Dirichlet series
| L(s) = 1 | − 321·5-s + 83·7-s + 111·11-s + 1.84e3·13-s − 48·17-s + 1.01e4·19-s − 1.91e4·23-s − 1.85e5·25-s − 6.04e3·29-s + 1.53e5·31-s − 2.66e4·35-s + 6.96e4·37-s − 4.46e5·41-s + 3.84e5·43-s − 2.98e5·47-s − 2.70e6·49-s − 4.54e5·53-s − 3.56e4·55-s − 2.61e6·59-s + 1.46e5·61-s − 5.92e5·65-s − 1.63e6·67-s − 4.35e6·71-s − 2.13e6·73-s + 9.21e3·77-s − 2.40e6·79-s − 1.29e7·83-s + ⋯ |
| L(s) = 1 | − 1.14·5-s + 0.0914·7-s + 0.0251·11-s + 0.233·13-s − 0.00236·17-s + 0.338·19-s − 0.327·23-s − 2.37·25-s − 0.0460·29-s + 0.922·31-s − 0.105·35-s + 0.226·37-s − 1.01·41-s + 0.737·43-s − 0.419·47-s − 3.28·49-s − 0.418·53-s − 0.0288·55-s − 1.66·59-s + 0.0824·61-s − 0.267·65-s − 0.665·67-s − 1.44·71-s − 0.641·73-s + 0.00229·77-s − 0.548·79-s − 2.48·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{28}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{14} \cdot 3^{28}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.08804\times 10^{14}\) |
| Root analytic conductor: | \(10.0604\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 2^{14} \cdot 3^{28} ,\ ( \ : [7/2]^{7} ),\ -1 )\) |
Particular Values
| \(L(4)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 + 321 T + 288269 T^{2} + 106954416 T^{3} + 10455325908 p T^{4} + 659093471676 p^{2} T^{5} + 47386126072646 p^{3} T^{6} + 2623402606812162 p^{4} T^{7} + 47386126072646 p^{10} T^{8} + 659093471676 p^{16} T^{9} + 10455325908 p^{22} T^{10} + 106954416 p^{28} T^{11} + 288269 p^{35} T^{12} + 321 p^{42} T^{13} + p^{49} T^{14} \) |
| 7 | \( 1 - 83 T + 2709907 T^{2} - 385423132 T^{3} + 3606258649724 T^{4} - 429452709766336 T^{5} + 3560929443717885546 T^{6} - \)\(28\!\cdots\!90\)\( T^{7} + 3560929443717885546 p^{7} T^{8} - 429452709766336 p^{14} T^{9} + 3606258649724 p^{21} T^{10} - 385423132 p^{28} T^{11} + 2709907 p^{35} T^{12} - 83 p^{42} T^{13} + p^{49} T^{14} \) | |
| 11 | \( 1 - 111 T + 59932778 T^{2} + 36463446207 T^{3} + 1639910271846171 T^{4} + 1605993202647680310 T^{5} + \)\(33\!\cdots\!69\)\( T^{6} + \)\(33\!\cdots\!71\)\( T^{7} + \)\(33\!\cdots\!69\)\( p^{7} T^{8} + 1605993202647680310 p^{14} T^{9} + 1639910271846171 p^{21} T^{10} + 36463446207 p^{28} T^{11} + 59932778 p^{35} T^{12} - 111 p^{42} T^{13} + p^{49} T^{14} \) | |
| 13 | \( 1 - 1847 T + 223901653 T^{2} - 408935370892 T^{3} + 1875489582629828 p T^{4} - 26010462661341181324 T^{5} + \)\(14\!\cdots\!58\)\( p T^{6} - \)\(56\!\cdots\!10\)\( p^{2} T^{7} + \)\(14\!\cdots\!58\)\( p^{8} T^{8} - 26010462661341181324 p^{14} T^{9} + 1875489582629828 p^{22} T^{10} - 408935370892 p^{28} T^{11} + 223901653 p^{35} T^{12} - 1847 p^{42} T^{13} + p^{49} T^{14} \) | |
| 17 | \( 1 + 48 T + 1485264692 T^{2} - 4004497121958 T^{3} + 1151318105645396862 T^{4} - \)\(47\!\cdots\!80\)\( T^{5} + \)\(62\!\cdots\!01\)\( T^{6} - \)\(26\!\cdots\!72\)\( T^{7} + \)\(62\!\cdots\!01\)\( p^{7} T^{8} - \)\(47\!\cdots\!80\)\( p^{14} T^{9} + 1151318105645396862 p^{21} T^{10} - 4004497121958 p^{28} T^{11} + 1485264692 p^{35} T^{12} + 48 p^{42} T^{13} + p^{49} T^{14} \) | |
| 19 | \( 1 - 10124 T + 3216032164 T^{2} - 38515523838916 T^{3} + 5806727150727743294 T^{4} - \)\(76\!\cdots\!24\)\( T^{5} + \)\(70\!\cdots\!27\)\( T^{6} - \)\(85\!\cdots\!28\)\( T^{7} + \)\(70\!\cdots\!27\)\( p^{7} T^{8} - \)\(76\!\cdots\!24\)\( p^{14} T^{9} + 5806727150727743294 p^{21} T^{10} - 38515523838916 p^{28} T^{11} + 3216032164 p^{35} T^{12} - 10124 p^{42} T^{13} + p^{49} T^{14} \) | |
| 23 | \( 1 + 19119 T + 12395611907 T^{2} + 20501741458620 p T^{3} + 82626308880711548340 T^{4} + \)\(37\!\cdots\!80\)\( T^{5} + \)\(38\!\cdots\!42\)\( T^{6} + \)\(16\!\cdots\!38\)\( T^{7} + \)\(38\!\cdots\!42\)\( p^{7} T^{8} + \)\(37\!\cdots\!80\)\( p^{14} T^{9} + 82626308880711548340 p^{21} T^{10} + 20501741458620 p^{29} T^{11} + 12395611907 p^{35} T^{12} + 19119 p^{42} T^{13} + p^{49} T^{14} \) | |
| 29 | \( 1 + 6045 T + 92317152389 T^{2} - 770047602381180 T^{3} + \)\(38\!\cdots\!16\)\( T^{4} - \)\(67\!\cdots\!48\)\( T^{5} + \)\(96\!\cdots\!90\)\( T^{6} - \)\(18\!\cdots\!42\)\( T^{7} + \)\(96\!\cdots\!90\)\( p^{7} T^{8} - \)\(67\!\cdots\!48\)\( p^{14} T^{9} + \)\(38\!\cdots\!16\)\( p^{21} T^{10} - 770047602381180 p^{28} T^{11} + 92317152389 p^{35} T^{12} + 6045 p^{42} T^{13} + p^{49} T^{14} \) | |
| 31 | \( 1 - 153089 T + 104686015495 T^{2} - 19671954356863408 T^{3} + \)\(63\!\cdots\!32\)\( T^{4} - \)\(10\!\cdots\!48\)\( T^{5} + \)\(26\!\cdots\!62\)\( T^{6} - \)\(34\!\cdots\!18\)\( T^{7} + \)\(26\!\cdots\!62\)\( p^{7} T^{8} - \)\(10\!\cdots\!48\)\( p^{14} T^{9} + \)\(63\!\cdots\!32\)\( p^{21} T^{10} - 19671954356863408 p^{28} T^{11} + 104686015495 p^{35} T^{12} - 153089 p^{42} T^{13} + p^{49} T^{14} \) | |
| 37 | \( 1 - 69674 T + 98906171491 T^{2} - 5553329906106652 T^{3} + \)\(18\!\cdots\!45\)\( T^{4} - \)\(27\!\cdots\!54\)\( T^{5} + \)\(20\!\cdots\!55\)\( T^{6} + \)\(40\!\cdots\!12\)\( T^{7} + \)\(20\!\cdots\!55\)\( p^{7} T^{8} - \)\(27\!\cdots\!54\)\( p^{14} T^{9} + \)\(18\!\cdots\!45\)\( p^{21} T^{10} - 5553329906106652 p^{28} T^{11} + 98906171491 p^{35} T^{12} - 69674 p^{42} T^{13} + p^{49} T^{14} \) | |
| 41 | \( 1 + 446631 T + 233564381576 T^{2} + 175033342951056513 T^{3} + \)\(70\!\cdots\!69\)\( T^{4} + \)\(65\!\cdots\!10\)\( p T^{5} + \)\(14\!\cdots\!45\)\( T^{6} + \)\(67\!\cdots\!29\)\( T^{7} + \)\(14\!\cdots\!45\)\( p^{7} T^{8} + \)\(65\!\cdots\!10\)\( p^{15} T^{9} + \)\(70\!\cdots\!69\)\( p^{21} T^{10} + 175033342951056513 p^{28} T^{11} + 233564381576 p^{35} T^{12} + 446631 p^{42} T^{13} + p^{49} T^{14} \) | |
| 43 | \( 1 - 384347 T + 902780169370 T^{2} - 94357708525406821 T^{3} + \)\(33\!\cdots\!07\)\( T^{4} + \)\(38\!\cdots\!38\)\( T^{5} + \)\(94\!\cdots\!37\)\( T^{6} + \)\(20\!\cdots\!35\)\( T^{7} + \)\(94\!\cdots\!37\)\( p^{7} T^{8} + \)\(38\!\cdots\!38\)\( p^{14} T^{9} + \)\(33\!\cdots\!07\)\( p^{21} T^{10} - 94357708525406821 p^{28} T^{11} + 902780169370 p^{35} T^{12} - 384347 p^{42} T^{13} + p^{49} T^{14} \) | |
| 47 | \( 1 + 298413 T + 1446129277331 T^{2} - 45823224450094956 T^{3} + \)\(12\!\cdots\!96\)\( T^{4} - \)\(16\!\cdots\!64\)\( T^{5} + \)\(77\!\cdots\!14\)\( T^{6} - \)\(16\!\cdots\!86\)\( T^{7} + \)\(77\!\cdots\!14\)\( p^{7} T^{8} - \)\(16\!\cdots\!64\)\( p^{14} T^{9} + \)\(12\!\cdots\!96\)\( p^{21} T^{10} - 45823224450094956 p^{28} T^{11} + 1446129277331 p^{35} T^{12} + 298413 p^{42} T^{13} + p^{49} T^{14} \) | |
| 53 | \( 1 + 454038 T + 2557714022723 T^{2} + 2043811430434037988 T^{3} + \)\(37\!\cdots\!25\)\( T^{4} + \)\(36\!\cdots\!54\)\( T^{5} + \)\(50\!\cdots\!87\)\( T^{6} + \)\(41\!\cdots\!48\)\( T^{7} + \)\(50\!\cdots\!87\)\( p^{7} T^{8} + \)\(36\!\cdots\!54\)\( p^{14} T^{9} + \)\(37\!\cdots\!25\)\( p^{21} T^{10} + 2043811430434037988 p^{28} T^{11} + 2557714022723 p^{35} T^{12} + 454038 p^{42} T^{13} + p^{49} T^{14} \) | |
| 59 | \( 1 + 2619543 T + 8908305252434 T^{2} + 9682021693400493249 T^{3} + \)\(25\!\cdots\!91\)\( T^{4} + \)\(23\!\cdots\!66\)\( T^{5} + \)\(93\!\cdots\!09\)\( T^{6} + \)\(16\!\cdots\!43\)\( p T^{7} + \)\(93\!\cdots\!09\)\( p^{7} T^{8} + \)\(23\!\cdots\!66\)\( p^{14} T^{9} + \)\(25\!\cdots\!91\)\( p^{21} T^{10} + 9682021693400493249 p^{28} T^{11} + 8908305252434 p^{35} T^{12} + 2619543 p^{42} T^{13} + p^{49} T^{14} \) | |
| 61 | \( 1 - 146231 T + 8567957992381 T^{2} + 373875458799836480 T^{3} + \)\(49\!\cdots\!32\)\( T^{4} + \)\(99\!\cdots\!32\)\( T^{5} + \)\(20\!\cdots\!10\)\( T^{6} + \)\(34\!\cdots\!10\)\( T^{7} + \)\(20\!\cdots\!10\)\( p^{7} T^{8} + \)\(99\!\cdots\!32\)\( p^{14} T^{9} + \)\(49\!\cdots\!32\)\( p^{21} T^{10} + 373875458799836480 p^{28} T^{11} + 8567957992381 p^{35} T^{12} - 146231 p^{42} T^{13} + p^{49} T^{14} \) | |
| 67 | \( 1 + 1637419 T + 13174866266386 T^{2} - 2143861682049311323 T^{3} + \)\(10\!\cdots\!75\)\( T^{4} - \)\(95\!\cdots\!86\)\( T^{5} + \)\(56\!\cdots\!45\)\( T^{6} - \)\(12\!\cdots\!27\)\( T^{7} + \)\(56\!\cdots\!45\)\( p^{7} T^{8} - \)\(95\!\cdots\!86\)\( p^{14} T^{9} + \)\(10\!\cdots\!75\)\( p^{21} T^{10} - 2143861682049311323 p^{28} T^{11} + 13174866266386 p^{35} T^{12} + 1637419 p^{42} T^{13} + p^{49} T^{14} \) | |
| 71 | \( 1 + 4353492 T + 40479771524033 T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(86\!\cdots\!57\)\( T^{4} + \)\(30\!\cdots\!88\)\( T^{5} + \)\(11\!\cdots\!13\)\( T^{6} + \)\(34\!\cdots\!08\)\( T^{7} + \)\(11\!\cdots\!13\)\( p^{7} T^{8} + \)\(30\!\cdots\!88\)\( p^{14} T^{9} + \)\(86\!\cdots\!57\)\( p^{21} T^{10} + \)\(16\!\cdots\!80\)\( p^{28} T^{11} + 40479771524033 p^{35} T^{12} + 4353492 p^{42} T^{13} + p^{49} T^{14} \) | |
| 73 | \( 1 + 2132260 T + 53531700838948 T^{2} + 99941694733232483810 T^{3} + \)\(13\!\cdots\!58\)\( T^{4} + \)\(22\!\cdots\!60\)\( T^{5} + \)\(22\!\cdots\!93\)\( T^{6} + \)\(30\!\cdots\!80\)\( T^{7} + \)\(22\!\cdots\!93\)\( p^{7} T^{8} + \)\(22\!\cdots\!60\)\( p^{14} T^{9} + \)\(13\!\cdots\!58\)\( p^{21} T^{10} + 99941694733232483810 p^{28} T^{11} + 53531700838948 p^{35} T^{12} + 2132260 p^{42} T^{13} + p^{49} T^{14} \) | |
| 79 | \( 1 + 2402185 T + 98308923096463 T^{2} + \)\(18\!\cdots\!76\)\( T^{3} + \)\(44\!\cdots\!24\)\( T^{4} + \)\(64\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!86\)\( T^{6} + \)\(14\!\cdots\!62\)\( T^{7} + \)\(12\!\cdots\!86\)\( p^{7} T^{8} + \)\(64\!\cdots\!00\)\( p^{14} T^{9} + \)\(44\!\cdots\!24\)\( p^{21} T^{10} + \)\(18\!\cdots\!76\)\( p^{28} T^{11} + 98308923096463 p^{35} T^{12} + 2402185 p^{42} T^{13} + p^{49} T^{14} \) | |
| 83 | \( 1 + 12936357 T + 188414065773779 T^{2} + \)\(17\!\cdots\!24\)\( T^{3} + \)\(14\!\cdots\!00\)\( T^{4} + \)\(99\!\cdots\!32\)\( T^{5} + \)\(60\!\cdots\!82\)\( T^{6} + \)\(33\!\cdots\!70\)\( T^{7} + \)\(60\!\cdots\!82\)\( p^{7} T^{8} + \)\(99\!\cdots\!32\)\( p^{14} T^{9} + \)\(14\!\cdots\!00\)\( p^{21} T^{10} + \)\(17\!\cdots\!24\)\( p^{28} T^{11} + 188414065773779 p^{35} T^{12} + 12936357 p^{42} T^{13} + p^{49} T^{14} \) | |
| 89 | \( 1 + 19684830 T + 381545252003171 T^{2} + \)\(46\!\cdots\!40\)\( T^{3} + \)\(53\!\cdots\!09\)\( T^{4} + \)\(46\!\cdots\!42\)\( T^{5} + \)\(39\!\cdots\!55\)\( T^{6} + \)\(26\!\cdots\!00\)\( T^{7} + \)\(39\!\cdots\!55\)\( p^{7} T^{8} + \)\(46\!\cdots\!42\)\( p^{14} T^{9} + \)\(53\!\cdots\!09\)\( p^{21} T^{10} + \)\(46\!\cdots\!40\)\( p^{28} T^{11} + 381545252003171 p^{35} T^{12} + 19684830 p^{42} T^{13} + p^{49} T^{14} \) | |
| 97 | \( 1 - 2853257 T + 276322439092480 T^{2} - \)\(14\!\cdots\!59\)\( T^{3} + \)\(26\!\cdots\!97\)\( T^{4} - \)\(27\!\cdots\!02\)\( T^{5} + \)\(12\!\cdots\!93\)\( T^{6} - \)\(29\!\cdots\!55\)\( T^{7} + \)\(12\!\cdots\!93\)\( p^{7} T^{8} - \)\(27\!\cdots\!02\)\( p^{14} T^{9} + \)\(26\!\cdots\!97\)\( p^{21} T^{10} - \)\(14\!\cdots\!59\)\( p^{28} T^{11} + 276322439092480 p^{35} T^{12} - 2853257 p^{42} T^{13} + p^{49} T^{14} \) | |
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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
−4.97709329864992507640886536937, −4.96060053074827594980097285122, −4.69294917192967288891406647014, −4.55364935876674709420569782218, −4.12178684445001678351541555725, −3.94724540962778523958731590458, −3.89767460348828513975779369293, −3.85557298385712610338127657162, −3.82794240647127882938863842840, −3.68474771181837190189849573237, −3.54058319309912023497367956376, −2.94712840989272236799438030878, −2.89675894051725167270304122111, −2.86224313390557568919578566309, −2.57276804065428693383591317047, −2.51251385238350712738129439661, −2.43685805270079556556249601482, −2.30596781207506100862531197702, −1.61880671532859283688459051685, −1.53153808751246876463087853030, −1.45801592957720685027873348361, −1.31592709680718044260344516262, −1.26832673896512344676996807191, −1.15190826852163754412297011824, −1.04871154262923754304690217883, 0, 0, 0, 0, 0, 0, 0, 1.04871154262923754304690217883, 1.15190826852163754412297011824, 1.26832673896512344676996807191, 1.31592709680718044260344516262, 1.45801592957720685027873348361, 1.53153808751246876463087853030, 1.61880671532859283688459051685, 2.30596781207506100862531197702, 2.43685805270079556556249601482, 2.51251385238350712738129439661, 2.57276804065428693383591317047, 2.86224313390557568919578566309, 2.89675894051725167270304122111, 2.94712840989272236799438030878, 3.54058319309912023497367956376, 3.68474771181837190189849573237, 3.82794240647127882938863842840, 3.85557298385712610338127657162, 3.89767460348828513975779369293, 3.94724540962778523958731590458, 4.12178684445001678351541555725, 4.55364935876674709420569782218, 4.69294917192967288891406647014, 4.96060053074827594980097285122, 4.97709329864992507640886536937
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.