Dirichlet series
| L(s) = 1 | − 2·2-s − 12·4-s − 35·5-s − 54·7-s + 24·8-s − 55·9-s + 70·10-s + 4·11-s + 108·14-s + 55·16-s + 50·17-s + 110·18-s − 312·19-s + 420·20-s − 8·22-s + 266·23-s + 700·25-s − 8·27-s + 648·28-s + 294·29-s + 270·31-s − 244·32-s − 100·34-s + 1.89e3·35-s + 660·36-s − 470·37-s + 624·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 3/2·4-s − 3.13·5-s − 2.91·7-s + 1.06·8-s − 2.03·9-s + 2.21·10-s + 0.109·11-s + 2.06·14-s + 0.859·16-s + 0.713·17-s + 1.44·18-s − 3.76·19-s + 4.69·20-s − 0.0775·22-s + 2.41·23-s + 28/5·25-s − 0.0570·27-s + 4.37·28-s + 1.88·29-s + 1.56·31-s − 1.34·32-s − 0.504·34-s + 9.12·35-s + 3.05·36-s − 2.08·37-s + 2.66·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{7} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{7} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(5^{7} \cdot 13^{14}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(7.65701\times 10^{11}\) |
| Root analytic conductor: | \(7.06092\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 5^{7} \cdot 13^{14} ,\ ( \ : [3/2]^{7} ),\ -1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + p T )^{7} \) |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + p T + p^{4} T^{2} + p^{5} T^{3} + 153 T^{4} + 55 p^{3} T^{5} + 207 p^{3} T^{6} + 153 p^{5} T^{7} + 207 p^{6} T^{8} + 55 p^{9} T^{9} + 153 p^{9} T^{10} + p^{17} T^{11} + p^{19} T^{12} + p^{19} T^{13} + p^{21} T^{14} \) |
| 3 | \( 1 + 55 T^{2} + 8 T^{3} + 2407 T^{4} + 668 T^{5} + 26359 p T^{6} - 21088 T^{7} + 26359 p^{4} T^{8} + 668 p^{6} T^{9} + 2407 p^{9} T^{10} + 8 p^{12} T^{11} + 55 p^{15} T^{12} + p^{21} T^{14} \) | |
| 7 | \( 1 + 54 T + 2333 T^{2} + 11476 p T^{3} + 2325761 T^{4} + 57742106 T^{5} + 1284512613 T^{6} + 25063331336 T^{7} + 1284512613 p^{3} T^{8} + 57742106 p^{6} T^{9} + 2325761 p^{9} T^{10} + 11476 p^{13} T^{11} + 2333 p^{15} T^{12} + 54 p^{18} T^{13} + p^{21} T^{14} \) | |
| 11 | \( 1 - 4 T + 6187 T^{2} + 6108 T^{3} + 17447495 T^{4} + 81051864 T^{5} + 31381382217 T^{6} + 175505622464 T^{7} + 31381382217 p^{3} T^{8} + 81051864 p^{6} T^{9} + 17447495 p^{9} T^{10} + 6108 p^{12} T^{11} + 6187 p^{15} T^{12} - 4 p^{18} T^{13} + p^{21} T^{14} \) | |
| 17 | \( 1 - 50 T + 19615 T^{2} - 589516 T^{3} + 188124445 T^{4} - 4171814830 T^{5} + 1277971791459 T^{6} - 24436487663272 T^{7} + 1277971791459 p^{3} T^{8} - 4171814830 p^{6} T^{9} + 188124445 p^{9} T^{10} - 589516 p^{12} T^{11} + 19615 p^{15} T^{12} - 50 p^{18} T^{13} + p^{21} T^{14} \) | |
| 19 | \( 1 + 312 T + 76991 T^{2} + 13042000 T^{3} + 1899198587 T^{4} + 11758764068 p T^{5} + 23252637022617 T^{6} + 2039193573417848 T^{7} + 23252637022617 p^{3} T^{8} + 11758764068 p^{7} T^{9} + 1899198587 p^{9} T^{10} + 13042000 p^{12} T^{11} + 76991 p^{15} T^{12} + 312 p^{18} T^{13} + p^{21} T^{14} \) | |
| 23 | \( 1 - 266 T + 3245 p T^{2} - 12319172 T^{3} + 2094084903 T^{4} - 11213415574 p T^{5} + 34199173846953 T^{6} - 3581101362135216 T^{7} + 34199173846953 p^{3} T^{8} - 11213415574 p^{7} T^{9} + 2094084903 p^{9} T^{10} - 12319172 p^{12} T^{11} + 3245 p^{16} T^{12} - 266 p^{18} T^{13} + p^{21} T^{14} \) | |
| 29 | \( 1 - 294 T + 109035 T^{2} - 20517020 T^{3} + 5002413109 T^{4} - 773140090362 T^{5} + 158882226378071 T^{6} - 21393846073167368 T^{7} + 158882226378071 p^{3} T^{8} - 773140090362 p^{6} T^{9} + 5002413109 p^{9} T^{10} - 20517020 p^{12} T^{11} + 109035 p^{15} T^{12} - 294 p^{18} T^{13} + p^{21} T^{14} \) | |
| 31 | \( 1 - 270 T + 161623 T^{2} - 33450916 T^{3} + 11838397719 T^{4} - 2009025040078 T^{5} + 530787896931989 T^{6} - 74408355383755944 T^{7} + 530787896931989 p^{3} T^{8} - 2009025040078 p^{6} T^{9} + 11838397719 p^{9} T^{10} - 33450916 p^{12} T^{11} + 161623 p^{15} T^{12} - 270 p^{18} T^{13} + p^{21} T^{14} \) | |
| 37 | \( 1 + 470 T + 318327 T^{2} + 102025324 T^{3} + 42061831113 T^{4} + 10683186485418 T^{5} + 3318516181175575 T^{6} + 681773941683913832 T^{7} + 3318516181175575 p^{3} T^{8} + 10683186485418 p^{6} T^{9} + 42061831113 p^{9} T^{10} + 102025324 p^{12} T^{11} + 318327 p^{15} T^{12} + 470 p^{18} T^{13} + p^{21} T^{14} \) | |
| 41 | \( 1 + 628 T + 332899 T^{2} + 129878648 T^{3} + 48377552377 T^{4} + 14979797940812 T^{5} + 4539940387350963 T^{6} + 1191673990309811984 T^{7} + 4539940387350963 p^{3} T^{8} + 14979797940812 p^{6} T^{9} + 48377552377 p^{9} T^{10} + 129878648 p^{12} T^{11} + 332899 p^{15} T^{12} + 628 p^{18} T^{13} + p^{21} T^{14} \) | |
| 43 | \( 1 + 364 T + 409899 T^{2} + 108280452 T^{3} + 75995797771 T^{4} + 16550883218560 T^{5} + 9008034180123333 T^{6} + 1638492885410440872 T^{7} + 9008034180123333 p^{3} T^{8} + 16550883218560 p^{6} T^{9} + 75995797771 p^{9} T^{10} + 108280452 p^{12} T^{11} + 409899 p^{15} T^{12} + 364 p^{18} T^{13} + p^{21} T^{14} \) | |
| 47 | \( 1 + 386 T + 471117 T^{2} + 120513092 T^{3} + 99001425209 T^{4} + 19759245910542 T^{5} + 14080854648355133 T^{6} + 2390748635870883736 T^{7} + 14080854648355133 p^{3} T^{8} + 19759245910542 p^{6} T^{9} + 99001425209 p^{9} T^{10} + 120513092 p^{12} T^{11} + 471117 p^{15} T^{12} + 386 p^{18} T^{13} + p^{21} T^{14} \) | |
| 53 | \( 1 - 520 T + 633367 T^{2} - 178061136 T^{3} + 134174378249 T^{4} - 8352532521528 T^{5} + 12956406035805639 T^{6} + 1974062645155978400 T^{7} + 12956406035805639 p^{3} T^{8} - 8352532521528 p^{6} T^{9} + 134174378249 p^{9} T^{10} - 178061136 p^{12} T^{11} + 633367 p^{15} T^{12} - 520 p^{18} T^{13} + p^{21} T^{14} \) | |
| 59 | \( 1 - 248 T + 859503 T^{2} - 213967344 T^{3} + 391522046995 T^{4} - 95741225678124 T^{5} + 115195120629180833 T^{6} - 24727524043128526856 T^{7} + 115195120629180833 p^{3} T^{8} - 95741225678124 p^{6} T^{9} + 391522046995 p^{9} T^{10} - 213967344 p^{12} T^{11} + 859503 p^{15} T^{12} - 248 p^{18} T^{13} + p^{21} T^{14} \) | |
| 61 | \( 1 + 730 T + 1242147 T^{2} + 831341412 T^{3} + 708901440253 T^{4} + 424396208710438 T^{5} + 244692387212101815 T^{6} + \)\(12\!\cdots\!08\)\( T^{7} + 244692387212101815 p^{3} T^{8} + 424396208710438 p^{6} T^{9} + 708901440253 p^{9} T^{10} + 831341412 p^{12} T^{11} + 1242147 p^{15} T^{12} + 730 p^{18} T^{13} + p^{21} T^{14} \) | |
| 67 | \( 1 + 1646 T + 2002545 T^{2} + 1573617780 T^{3} + 1066920817153 T^{4} + 538266837503922 T^{5} + 275569421024656297 T^{6} + \)\(13\!\cdots\!40\)\( T^{7} + 275569421024656297 p^{3} T^{8} + 538266837503922 p^{6} T^{9} + 1066920817153 p^{9} T^{10} + 1573617780 p^{12} T^{11} + 2002545 p^{15} T^{12} + 1646 p^{18} T^{13} + p^{21} T^{14} \) | |
| 71 | \( 1 + 1718 T + 2619483 T^{2} + 2737786960 T^{3} + 2696389632291 T^{4} + 2118817282276062 T^{5} + 1562160325887066125 T^{6} + \)\(96\!\cdots\!60\)\( T^{7} + 1562160325887066125 p^{3} T^{8} + 2118817282276062 p^{6} T^{9} + 2696389632291 p^{9} T^{10} + 2737786960 p^{12} T^{11} + 2619483 p^{15} T^{12} + 1718 p^{18} T^{13} + p^{21} T^{14} \) | |
| 73 | \( 1 - 1610 T + 2792139 T^{2} - 37934564 p T^{3} + 2944910965601 T^{4} - 2243019397603590 T^{5} + 1811447209789220275 T^{6} - \)\(11\!\cdots\!16\)\( T^{7} + 1811447209789220275 p^{3} T^{8} - 2243019397603590 p^{6} T^{9} + 2944910965601 p^{9} T^{10} - 37934564 p^{13} T^{11} + 2792139 p^{15} T^{12} - 1610 p^{18} T^{13} + p^{21} T^{14} \) | |
| 79 | \( 1 + 512 T + 2045465 T^{2} + 1210402496 T^{3} + 2161877892581 T^{4} + 1326231242967808 T^{5} + 1498346935045947885 T^{6} + \)\(83\!\cdots\!48\)\( T^{7} + 1498346935045947885 p^{3} T^{8} + 1326231242967808 p^{6} T^{9} + 2161877892581 p^{9} T^{10} + 1210402496 p^{12} T^{11} + 2045465 p^{15} T^{12} + 512 p^{18} T^{13} + p^{21} T^{14} \) | |
| 83 | \( 1 - 226 T + 551697 T^{2} + 511708068 T^{3} + 326757867457 T^{4} + 337380082800738 T^{5} + 214284091113451049 T^{6} + \)\(30\!\cdots\!84\)\( T^{7} + 214284091113451049 p^{3} T^{8} + 337380082800738 p^{6} T^{9} + 326757867457 p^{9} T^{10} + 511708068 p^{12} T^{11} + 551697 p^{15} T^{12} - 226 p^{18} T^{13} + p^{21} T^{14} \) | |
| 89 | \( 1 + 3420 T + 8403167 T^{2} + 14959979688 T^{3} + 22016917876485 T^{4} + 26800970303041188 T^{5} + 28146829037139860779 T^{6} + \)\(25\!\cdots\!40\)\( T^{7} + 28146829037139860779 p^{3} T^{8} + 26800970303041188 p^{6} T^{9} + 22016917876485 p^{9} T^{10} + 14959979688 p^{12} T^{11} + 8403167 p^{15} T^{12} + 3420 p^{18} T^{13} + p^{21} T^{14} \) | |
| 97 | \( 1 + 1982 T + 4421403 T^{2} + 5685747788 T^{3} + 7421769693833 T^{4} + 7694710573386402 T^{5} + 7739121271091709979 T^{6} + \)\(74\!\cdots\!04\)\( T^{7} + 7739121271091709979 p^{3} T^{8} + 7694710573386402 p^{6} T^{9} + 7421769693833 p^{9} T^{10} + 5685747788 p^{12} T^{11} + 4421403 p^{15} T^{12} + 1982 p^{18} T^{13} + p^{21} T^{14} \) | |
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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
−4.83205693308422417515097848120, −4.77141092046564117087323762389, −4.59356438138504123491276884545, −4.45016881235923005950558836282, −4.24751939348775659914024652415, −4.18561092634141280017893749586, −4.07617084475044636052413768686, −3.83780047170968249715622510418, −3.70406140023946739680249164446, −3.44852093223198914662275651442, −3.41614687401982440192629431620, −3.25889635565543021201752253144, −3.12398284354755118978031639975, −3.03821589961895574336020042337, −3.01257768319468470296790761411, −2.72644311165557228357597296248, −2.52438363034067589952188514441, −2.39330450078489858336303298781, −2.36053317112198597909355866904, −1.59601027681417332896968460747, −1.56254481572502820800695790331, −1.43623333421666201157602415082, −1.06019302791896348590049789358, −1.05058696348213667448897677398, −0.69199376421057972828994997013, 0, 0, 0, 0, 0, 0, 0, 0.69199376421057972828994997013, 1.05058696348213667448897677398, 1.06019302791896348590049789358, 1.43623333421666201157602415082, 1.56254481572502820800695790331, 1.59601027681417332896968460747, 2.36053317112198597909355866904, 2.39330450078489858336303298781, 2.52438363034067589952188514441, 2.72644311165557228357597296248, 3.01257768319468470296790761411, 3.03821589961895574336020042337, 3.12398284354755118978031639975, 3.25889635565543021201752253144, 3.41614687401982440192629431620, 3.44852093223198914662275651442, 3.70406140023946739680249164446, 3.83780047170968249715622510418, 4.07617084475044636052413768686, 4.18561092634141280017893749586, 4.24751939348775659914024652415, 4.45016881235923005950558836282, 4.59356438138504123491276884545, 4.77141092046564117087323762389, 4.83205693308422417515097848120
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.