Properties

Label 14-354e7-1.1-c7e7-0-0
Degree $14$
Conductor $6.967\times 10^{17}$
Sign $-1$
Analytic cond. $2.02234\times 10^{14}$
Root an. cond. $10.5159$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $7$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 56·2-s + 189·3-s + 1.79e3·4-s − 158·5-s − 1.05e4·6-s − 581·7-s − 4.30e4·8-s + 2.04e4·9-s + 8.84e3·10-s − 2.20e3·11-s + 3.38e5·12-s − 8.42e3·13-s + 3.25e4·14-s − 2.98e4·15-s + 8.60e5·16-s − 2.42e3·17-s − 1.14e6·18-s − 3.70e4·19-s − 2.83e5·20-s − 1.09e5·21-s + 1.23e5·22-s + 9.93e4·23-s − 8.12e6·24-s − 2.10e5·25-s + 4.71e5·26-s + 1.65e6·27-s − 1.04e6·28-s + ⋯
L(s)  = 1  − 4.94·2-s + 4.04·3-s + 14·4-s − 0.565·5-s − 20.0·6-s − 0.640·7-s − 29.6·8-s + 28/3·9-s + 2.79·10-s − 0.498·11-s + 56.5·12-s − 1.06·13-s + 3.16·14-s − 2.28·15-s + 52.5·16-s − 0.119·17-s − 46.1·18-s − 1.24·19-s − 7.91·20-s − 2.58·21-s + 2.46·22-s + 1.70·23-s − 120.·24-s − 2.69·25-s + 5.26·26-s + 16.1·27-s − 8.96·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{7} \cdot 59^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{7} \cdot 59^{7}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(2^{7} \cdot 3^{7} \cdot 59^{7}\)
Sign: $-1$
Analytic conductor: \(2.02234\times 10^{14}\)
Root analytic conductor: \(10.5159\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(7\)
Selberg data: \((14,\ 2^{7} \cdot 3^{7} \cdot 59^{7} ,\ ( \ : [7/2]^{7} ),\ -1 )\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p^{3} T )^{7} \)
3 \( ( 1 - p^{3} T )^{7} \)
59 \( ( 1 + p^{3} T )^{7} \)
good5 \( 1 + 158 T + 235239 T^{2} + 9358534 p T^{3} + 6319919628 p T^{4} + 283722643546 p^{2} T^{5} + 24186077427556 p^{3} T^{6} + 217293569331748 p^{5} T^{7} + 24186077427556 p^{10} T^{8} + 283722643546 p^{16} T^{9} + 6319919628 p^{22} T^{10} + 9358534 p^{29} T^{11} + 235239 p^{35} T^{12} + 158 p^{42} T^{13} + p^{49} T^{14} \)
7 \( 1 + 83 p T + 4974358 T^{2} + 355859233 p T^{3} + 11164338373569 T^{4} + 674847316192888 p T^{5} + 14611666172746373240 T^{6} + \)\(10\!\cdots\!28\)\( p^{2} T^{7} + 14611666172746373240 p^{7} T^{8} + 674847316192888 p^{15} T^{9} + 11164338373569 p^{21} T^{10} + 355859233 p^{29} T^{11} + 4974358 p^{35} T^{12} + 83 p^{43} T^{13} + p^{49} T^{14} \)
11 \( 1 + 2201 T + 80197215 T^{2} + 264804905612 T^{3} + 3392032260894312 T^{4} + 11071204651313191412 T^{5} + \)\(99\!\cdots\!38\)\( T^{6} + \)\(26\!\cdots\!62\)\( T^{7} + \)\(99\!\cdots\!38\)\( p^{7} T^{8} + 11071204651313191412 p^{14} T^{9} + 3392032260894312 p^{21} T^{10} + 264804905612 p^{28} T^{11} + 80197215 p^{35} T^{12} + 2201 p^{42} T^{13} + p^{49} T^{14} \)
13 \( 1 + 8421 T + 368372453 T^{2} + 2776782267060 T^{3} + 61879354118304136 T^{4} + \)\(40\!\cdots\!96\)\( T^{5} + \)\(61\!\cdots\!06\)\( T^{6} + \)\(33\!\cdots\!10\)\( T^{7} + \)\(61\!\cdots\!06\)\( p^{7} T^{8} + \)\(40\!\cdots\!96\)\( p^{14} T^{9} + 61879354118304136 p^{21} T^{10} + 2776782267060 p^{28} T^{11} + 368372453 p^{35} T^{12} + 8421 p^{42} T^{13} + p^{49} T^{14} \)
17 \( 1 + 2425 T + 1524038434 T^{2} + 10834173154613 T^{3} + 1218275774478892739 T^{4} + \)\(96\!\cdots\!84\)\( T^{5} + \)\(70\!\cdots\!86\)\( T^{6} + \)\(46\!\cdots\!24\)\( T^{7} + \)\(70\!\cdots\!86\)\( p^{7} T^{8} + \)\(96\!\cdots\!84\)\( p^{14} T^{9} + 1218275774478892739 p^{21} T^{10} + 10834173154613 p^{28} T^{11} + 1524038434 p^{35} T^{12} + 2425 p^{42} T^{13} + p^{49} T^{14} \)
19 \( 1 + 37084 T + 4609354014 T^{2} + 143147997774952 T^{3} + 10056384199915257849 T^{4} + \)\(26\!\cdots\!40\)\( T^{5} + \)\(13\!\cdots\!56\)\( T^{6} + \)\(29\!\cdots\!84\)\( T^{7} + \)\(13\!\cdots\!56\)\( p^{7} T^{8} + \)\(26\!\cdots\!40\)\( p^{14} T^{9} + 10056384199915257849 p^{21} T^{10} + 143147997774952 p^{28} T^{11} + 4609354014 p^{35} T^{12} + 37084 p^{42} T^{13} + p^{49} T^{14} \)
23 \( 1 - 99364 T + 16736276482 T^{2} - 1361780617703334 T^{3} + \)\(13\!\cdots\!91\)\( T^{4} - \)\(90\!\cdots\!30\)\( T^{5} + \)\(65\!\cdots\!22\)\( T^{6} - \)\(37\!\cdots\!52\)\( T^{7} + \)\(65\!\cdots\!22\)\( p^{7} T^{8} - \)\(90\!\cdots\!30\)\( p^{14} T^{9} + \)\(13\!\cdots\!91\)\( p^{21} T^{10} - 1361780617703334 p^{28} T^{11} + 16736276482 p^{35} T^{12} - 99364 p^{42} T^{13} + p^{49} T^{14} \)
29 \( 1 - 2498 T + 30587733462 T^{2} + 2183125876353516 T^{3} + \)\(11\!\cdots\!51\)\( T^{4} + \)\(35\!\cdots\!52\)\( T^{5} + \)\(23\!\cdots\!66\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!66\)\( p^{7} T^{8} + \)\(35\!\cdots\!52\)\( p^{14} T^{9} + \)\(11\!\cdots\!51\)\( p^{21} T^{10} + 2183125876353516 p^{28} T^{11} + 30587733462 p^{35} T^{12} - 2498 p^{42} T^{13} + p^{49} T^{14} \)
31 \( 1 + 57962 T + 105024079088 T^{2} + 7756782302404818 T^{3} + \)\(56\!\cdots\!47\)\( T^{4} + \)\(38\!\cdots\!94\)\( T^{5} + \)\(21\!\cdots\!72\)\( T^{6} + \)\(12\!\cdots\!16\)\( T^{7} + \)\(21\!\cdots\!72\)\( p^{7} T^{8} + \)\(38\!\cdots\!94\)\( p^{14} T^{9} + \)\(56\!\cdots\!47\)\( p^{21} T^{10} + 7756782302404818 p^{28} T^{11} + 105024079088 p^{35} T^{12} + 57962 p^{42} T^{13} + p^{49} T^{14} \)
37 \( 1 + 6497 T + 349667776944 T^{2} - 31178823870538053 T^{3} + \)\(58\!\cdots\!83\)\( T^{4} - \)\(92\!\cdots\!60\)\( T^{5} + \)\(69\!\cdots\!08\)\( T^{6} - \)\(11\!\cdots\!64\)\( T^{7} + \)\(69\!\cdots\!08\)\( p^{7} T^{8} - \)\(92\!\cdots\!60\)\( p^{14} T^{9} + \)\(58\!\cdots\!83\)\( p^{21} T^{10} - 31178823870538053 p^{28} T^{11} + 349667776944 p^{35} T^{12} + 6497 p^{42} T^{13} + p^{49} T^{14} \)
41 \( 1 + 319165 T + 640384920462 T^{2} + 178200859542970887 T^{3} + \)\(20\!\cdots\!05\)\( T^{4} + \)\(51\!\cdots\!58\)\( T^{5} + \)\(45\!\cdots\!12\)\( T^{6} + \)\(10\!\cdots\!40\)\( T^{7} + \)\(45\!\cdots\!12\)\( p^{7} T^{8} + \)\(51\!\cdots\!58\)\( p^{14} T^{9} + \)\(20\!\cdots\!05\)\( p^{21} T^{10} + 178200859542970887 p^{28} T^{11} + 640384920462 p^{35} T^{12} + 319165 p^{42} T^{13} + p^{49} T^{14} \)
43 \( 1 + 633743 T + 1600036161871 T^{2} + 781429665670081836 T^{3} + \)\(11\!\cdots\!44\)\( T^{4} + \)\(45\!\cdots\!28\)\( T^{5} + \)\(48\!\cdots\!34\)\( T^{6} + \)\(15\!\cdots\!98\)\( T^{7} + \)\(48\!\cdots\!34\)\( p^{7} T^{8} + \)\(45\!\cdots\!28\)\( p^{14} T^{9} + \)\(11\!\cdots\!44\)\( p^{21} T^{10} + 781429665670081836 p^{28} T^{11} + 1600036161871 p^{35} T^{12} + 633743 p^{42} T^{13} + p^{49} T^{14} \)
47 \( 1 + 1626560 T + 3423582356188 T^{2} + 3890465167890791954 T^{3} + \)\(48\!\cdots\!89\)\( T^{4} + \)\(42\!\cdots\!24\)\( T^{5} + \)\(38\!\cdots\!74\)\( T^{6} + \)\(27\!\cdots\!64\)\( T^{7} + \)\(38\!\cdots\!74\)\( p^{7} T^{8} + \)\(42\!\cdots\!24\)\( p^{14} T^{9} + \)\(48\!\cdots\!89\)\( p^{21} T^{10} + 3890465167890791954 p^{28} T^{11} + 3423582356188 p^{35} T^{12} + 1626560 p^{42} T^{13} + p^{49} T^{14} \)
53 \( 1 + 1215602 T + 4659661750195 T^{2} + 2948266795517225518 T^{3} + \)\(75\!\cdots\!28\)\( T^{4} + \)\(88\!\cdots\!94\)\( T^{5} + \)\(57\!\cdots\!88\)\( T^{6} - \)\(40\!\cdots\!16\)\( T^{7} + \)\(57\!\cdots\!88\)\( p^{7} T^{8} + \)\(88\!\cdots\!94\)\( p^{14} T^{9} + \)\(75\!\cdots\!28\)\( p^{21} T^{10} + 2948266795517225518 p^{28} T^{11} + 4659661750195 p^{35} T^{12} + 1215602 p^{42} T^{13} + p^{49} T^{14} \)
61 \( 1 + 3180086 T + 20783843786424 T^{2} + 54703227712575894954 T^{3} + \)\(19\!\cdots\!19\)\( T^{4} + \)\(40\!\cdots\!28\)\( T^{5} + \)\(99\!\cdots\!48\)\( T^{6} + \)\(16\!\cdots\!80\)\( T^{7} + \)\(99\!\cdots\!48\)\( p^{7} T^{8} + \)\(40\!\cdots\!28\)\( p^{14} T^{9} + \)\(19\!\cdots\!19\)\( p^{21} T^{10} + 54703227712575894954 p^{28} T^{11} + 20783843786424 p^{35} T^{12} + 3180086 p^{42} T^{13} + p^{49} T^{14} \)
67 \( 1 + 5349632 T + 17741220232675 T^{2} + 40833406529352239828 T^{3} + \)\(83\!\cdots\!00\)\( T^{4} + \)\(25\!\cdots\!96\)\( T^{5} + \)\(89\!\cdots\!78\)\( T^{6} + \)\(26\!\cdots\!44\)\( T^{7} + \)\(89\!\cdots\!78\)\( p^{7} T^{8} + \)\(25\!\cdots\!96\)\( p^{14} T^{9} + \)\(83\!\cdots\!00\)\( p^{21} T^{10} + 40833406529352239828 p^{28} T^{11} + 17741220232675 p^{35} T^{12} + 5349632 p^{42} T^{13} + p^{49} T^{14} \)
71 \( 1 - 1752423 T + 9694865562131 T^{2} - 21381318343244992116 T^{3} + \)\(16\!\cdots\!52\)\( T^{4} - \)\(44\!\cdots\!12\)\( T^{5} + \)\(19\!\cdots\!98\)\( T^{6} - \)\(25\!\cdots\!58\)\( T^{7} + \)\(19\!\cdots\!98\)\( p^{7} T^{8} - \)\(44\!\cdots\!12\)\( p^{14} T^{9} + \)\(16\!\cdots\!52\)\( p^{21} T^{10} - 21381318343244992116 p^{28} T^{11} + 9694865562131 p^{35} T^{12} - 1752423 p^{42} T^{13} + p^{49} T^{14} \)
73 \( 1 + 1843424 T + 44028150803908 T^{2} + 72977107717541007504 T^{3} + \)\(10\!\cdots\!39\)\( T^{4} + \)\(16\!\cdots\!68\)\( T^{5} + \)\(17\!\cdots\!72\)\( T^{6} + \)\(21\!\cdots\!52\)\( T^{7} + \)\(17\!\cdots\!72\)\( p^{7} T^{8} + \)\(16\!\cdots\!68\)\( p^{14} T^{9} + \)\(10\!\cdots\!39\)\( p^{21} T^{10} + 72977107717541007504 p^{28} T^{11} + 44028150803908 p^{35} T^{12} + 1843424 p^{42} T^{13} + p^{49} T^{14} \)
79 \( 1 + 4769243 T + 88137963958767 T^{2} + \)\(42\!\cdots\!00\)\( T^{3} + \)\(38\!\cdots\!92\)\( T^{4} + \)\(17\!\cdots\!92\)\( T^{5} + \)\(10\!\cdots\!50\)\( T^{6} + \)\(42\!\cdots\!34\)\( T^{7} + \)\(10\!\cdots\!50\)\( p^{7} T^{8} + \)\(17\!\cdots\!92\)\( p^{14} T^{9} + \)\(38\!\cdots\!92\)\( p^{21} T^{10} + \)\(42\!\cdots\!00\)\( p^{28} T^{11} + 88137963958767 p^{35} T^{12} + 4769243 p^{42} T^{13} + p^{49} T^{14} \)
83 \( 1 - 5154441 T + 75265149017952 T^{2} - \)\(33\!\cdots\!31\)\( T^{3} + \)\(39\!\cdots\!67\)\( T^{4} - \)\(18\!\cdots\!06\)\( T^{5} + \)\(14\!\cdots\!88\)\( T^{6} - \)\(54\!\cdots\!40\)\( T^{7} + \)\(14\!\cdots\!88\)\( p^{7} T^{8} - \)\(18\!\cdots\!06\)\( p^{14} T^{9} + \)\(39\!\cdots\!67\)\( p^{21} T^{10} - \)\(33\!\cdots\!31\)\( p^{28} T^{11} + 75265149017952 p^{35} T^{12} - 5154441 p^{42} T^{13} + p^{49} T^{14} \)
89 \( 1 - 20086462 T + 351645976144580 T^{2} - \)\(42\!\cdots\!64\)\( T^{3} + \)\(44\!\cdots\!61\)\( T^{4} - \)\(39\!\cdots\!10\)\( T^{5} + \)\(31\!\cdots\!14\)\( T^{6} - \)\(21\!\cdots\!68\)\( T^{7} + \)\(31\!\cdots\!14\)\( p^{7} T^{8} - \)\(39\!\cdots\!10\)\( p^{14} T^{9} + \)\(44\!\cdots\!61\)\( p^{21} T^{10} - \)\(42\!\cdots\!64\)\( p^{28} T^{11} + 351645976144580 p^{35} T^{12} - 20086462 p^{42} T^{13} + p^{49} T^{14} \)
97 \( 1 - 6244248 T + 289051776707353 T^{2} - \)\(11\!\cdots\!44\)\( T^{3} + \)\(37\!\cdots\!88\)\( T^{4} - \)\(50\!\cdots\!12\)\( T^{5} + \)\(32\!\cdots\!94\)\( T^{6} + \)\(16\!\cdots\!28\)\( T^{7} + \)\(32\!\cdots\!94\)\( p^{7} T^{8} - \)\(50\!\cdots\!12\)\( p^{14} T^{9} + \)\(37\!\cdots\!88\)\( p^{21} T^{10} - \)\(11\!\cdots\!44\)\( p^{28} T^{11} + 289051776707353 p^{35} T^{12} - 6244248 p^{42} T^{13} + p^{49} T^{14} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.93102611442050756353652906056, −4.68509259028669757062422942284, −4.59958130364801107305080550014, −4.53179259142080000717820134205, −3.73188141498250060381067070899, −3.64180142749328335050264357604, −3.58027609163301513316299285228, −3.48494799077439790119742572845, −3.45059688085521716860597355868, −3.42852830524443739199824395804, −3.36044669428276146621764742734, −2.75496191922328816353634627861, −2.62231341084741396951512405888, −2.50085066162538318008757041273, −2.39800009669369716647050393469, −2.38951863956471379123836920100, −2.26193663834795188260337642259, −2.19041468972857325583916451421, −1.60605813361317420938808200797, −1.57520673320971122526197718853, −1.42977823233667579174145630406, −1.30394201877407684963316962333, −1.24856367866033056121944560423, −1.22368504628631851236296697738, −1.00506809067704478133099818718, 0, 0, 0, 0, 0, 0, 0, 1.00506809067704478133099818718, 1.22368504628631851236296697738, 1.24856367866033056121944560423, 1.30394201877407684963316962333, 1.42977823233667579174145630406, 1.57520673320971122526197718853, 1.60605813361317420938808200797, 2.19041468972857325583916451421, 2.26193663834795188260337642259, 2.38951863956471379123836920100, 2.39800009669369716647050393469, 2.50085066162538318008757041273, 2.62231341084741396951512405888, 2.75496191922328816353634627861, 3.36044669428276146621764742734, 3.42852830524443739199824395804, 3.45059688085521716860597355868, 3.48494799077439790119742572845, 3.58027609163301513316299285228, 3.64180142749328335050264357604, 3.73188141498250060381067070899, 4.53179259142080000717820134205, 4.59958130364801107305080550014, 4.68509259028669757062422942284, 4.93102611442050756353652906056

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

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