Dirichlet series
| L(s) = 1 | + 6·5-s + 49·7-s − 44·9-s + 77·11-s + 88·13-s + 134·17-s − 14·19-s − 42·23-s − 248·25-s + 10·27-s + 482·29-s − 50·31-s + 294·35-s + 152·37-s + 234·41-s − 472·43-s − 264·45-s − 728·47-s + 1.37e3·49-s − 102·53-s + 462·55-s − 1.70e3·59-s + 656·61-s − 2.15e3·63-s + 528·65-s − 1.12e3·67-s + 918·71-s + ⋯ |
| L(s) = 1 | + 0.536·5-s + 2.64·7-s − 1.62·9-s + 2.11·11-s + 1.87·13-s + 1.91·17-s − 0.169·19-s − 0.380·23-s − 1.98·25-s + 0.0712·27-s + 3.08·29-s − 0.289·31-s + 1.41·35-s + 0.675·37-s + 0.891·41-s − 1.67·43-s − 0.874·45-s − 2.25·47-s + 4·49-s − 0.264·53-s + 1.13·55-s − 3.76·59-s + 1.37·61-s − 4.31·63-s + 1.00·65-s − 2.05·67-s + 1.53·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{7} \cdot 11^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{7} \cdot 11^{7}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{28} \cdot 7^{7} \cdot 11^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.07235\times 10^{13}\) |
| Root analytic conductor: | \(8.52586\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{28} \cdot 7^{7} \cdot 11^{7} ,\ ( \ : [3/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(61.42301117\) |
| \(L(\frac12)\) | \(\approx\) | \(61.42301117\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 - p T )^{7} \) | |
| 11 | \( ( 1 - p T )^{7} \) | |
| good | 3 | \( 1 + 44 T^{2} - 10 T^{3} + 524 T^{4} - 3532 T^{5} + 18349 T^{6} - 175828 T^{7} + 18349 p^{3} T^{8} - 3532 p^{6} T^{9} + 524 p^{9} T^{10} - 10 p^{12} T^{11} + 44 p^{15} T^{12} + p^{21} T^{14} \) |
| 5 | \( 1 - 6 T + 284 T^{2} - 2272 T^{3} + 65826 T^{4} - 404002 T^{5} + 2026369 p T^{6} - 67987536 T^{7} + 2026369 p^{4} T^{8} - 404002 p^{6} T^{9} + 65826 p^{9} T^{10} - 2272 p^{12} T^{11} + 284 p^{15} T^{12} - 6 p^{18} T^{13} + p^{21} T^{14} \) | |
| 13 | \( 1 - 88 T + 9809 T^{2} - 715488 T^{3} + 54045339 T^{4} - 3006894808 T^{5} + 175980233971 T^{6} - 8250801499552 T^{7} + 175980233971 p^{3} T^{8} - 3006894808 p^{6} T^{9} + 54045339 p^{9} T^{10} - 715488 p^{12} T^{11} + 9809 p^{15} T^{12} - 88 p^{18} T^{13} + p^{21} T^{14} \) | |
| 17 | \( 1 - 134 T + 27305 T^{2} - 2485848 T^{3} + 290356439 T^{4} - 20303562618 T^{5} + 1833806643903 T^{6} - 111127946382800 T^{7} + 1833806643903 p^{3} T^{8} - 20303562618 p^{6} T^{9} + 290356439 p^{9} T^{10} - 2485848 p^{12} T^{11} + 27305 p^{15} T^{12} - 134 p^{18} T^{13} + p^{21} T^{14} \) | |
| 19 | \( 1 + 14 T + 15445 T^{2} + 370116 T^{3} + 221820981 T^{4} + 3362375266 T^{5} + 1940890827609 T^{6} + 36078670195896 T^{7} + 1940890827609 p^{3} T^{8} + 3362375266 p^{6} T^{9} + 221820981 p^{9} T^{10} + 370116 p^{12} T^{11} + 15445 p^{15} T^{12} + 14 p^{18} T^{13} + p^{21} T^{14} \) | |
| 23 | \( 1 + 42 T + 27842 T^{2} - 501852 T^{3} + 488798080 T^{4} - 10372070314 T^{5} + 8502794586935 T^{6} - 72704581429128 T^{7} + 8502794586935 p^{3} T^{8} - 10372070314 p^{6} T^{9} + 488798080 p^{9} T^{10} - 501852 p^{12} T^{11} + 27842 p^{15} T^{12} + 42 p^{18} T^{13} + p^{21} T^{14} \) | |
| 29 | \( 1 - 482 T + 228623 T^{2} - 69234996 T^{3} + 19038406729 T^{4} - 4147230797646 T^{5} + 814089784958751 T^{6} - 133910789584078360 T^{7} + 814089784958751 p^{3} T^{8} - 4147230797646 p^{6} T^{9} + 19038406729 p^{9} T^{10} - 69234996 p^{12} T^{11} + 228623 p^{15} T^{12} - 482 p^{18} T^{13} + p^{21} T^{14} \) | |
| 31 | \( 1 + 50 T + 165036 T^{2} + 11906720 T^{3} + 12248821132 T^{4} + 1007923922542 T^{5} + 17662703433551 p T^{6} + 41569550168789568 T^{7} + 17662703433551 p^{4} T^{8} + 1007923922542 p^{6} T^{9} + 12248821132 p^{9} T^{10} + 11906720 p^{12} T^{11} + 165036 p^{15} T^{12} + 50 p^{18} T^{13} + p^{21} T^{14} \) | |
| 37 | \( 1 - 152 T + 48644 T^{2} + 3370374 T^{3} + 914590226 T^{4} + 238147307416 T^{5} + 231060901239309 T^{6} - 34349523495575916 T^{7} + 231060901239309 p^{3} T^{8} + 238147307416 p^{6} T^{9} + 914590226 p^{9} T^{10} + 3370374 p^{12} T^{11} + 48644 p^{15} T^{12} - 152 p^{18} T^{13} + p^{21} T^{14} \) | |
| 41 | \( 1 - 234 T + 5961 p T^{2} - 32812360 T^{3} + 26027650695 T^{4} - 2322405515318 T^{5} + 2098716471450151 T^{6} - 166131275752043120 T^{7} + 2098716471450151 p^{3} T^{8} - 2322405515318 p^{6} T^{9} + 26027650695 p^{9} T^{10} - 32812360 p^{12} T^{11} + 5961 p^{16} T^{12} - 234 p^{18} T^{13} + p^{21} T^{14} \) | |
| 43 | \( 1 + 472 T + 299249 T^{2} + 141073184 T^{3} + 54709149297 T^{4} + 20732706560040 T^{5} + 6662097845348473 T^{6} + 1954933217456521792 T^{7} + 6662097845348473 p^{3} T^{8} + 20732706560040 p^{6} T^{9} + 54709149297 p^{9} T^{10} + 141073184 p^{12} T^{11} + 299249 p^{15} T^{12} + 472 p^{18} T^{13} + p^{21} T^{14} \) | |
| 47 | \( 1 + 728 T + 531007 T^{2} + 279637504 T^{3} + 137101200639 T^{4} + 56205275522408 T^{5} + 21393386805952065 T^{6} + 7209517394974785920 T^{7} + 21393386805952065 p^{3} T^{8} + 56205275522408 p^{6} T^{9} + 137101200639 p^{9} T^{10} + 279637504 p^{12} T^{11} + 531007 p^{15} T^{12} + 728 p^{18} T^{13} + p^{21} T^{14} \) | |
| 53 | \( 1 + 102 T + 362147 T^{2} + 58937924 T^{3} + 66192367093 T^{4} + 16550013323178 T^{5} + 11223024004883295 T^{6} + 3035593503241377848 T^{7} + 11223024004883295 p^{3} T^{8} + 16550013323178 p^{6} T^{9} + 66192367093 p^{9} T^{10} + 58937924 p^{12} T^{11} + 362147 p^{15} T^{12} + 102 p^{18} T^{13} + p^{21} T^{14} \) | |
| 59 | \( 1 + 1704 T + 1862148 T^{2} + 1368305002 T^{3} + 874437219132 T^{4} + 491863769450180 T^{5} + 4551080320375079 p T^{6} + \)\(12\!\cdots\!64\)\( T^{7} + 4551080320375079 p^{4} T^{8} + 491863769450180 p^{6} T^{9} + 874437219132 p^{9} T^{10} + 1368305002 p^{12} T^{11} + 1862148 p^{15} T^{12} + 1704 p^{18} T^{13} + p^{21} T^{14} \) | |
| 61 | \( 1 - 656 T + 1183501 T^{2} - 579578048 T^{3} + 602414873903 T^{4} - 232960751490176 T^{5} + 187633183287067091 T^{6} - 61372942309294973152 T^{7} + 187633183287067091 p^{3} T^{8} - 232960751490176 p^{6} T^{9} + 602414873903 p^{9} T^{10} - 579578048 p^{12} T^{11} + 1183501 p^{15} T^{12} - 656 p^{18} T^{13} + p^{21} T^{14} \) | |
| 67 | \( 1 + 1126 T + 1986742 T^{2} + 1557736324 T^{3} + 24148623212 p T^{4} + 983663348300682 T^{5} + 757552257136066331 T^{6} + \)\(37\!\cdots\!20\)\( T^{7} + 757552257136066331 p^{3} T^{8} + 983663348300682 p^{6} T^{9} + 24148623212 p^{10} T^{10} + 1557736324 p^{12} T^{11} + 1986742 p^{15} T^{12} + 1126 p^{18} T^{13} + p^{21} T^{14} \) | |
| 71 | \( 1 - 918 T + 1410226 T^{2} - 1127188880 T^{3} + 1024999468960 T^{4} - 707246195335354 T^{5} + 489168566062675767 T^{6} - \)\(30\!\cdots\!88\)\( T^{7} + 489168566062675767 p^{3} T^{8} - 707246195335354 p^{6} T^{9} + 1024999468960 p^{9} T^{10} - 1127188880 p^{12} T^{11} + 1410226 p^{15} T^{12} - 918 p^{18} T^{13} + p^{21} T^{14} \) | |
| 73 | \( 1 - 1094 T + 956505 T^{2} + 29360648 T^{3} - 154335549345 T^{4} + 138405554077542 T^{5} + 169049507742265063 T^{6} - \)\(12\!\cdots\!00\)\( T^{7} + 169049507742265063 p^{3} T^{8} + 138405554077542 p^{6} T^{9} - 154335549345 p^{9} T^{10} + 29360648 p^{12} T^{11} + 956505 p^{15} T^{12} - 1094 p^{18} T^{13} + p^{21} T^{14} \) | |
| 79 | \( 1 + 672 T + 1991289 T^{2} + 1005883200 T^{3} + 1879401075445 T^{4} + 854908116670688 T^{5} + 1258365779798928861 T^{6} + \)\(51\!\cdots\!88\)\( T^{7} + 1258365779798928861 p^{3} T^{8} + 854908116670688 p^{6} T^{9} + 1879401075445 p^{9} T^{10} + 1005883200 p^{12} T^{11} + 1991289 p^{15} T^{12} + 672 p^{18} T^{13} + p^{21} T^{14} \) | |
| 83 | \( 1 + 782 T + 2973761 T^{2} + 1635268700 T^{3} + 3819085657785 T^{4} + 1534814206273218 T^{5} + 3016850158169180577 T^{6} + \)\(97\!\cdots\!64\)\( T^{7} + 3016850158169180577 p^{3} T^{8} + 1534814206273218 p^{6} T^{9} + 3819085657785 p^{9} T^{10} + 1635268700 p^{12} T^{11} + 2973761 p^{15} T^{12} + 782 p^{18} T^{13} + p^{21} T^{14} \) | |
| 89 | \( 1 + 464 T + 2174152 T^{2} + 844413518 T^{3} + 2524441633154 T^{4} + 912233491401400 T^{5} + 2197789561954212749 T^{6} + \)\(78\!\cdots\!20\)\( T^{7} + 2197789561954212749 p^{3} T^{8} + 912233491401400 p^{6} T^{9} + 2524441633154 p^{9} T^{10} + 844413518 p^{12} T^{11} + 2174152 p^{15} T^{12} + 464 p^{18} T^{13} + p^{21} T^{14} \) | |
| 97 | \( 1 - 3000 T + 6069912 T^{2} - 9347999414 T^{3} + 12452984432090 T^{4} - 14784931980343936 T^{5} + 16273667759607265133 T^{6} - \)\(16\!\cdots\!40\)\( T^{7} + 16273667759607265133 p^{3} T^{8} - 14784931980343936 p^{6} T^{9} + 12452984432090 p^{9} T^{10} - 9347999414 p^{12} T^{11} + 6069912 p^{15} T^{12} - 3000 p^{18} T^{13} + p^{21} T^{14} \) | |
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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
−4.26152928310261404084630890402, −3.94382895927978185697756710197, −3.93975732662952024106181305291, −3.82314871952104351976285773287, −3.40037582810331154894625330305, −3.35437390901574907664249860263, −3.18740671825920417364786833311, −3.14577510364710213365497718125, −3.09908034623271819435335173175, −3.08409610056267059535333611063, −2.78723704626474249035089916959, −2.21802647403040074042824207171, −2.17231899093746881109069787923, −2.12858825219902819317042048998, −1.94516317992995127595829630777, −1.85791546008814427942007511280, −1.66143365173285989362741250358, −1.55151143276967361652105575604, −1.34998767291338910683915493150, −1.02092803072629019538741852465, −0.927199658352570073717193443460, −0.918957780529917934633803546381, −0.65289168032938232957733969051, −0.36606981098494498423112312431, −0.35465166173679589279114760821, 0.35465166173679589279114760821, 0.36606981098494498423112312431, 0.65289168032938232957733969051, 0.918957780529917934633803546381, 0.927199658352570073717193443460, 1.02092803072629019538741852465, 1.34998767291338910683915493150, 1.55151143276967361652105575604, 1.66143365173285989362741250358, 1.85791546008814427942007511280, 1.94516317992995127595829630777, 2.12858825219902819317042048998, 2.17231899093746881109069787923, 2.21802647403040074042824207171, 2.78723704626474249035089916959, 3.08409610056267059535333611063, 3.09908034623271819435335173175, 3.14577510364710213365497718125, 3.18740671825920417364786833311, 3.35437390901574907664249860263, 3.40037582810331154894625330305, 3.82314871952104351976285773287, 3.93975732662952024106181305291, 3.94382895927978185697756710197, 4.26152928310261404084630890402
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.