - 313908 p^{42} T^{15} + p^{48} T^{16} \)
53 \( 1 - 278484 T - 12243422506 T^{2} + 6978883270647576 T^{3} + \)\(54\!\cdots\!05\)\( T^{4} - \)\(81\!\cdots\!24\)\( T^{5} - \)\(30\!\cdots\!46\)\( T^{6} - \)\(74\!\cdots\!84\)\( T^{7} + \)\(12\!\cdots\!48\)\( T^{8} - \)\(74\!\cdots\!84\)\( p^{6} T^{9} - \)\(30\!\cdots\!46\)\( p^{12} T^{10} - \)\(81\!\cdots\!24\)\( p^{18} T^{11} + \)\(54\!\cdots\!05\)\( p^{24} T^{12} + 6978883270647576 p^{30} T^{13} - 12243422506 p^{36} T^{14} - 278484 p^{42} T^{15} + p^{48} T^{16} \)
59 \( 1 + 835464 T + 435025459648 T^{2} + 169063460080564224 T^{3} + \)\(54\!\cdots\!31\)\( T^{4} + \)\(26\!\cdots\!00\)\( p T^{5} + \)\(39\!\cdots\!40\)\( T^{6} + \)\(15\!\cdots\!40\)\( p T^{7} + \)\(20\!\cdots\!20\)\( T^{8} + \)\(15\!\cdots\!40\)\( p^{7} T^{9} + \)\(39\!\cdots\!40\)\( p^{12} T^{10} + \)\(26\!\cdots\!00\)\( p^{19} T^{11} + \)\(54\!\cdots\!31\)\( p^{24} T^{12} + 169063460080564224 p^{30} T^{13} + 435025459648 p^{36} T^{14} + 835464 p^{42} T^{15} + p^{48} T^{16} \)
61 \( 1 + 995316 T + 651608313382 T^{2} + 319884941108213880 T^{3} + \)\(13\!\cdots\!29\)\( T^{4} + \)\(45\!\cdots\!36\)\( T^{5} + \)\(13\!\cdots\!74\)\( T^{6} + \)\(37\!\cdots\!40\)\( T^{7} + \)\(90\!\cdots\!88\)\( T^{8} + \)\(37\!\cdots\!40\)\( p^{6} T^{9} + \)\(13\!\cdots\!74\)\( p^{12} T^{10} + \)\(45\!\cdots\!36\)\( p^{18} T^{11} + \)\(13\!\cdots\!29\)\( p^{24} T^{12} + 319884941108213880 p^{30} T^{13} + 651608313382 p^{36} T^{14} + 995316 p^{42} T^{15} + p^{48} T^{16} \)
67 \( 1 - 648808 T + 6714089688 T^{2} + 44922752496936664 T^{3} + \)\(11\!\cdots\!35\)\( T^{4} - \)\(40\!\cdots\!72\)\( T^{5} - \)\(17\!\cdots\!32\)\( T^{6} - \)\(18\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!80\)\( T^{8} - \)\(18\!\cdots\!00\)\( p^{6} T^{9} - \)\(17\!\cdots\!32\)\( p^{12} T^{10} - \)\(40\!\cdots\!72\)\( p^{18} T^{11} + \)\(11\!\cdots\!35\)\( p^{24} T^{12} + 44922752496936664 p^{30} T^{13} + 6714089688 p^{36} T^{14} - 648808 p^{42} T^{15} + p^{48} T^{16} \)
71 \( ( 1 - 95064 T + 220277516404 T^{2} + 16633830011187384 T^{3} + \)\(29\!\cdots\!42\)\( T^{4} + 16633830011187384 p^{6} T^{5} + 220277516404 p^{12} T^{6} - 95064 p^{18} T^{7} + p^{24} T^{8} )^{2} \)
73 \( 1 + 1617084 T + 1664708471134 T^{2} + 1282436417049503688 T^{3} + \)\(82\!\cdots\!85\)\( T^{4} + \)\(46\!\cdots\!92\)\( T^{5} + \)\(23\!\cdots\!42\)\( T^{6} + \)\(10\!\cdots\!80\)\( T^{7} + \)\(42\!\cdots\!88\)\( T^{8} + \)\(10\!\cdots\!80\)\( p^{6} T^{9} + \)\(23\!\cdots\!42\)\( p^{12} T^{10} + \)\(46\!\cdots\!92\)\( p^{18} T^{11} + \)\(82\!\cdots\!85\)\( p^{24} T^{12} + 1282436417049503688 p^{30} T^{13} + 1664708471134 p^{36} T^{14} + 1617084 p^{42} T^{15} + p^{48} T^{16} \)
79 \( 1 - 70096 T - 570937949040 T^{2} - 91349453058129800 T^{3} + \)\(16\!\cdots\!43\)\( T^{4} + \)\(44\!\cdots\!64\)\( T^{5} - \)\(29\!\cdots\!92\)\( T^{6} - \)\(61\!\cdots\!56\)\( T^{7} + \)\(61\!\cdots\!96\)\( T^{8} - \)\(61\!\cdots\!56\)\( p^{6} T^{9} - \)\(29\!\cdots\!92\)\( p^{12} T^{10} + \)\(44\!\cdots\!64\)\( p^{18} T^{11} + \)\(16\!\cdots\!43\)\( p^{24} T^{12} - 91349453058129800 p^{30} T^{13} - 570937949040 p^{36} T^{14} - 70096 p^{42} T^{15} + p^{48} T^{16} \)
83 \( 1 - 1511767571240 T^{2} + \)\(11\!\cdots\!56\)\( T^{4} - \)\(60\!\cdots\!20\)\( T^{6} + \)\(23\!\cdots\!90\)\( T^{8} - \)\(60\!\cdots\!20\)\( p^{12} T^{10} + \)\(11\!\cdots\!56\)\( p^{24} T^{12} - 1511767571240 p^{36} T^{14} + p^{48} T^{16} \)
89 \( 1 - 739116 T + 1962144789862 T^{2} - 1315661442189804360 T^{3} + \)\(21\!\cdots\!93\)\( T^{4} - \)\(11\!\cdots\!40\)\( T^{5} + \)\(15\!\cdots\!30\)\( T^{6} - \)\(72\!\cdots\!64\)\( T^{7} + \)\(84\!\cdots\!64\)\( T^{8} - \)\(72\!\cdots\!64\)\( p^{6} T^{9} + \)\(15\!\cdots\!30\)\( p^{12} T^{10} - \)\(11\!\cdots\!40\)\( p^{18} T^{11} + \)\(21\!\cdots\!93\)\( p^{24} T^{12} - 1315661442189804360 p^{30} T^{13} + 1962144789862 p^{36} T^{14} - 739116 p^{42} T^{15} + p^{48} T^{16} \)
97 \( 1 - 2264519987840 T^{2} + \)\(25\!\cdots\!68\)\( T^{4} - \)\(26\!\cdots\!00\)\( T^{6} + \)\(24\!\cdots\!18\)\( T^{8} - \)\(26\!\cdots\!00\)\( p^{12} T^{10} + \)\(25\!\cdots\!68\)\( p^{24} T^{12} - 2264519987840 p^{36} T^{14} + p^{48} T^{16} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.479157816787994468742971312205, −8.440769700455981893589758909666, −8.052674674244446380541883275511, −7.997149627071705271107016265200, −7.85178318415484875848240046217, −7.55035587508063884003396003026, −7.23144016802011448639306366245, −6.79581831359207248185149357452, −6.69292121467620542601053896388, −6.40064284635542909788011158859, −5.88485815721681064297071919201, −5.66345889003333732125204759490, −5.58322959317473051261828330436, −4.57558884213832782254869380218, −4.56593239874600100182625789833, −4.54020170766729678799218717100, −4.30491369981107068926182868784, −4.26718284075360051888313407319, −3.37708825232127412988317115454, −3.35840157923930725669284949627, −2.25381316288422032692063158545, −2.13848555479060199688344947858, −1.89994797709191889328601166265, −0.07782869679193240682227583333, −0.06301963162225269464454508655, 0.06301963162225269464454508655, 0.07782869679193240682227583333, 1.89994797709191889328601166265, 2.13848555479060199688344947858, 2.25381316288422032692063158545, 3.35840157923930725669284949627, 3.37708825232127412988317115454, 4.26718284075360051888313407319, 4.30491369981107068926182868784, 4.54020170766729678799218717100, 4.56593239874600100182625789833, 4.57558884213832782254869380218, 5.58322959317473051261828330436, 5.66345889003333732125204759490, 5.88485815721681064297071919201, 6.40064284635542909788011158859, 6.69292121467620542601053896388, 6.79581831359207248185149357452, 7.23144016802011448639306366245, 7.55035587508063884003396003026, 7.85178318415484875848240046217, 7.997149627071705271107016265200, 8.052674674244446380541883275511, 8.440769700455981893589758909666, 8.479157816787994468742971312205

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

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