Dirichlet series
| L(s) = 1 | − 2-s − 63·3-s − 14·4-s + 63·6-s − 26·8-s + 2.26e3·9-s + 847·11-s + 882·12-s − 1.41e3·13-s − 497·16-s − 630·17-s − 2.26e3·18-s + 2.57e3·19-s − 847·22-s − 536·23-s + 1.63e3·24-s + 1.41e3·26-s − 6.12e4·27-s − 1.03e3·29-s + 1.87e3·31-s + 3.58e3·32-s − 5.33e4·33-s + 630·34-s − 3.17e4·36-s − 2.42e4·37-s − 2.57e3·38-s + 8.93e4·39-s + ⋯ |
| L(s) = 1 | − 0.176·2-s − 4.04·3-s − 0.437·4-s + 0.714·6-s − 0.143·8-s + 28/3·9-s + 2.11·11-s + 1.76·12-s − 2.32·13-s − 0.485·16-s − 0.528·17-s − 1.64·18-s + 1.63·19-s − 0.373·22-s − 0.211·23-s + 0.580·24-s + 0.411·26-s − 16.1·27-s − 0.229·29-s + 0.349·31-s + 0.618·32-s − 8.52·33-s + 0.0934·34-s − 4.08·36-s − 2.91·37-s − 0.288·38-s + 9.40·39-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 5^{14} \cdot 11^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 5^{14} \cdot 11^{7}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{7} \cdot 5^{14} \cdot 11^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.10070\times 10^{14}\) |
| Root analytic conductor: | \(11.5028\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 3^{7} \cdot 5^{14} \cdot 11^{7} ,\ ( \ : [5/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(1.875927170\) |
| \(L(\frac12)\) | \(\approx\) | \(1.875927170\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p^{2} T )^{7} \) |
| 5 | \( 1 \) | |
| 11 | \( ( 1 - p^{2} T )^{7} \) | |
| good | 2 | \( 1 + T + 15 T^{2} + 55 T^{3} + 197 p^{2} T^{4} - 71 p^{4} T^{5} + 313 p^{5} T^{6} - 6939 p^{4} T^{7} + 313 p^{10} T^{8} - 71 p^{14} T^{9} + 197 p^{17} T^{10} + 55 p^{20} T^{11} + 15 p^{25} T^{12} + p^{30} T^{13} + p^{35} T^{14} \) |
| 7 | \( 1 + 19457 T^{2} + 1543680 T^{3} + 275480981 T^{4} + 42381546240 T^{5} + 8795655702885 T^{6} + 360227099485696 T^{7} + 8795655702885 p^{5} T^{8} + 42381546240 p^{10} T^{9} + 275480981 p^{15} T^{10} + 1543680 p^{20} T^{11} + 19457 p^{25} T^{12} + p^{35} T^{14} \) | |
| 13 | \( 1 + 1418 T + 1595071 T^{2} + 1350750356 T^{3} + 1220129836897 T^{4} + 901027067583814 T^{5} + 48721283681298979 p T^{6} + \)\(37\!\cdots\!92\)\( T^{7} + 48721283681298979 p^{6} T^{8} + 901027067583814 p^{10} T^{9} + 1220129836897 p^{15} T^{10} + 1350750356 p^{20} T^{11} + 1595071 p^{25} T^{12} + 1418 p^{30} T^{13} + p^{35} T^{14} \) | |
| 17 | \( 1 + 630 T + 4041755 T^{2} - 22141988 T^{3} + 8274797262105 T^{4} - 2001695593221206 T^{5} + 16357062045068342827 T^{6} - \)\(19\!\cdots\!44\)\( T^{7} + 16357062045068342827 p^{5} T^{8} - 2001695593221206 p^{10} T^{9} + 8274797262105 p^{15} T^{10} - 22141988 p^{20} T^{11} + 4041755 p^{25} T^{12} + 630 p^{30} T^{13} + p^{35} T^{14} \) | |
| 19 | \( 1 - 2572 T + 11466965 T^{2} - 18948391832 T^{3} + 53943899418685 T^{4} - 68608805819713364 T^{5} + \)\(16\!\cdots\!13\)\( T^{6} - \)\(18\!\cdots\!64\)\( T^{7} + \)\(16\!\cdots\!13\)\( p^{5} T^{8} - 68608805819713364 p^{10} T^{9} + 53943899418685 p^{15} T^{10} - 18948391832 p^{20} T^{11} + 11466965 p^{25} T^{12} - 2572 p^{30} T^{13} + p^{35} T^{14} \) | |
| 23 | \( 1 + 536 T + 29536769 T^{2} + 24094911216 T^{3} + 429194292125541 T^{4} + 360544007702380520 T^{5} + \)\(40\!\cdots\!01\)\( T^{6} + \)\(29\!\cdots\!52\)\( T^{7} + \)\(40\!\cdots\!01\)\( p^{5} T^{8} + 360544007702380520 p^{10} T^{9} + 429194292125541 p^{15} T^{10} + 24094911216 p^{20} T^{11} + 29536769 p^{25} T^{12} + 536 p^{30} T^{13} + p^{35} T^{14} \) | |
| 29 | \( 1 + 1038 T + 35454591 T^{2} + 5646183292 T^{3} + 1422057427401233 T^{4} + 217034317637533218 T^{5} + \)\(33\!\cdots\!59\)\( T^{6} - \)\(45\!\cdots\!96\)\( T^{7} + \)\(33\!\cdots\!59\)\( p^{5} T^{8} + 217034317637533218 p^{10} T^{9} + 1422057427401233 p^{15} T^{10} + 5646183292 p^{20} T^{11} + 35454591 p^{25} T^{12} + 1038 p^{30} T^{13} + p^{35} T^{14} \) | |
| 31 | \( 1 - 1872 T + 141335129 T^{2} - 136243753504 T^{3} + 9497411713703893 T^{4} - 5647217921703053744 T^{5} + \)\(40\!\cdots\!21\)\( T^{6} - \)\(18\!\cdots\!48\)\( T^{7} + \)\(40\!\cdots\!21\)\( p^{5} T^{8} - 5647217921703053744 p^{10} T^{9} + 9497411713703893 p^{15} T^{10} - 136243753504 p^{20} T^{11} + 141335129 p^{25} T^{12} - 1872 p^{30} T^{13} + p^{35} T^{14} \) | |
| 37 | \( 1 + 24298 T + 583703943 T^{2} + 8352987511604 T^{3} + 117647604915595953 T^{4} + \)\(12\!\cdots\!58\)\( T^{5} + \)\(12\!\cdots\!91\)\( T^{6} + \)\(10\!\cdots\!44\)\( T^{7} + \)\(12\!\cdots\!91\)\( p^{5} T^{8} + \)\(12\!\cdots\!58\)\( p^{10} T^{9} + 117647604915595953 p^{15} T^{10} + 8352987511604 p^{20} T^{11} + 583703943 p^{25} T^{12} + 24298 p^{30} T^{13} + p^{35} T^{14} \) | |
| 41 | \( 1 + 17658 T + 290769059 T^{2} + 54839960644 p T^{3} + 16844104455106473 T^{4} + 33121742967855413510 T^{5} + \)\(11\!\cdots\!47\)\( T^{6} + \)\(29\!\cdots\!96\)\( T^{7} + \)\(11\!\cdots\!47\)\( p^{5} T^{8} + 33121742967855413510 p^{10} T^{9} + 16844104455106473 p^{15} T^{10} + 54839960644 p^{21} T^{11} + 290769059 p^{25} T^{12} + 17658 p^{30} T^{13} + p^{35} T^{14} \) | |
| 43 | \( 1 + 7244 T + 611861789 T^{2} + 2690133741720 T^{3} + 171605518487866637 T^{4} + \)\(41\!\cdots\!76\)\( T^{5} + \)\(31\!\cdots\!17\)\( T^{6} + \)\(50\!\cdots\!92\)\( T^{7} + \)\(31\!\cdots\!17\)\( p^{5} T^{8} + \)\(41\!\cdots\!76\)\( p^{10} T^{9} + 171605518487866637 p^{15} T^{10} + 2690133741720 p^{20} T^{11} + 611861789 p^{25} T^{12} + 7244 p^{30} T^{13} + p^{35} T^{14} \) | |
| 47 | \( 1 + 34560 T + 1075711113 T^{2} + 25088641241600 T^{3} + 602053451478901173 T^{4} + \)\(11\!\cdots\!88\)\( T^{5} + \)\(20\!\cdots\!85\)\( T^{6} + \)\(31\!\cdots\!60\)\( T^{7} + \)\(20\!\cdots\!85\)\( p^{5} T^{8} + \)\(11\!\cdots\!88\)\( p^{10} T^{9} + 602053451478901173 p^{15} T^{10} + 25088641241600 p^{20} T^{11} + 1075711113 p^{25} T^{12} + 34560 p^{30} T^{13} + p^{35} T^{14} \) | |
| 53 | \( 1 - 10214 T + 1725998775 T^{2} - 15894800856492 T^{3} + 1423047605516022417 T^{4} - \)\(13\!\cdots\!90\)\( T^{5} + \)\(79\!\cdots\!47\)\( T^{6} - \)\(74\!\cdots\!28\)\( T^{7} + \)\(79\!\cdots\!47\)\( p^{5} T^{8} - \)\(13\!\cdots\!90\)\( p^{10} T^{9} + 1423047605516022417 p^{15} T^{10} - 15894800856492 p^{20} T^{11} + 1725998775 p^{25} T^{12} - 10214 p^{30} T^{13} + p^{35} T^{14} \) | |
| 59 | \( 1 - 94676 T + 7017914909 T^{2} - 343729732191464 T^{3} + 14929237397381406109 T^{4} - \)\(51\!\cdots\!24\)\( T^{5} + \)\(16\!\cdots\!01\)\( T^{6} - \)\(46\!\cdots\!72\)\( T^{7} + \)\(16\!\cdots\!01\)\( p^{5} T^{8} - \)\(51\!\cdots\!24\)\( p^{10} T^{9} + 14929237397381406109 p^{15} T^{10} - 343729732191464 p^{20} T^{11} + 7017914909 p^{25} T^{12} - 94676 p^{30} T^{13} + p^{35} T^{14} \) | |
| 61 | \( 1 - 69538 T + 4803291887 T^{2} - 153803773950212 T^{3} + 4599437927613023073 T^{4} - \)\(13\!\cdots\!02\)\( T^{5} - \)\(14\!\cdots\!73\)\( T^{6} + \)\(11\!\cdots\!28\)\( T^{7} - \)\(14\!\cdots\!73\)\( p^{5} T^{8} - \)\(13\!\cdots\!02\)\( p^{10} T^{9} + 4599437927613023073 p^{15} T^{10} - 153803773950212 p^{20} T^{11} + 4803291887 p^{25} T^{12} - 69538 p^{30} T^{13} + p^{35} T^{14} \) | |
| 67 | \( 1 + 64908 T + 1918145909 T^{2} - 40955107242984 T^{3} - 4452342775263578547 T^{4} - \)\(11\!\cdots\!96\)\( T^{5} + \)\(43\!\cdots\!25\)\( T^{6} + \)\(29\!\cdots\!20\)\( T^{7} + \)\(43\!\cdots\!25\)\( p^{5} T^{8} - \)\(11\!\cdots\!96\)\( p^{10} T^{9} - 4452342775263578547 p^{15} T^{10} - 40955107242984 p^{20} T^{11} + 1918145909 p^{25} T^{12} + 64908 p^{30} T^{13} + p^{35} T^{14} \) | |
| 71 | \( 1 - 61816 T + 7953270993 T^{2} - 328579815879088 T^{3} + 23167675575546054341 T^{4} - \)\(66\!\cdots\!68\)\( T^{5} + \)\(37\!\cdots\!13\)\( T^{6} - \)\(97\!\cdots\!52\)\( T^{7} + \)\(37\!\cdots\!13\)\( p^{5} T^{8} - \)\(66\!\cdots\!68\)\( p^{10} T^{9} + 23167675575546054341 p^{15} T^{10} - 328579815879088 p^{20} T^{11} + 7953270993 p^{25} T^{12} - 61816 p^{30} T^{13} + p^{35} T^{14} \) | |
| 73 | \( 1 - 11890 T + 8245642355 T^{2} - 92718062877108 T^{3} + 35402827961316906713 T^{4} - \)\(29\!\cdots\!30\)\( T^{5} + \)\(10\!\cdots\!47\)\( T^{6} - \)\(68\!\cdots\!84\)\( T^{7} + \)\(10\!\cdots\!47\)\( p^{5} T^{8} - \)\(29\!\cdots\!30\)\( p^{10} T^{9} + 35402827961316906713 p^{15} T^{10} - 92718062877108 p^{20} T^{11} + 8245642355 p^{25} T^{12} - 11890 p^{30} T^{13} + p^{35} T^{14} \) | |
| 79 | \( 1 - 18928 T + 10938024409 T^{2} + 26153695667808 T^{3} + 62704049001623281605 T^{4} + \)\(52\!\cdots\!92\)\( T^{5} + \)\(27\!\cdots\!93\)\( T^{6} + \)\(18\!\cdots\!56\)\( T^{7} + \)\(27\!\cdots\!93\)\( p^{5} T^{8} + \)\(52\!\cdots\!92\)\( p^{10} T^{9} + 62704049001623281605 p^{15} T^{10} + 26153695667808 p^{20} T^{11} + 10938024409 p^{25} T^{12} - 18928 p^{30} T^{13} + p^{35} T^{14} \) | |
| 83 | \( 1 + 17492 T + 10880837541 T^{2} - 372633420055192 T^{3} + 54594985575140428013 T^{4} - \)\(35\!\cdots\!56\)\( T^{5} + \)\(32\!\cdots\!77\)\( T^{6} - \)\(13\!\cdots\!20\)\( T^{7} + \)\(32\!\cdots\!77\)\( p^{5} T^{8} - \)\(35\!\cdots\!56\)\( p^{10} T^{9} + 54594985575140428013 p^{15} T^{10} - 372633420055192 p^{20} T^{11} + 10880837541 p^{25} T^{12} + 17492 p^{30} T^{13} + p^{35} T^{14} \) | |
| 89 | \( 1 - 25302 T + 16317914051 T^{2} - 498414980970716 T^{3} + \)\(15\!\cdots\!21\)\( T^{4} - \)\(61\!\cdots\!54\)\( T^{5} + \)\(10\!\cdots\!55\)\( T^{6} - \)\(45\!\cdots\!56\)\( T^{7} + \)\(10\!\cdots\!55\)\( p^{5} T^{8} - \)\(61\!\cdots\!54\)\( p^{10} T^{9} + \)\(15\!\cdots\!21\)\( p^{15} T^{10} - 498414980970716 p^{20} T^{11} + 16317914051 p^{25} T^{12} - 25302 p^{30} T^{13} + p^{35} T^{14} \) | |
| 97 | \( 1 - 172546 T + 36282824443 T^{2} - 3533626468305076 T^{3} + \)\(55\!\cdots\!85\)\( T^{4} - \)\(48\!\cdots\!78\)\( T^{5} + \)\(65\!\cdots\!07\)\( T^{6} - \)\(49\!\cdots\!32\)\( T^{7} + \)\(65\!\cdots\!07\)\( p^{5} T^{8} - \)\(48\!\cdots\!78\)\( p^{10} T^{9} + \)\(55\!\cdots\!85\)\( p^{15} T^{10} - 3533626468305076 p^{20} T^{11} + 36282824443 p^{25} T^{12} - 172546 p^{30} T^{13} + p^{35} T^{14} \) | |
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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
−4.23218693547831422758166387640, −4.03345279462945993360666020092, −3.97403027418234609579279000746, −3.76758536036616173462947423412, −3.36439895116074411609613628198, −3.36263330560820247321006727422, −3.24698809213074301522764452538, −3.22854222492915645762120126479, −2.79885202835973317957859713481, −2.79039022984714922069021761138, −2.27282531156768051699541464589, −2.18121050550581449311156144962, −2.14414999482247910137127548560, −1.84618915005560917780114568201, −1.74158580162678695201025243806, −1.60881395679846757565073384401, −1.47712813476581141600464551575, −1.18554435880764262937615693245, −1.15848924112631424389739746391, −0.829391878551609159693798676385, −0.64538269502003239427710901636, −0.44727689307770138813179281320, −0.40948002771626902222055008645, −0.32236582105885751886746046793, −0.26233103596358636065280223081, 0.26233103596358636065280223081, 0.32236582105885751886746046793, 0.40948002771626902222055008645, 0.44727689307770138813179281320, 0.64538269502003239427710901636, 0.829391878551609159693798676385, 1.15848924112631424389739746391, 1.18554435880764262937615693245, 1.47712813476581141600464551575, 1.60881395679846757565073384401, 1.74158580162678695201025243806, 1.84618915005560917780114568201, 2.14414999482247910137127548560, 2.18121050550581449311156144962, 2.27282531156768051699541464589, 2.79039022984714922069021761138, 2.79885202835973317957859713481, 3.22854222492915645762120126479, 3.24698809213074301522764452538, 3.36263330560820247321006727422, 3.36439895116074411609613628198, 3.76758536036616173462947423412, 3.97403027418234609579279000746, 4.03345279462945993360666020092, 4.23218693547831422758166387640
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.