Dirichlet series
| L(s) = 1 | − 3·3-s + 5·5-s − 7·7-s − 2·9-s − 3·11-s + 16·13-s − 15·15-s + 8·17-s + 7·19-s + 21·21-s + 10·23-s − 25-s + 16·27-s + 11·29-s + 14·31-s + 9·33-s − 35·35-s + 13·37-s − 48·39-s + 11·41-s − 11·43-s − 10·45-s + 7·47-s + 28·49-s − 24·51-s + 9·53-s − 15·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2.23·5-s − 2.64·7-s − 2/3·9-s − 0.904·11-s + 4.43·13-s − 3.87·15-s + 1.94·17-s + 1.60·19-s + 4.58·21-s + 2.08·23-s − 1/5·25-s + 3.07·27-s + 2.04·29-s + 2.51·31-s + 1.56·33-s − 5.91·35-s + 2.13·37-s − 7.68·39-s + 1.71·41-s − 1.67·43-s − 1.49·45-s + 1.02·47-s + 4·49-s − 3.36·51-s + 1.23·53-s − 2.02·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 7^{7} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 7^{7} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{42} \cdot 7^{7} \cdot 19^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.70133\times 10^{12}\) |
| Root analytic conductor: | \(8.24431\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{42} \cdot 7^{7} \cdot 19^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(28.80284204\) |
| \(L(\frac12)\) | \(\approx\) | \(28.80284204\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 + T )^{7} \) | |
| 19 | \( ( 1 - T )^{7} \) | |
| good | 3 | \( 1 + p T + 11 T^{2} + 23 T^{3} + 17 p T^{4} + 26 p T^{5} + 46 p T^{6} + 208 T^{7} + 46 p^{2} T^{8} + 26 p^{3} T^{9} + 17 p^{4} T^{10} + 23 p^{4} T^{11} + 11 p^{5} T^{12} + p^{7} T^{13} + p^{7} T^{14} \) |
| 5 | \( 1 - p T + 26 T^{2} - 86 T^{3} + 261 T^{4} - 668 T^{5} + 311 p T^{6} - 3612 T^{7} + 311 p^{2} T^{8} - 668 p^{2} T^{9} + 261 p^{3} T^{10} - 86 p^{4} T^{11} + 26 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 3 T + 31 T^{2} + 75 T^{3} + 373 T^{4} + 1040 T^{5} + 3510 T^{6} + 12812 T^{7} + 3510 p T^{8} + 1040 p^{2} T^{9} + 373 p^{3} T^{10} + 75 p^{4} T^{11} + 31 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 16 T + p^{2} T^{2} - 1310 T^{3} + 8276 T^{4} - 3346 p T^{5} + 196006 T^{6} - 758280 T^{7} + 196006 p T^{8} - 3346 p^{3} T^{9} + 8276 p^{3} T^{10} - 1310 p^{4} T^{11} + p^{7} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 8 T + 76 T^{2} - 22 p T^{3} + 2083 T^{4} - 6474 T^{5} + 29692 T^{6} - 82648 T^{7} + 29692 p T^{8} - 6474 p^{2} T^{9} + 2083 p^{3} T^{10} - 22 p^{5} T^{11} + 76 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 10 T + 103 T^{2} - 860 T^{3} + 5944 T^{4} - 37156 T^{5} + 211698 T^{6} - 1032988 T^{7} + 211698 p T^{8} - 37156 p^{2} T^{9} + 5944 p^{3} T^{10} - 860 p^{4} T^{11} + 103 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 11 T + 135 T^{2} - 1157 T^{3} + 9497 T^{4} - 63778 T^{5} + 407888 T^{6} - 2297366 T^{7} + 407888 p T^{8} - 63778 p^{2} T^{9} + 9497 p^{3} T^{10} - 1157 p^{4} T^{11} + 135 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 14 T + 110 T^{2} - 656 T^{3} + 3701 T^{4} - 23426 T^{5} + 161432 T^{6} - 998704 T^{7} + 161432 p T^{8} - 23426 p^{2} T^{9} + 3701 p^{3} T^{10} - 656 p^{4} T^{11} + 110 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 13 T + 238 T^{2} - 2462 T^{3} + 25303 T^{4} - 207100 T^{5} + 1532395 T^{6} - 9895860 T^{7} + 1532395 p T^{8} - 207100 p^{2} T^{9} + 25303 p^{3} T^{10} - 2462 p^{4} T^{11} + 238 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 11 T + 273 T^{2} - 2123 T^{3} + 30115 T^{4} - 179926 T^{5} + 1896616 T^{6} - 9138656 T^{7} + 1896616 p T^{8} - 179926 p^{2} T^{9} + 30115 p^{3} T^{10} - 2123 p^{4} T^{11} + 273 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 11 T + 290 T^{2} + 2630 T^{3} + 36844 T^{4} + 269373 T^{5} + 2627943 T^{6} + 15189684 T^{7} + 2627943 p T^{8} + 269373 p^{2} T^{9} + 36844 p^{3} T^{10} + 2630 p^{4} T^{11} + 290 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 7 T + 150 T^{2} - 684 T^{3} + 11537 T^{4} - 48144 T^{5} + 665813 T^{6} - 2449114 T^{7} + 665813 p T^{8} - 48144 p^{2} T^{9} + 11537 p^{3} T^{10} - 684 p^{4} T^{11} + 150 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 9 T + 179 T^{2} - 1071 T^{3} + 17801 T^{4} - 95742 T^{5} + 1232232 T^{6} - 5368354 T^{7} + 1232232 p T^{8} - 95742 p^{2} T^{9} + 17801 p^{3} T^{10} - 1071 p^{4} T^{11} + 179 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 23 T + 480 T^{2} + 6512 T^{3} + 79483 T^{4} + 787808 T^{5} + 7161955 T^{6} + 57108198 T^{7} + 7161955 p T^{8} + 787808 p^{2} T^{9} + 79483 p^{3} T^{10} + 6512 p^{4} T^{11} + 480 p^{5} T^{12} + 23 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 23 T + 322 T^{2} - 2866 T^{3} + 22921 T^{4} - 170048 T^{5} + 1493483 T^{6} - 11657376 T^{7} + 1493483 p T^{8} - 170048 p^{2} T^{9} + 22921 p^{3} T^{10} - 2866 p^{4} T^{11} + 322 p^{5} T^{12} - 23 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 16 T + 368 T^{2} + 4132 T^{3} + 61939 T^{4} + 564738 T^{5} + 94812 p T^{6} + 46796796 T^{7} + 94812 p^{2} T^{8} + 564738 p^{2} T^{9} + 61939 p^{3} T^{10} + 4132 p^{4} T^{11} + 368 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 3 T + 300 T^{2} - 858 T^{3} + 39473 T^{4} - 115292 T^{5} + 3352939 T^{6} - 9865672 T^{7} + 3352939 p T^{8} - 115292 p^{2} T^{9} + 39473 p^{3} T^{10} - 858 p^{4} T^{11} + 300 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 4 T + 408 T^{2} - 2002 T^{3} + 75235 T^{4} - 388580 T^{5} + 8332424 T^{6} - 38507748 T^{7} + 8332424 p T^{8} - 388580 p^{2} T^{9} + 75235 p^{3} T^{10} - 2002 p^{4} T^{11} + 408 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 29 T + 504 T^{2} - 7008 T^{3} + 94534 T^{4} - 1118691 T^{5} + 11315279 T^{6} - 102511648 T^{7} + 11315279 p T^{8} - 1118691 p^{2} T^{9} + 94534 p^{3} T^{10} - 7008 p^{4} T^{11} + 504 p^{5} T^{12} - 29 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 6 T + 394 T^{2} + 1224 T^{3} + 66385 T^{4} + 71014 T^{5} + 7085124 T^{6} + 1904344 T^{7} + 7085124 p T^{8} + 71014 p^{2} T^{9} + 66385 p^{3} T^{10} + 1224 p^{4} T^{11} + 394 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 15 T + 404 T^{2} - 3642 T^{3} + 66590 T^{4} - 453265 T^{5} + 7397221 T^{6} - 42840108 T^{7} + 7397221 p T^{8} - 453265 p^{2} T^{9} + 66590 p^{3} T^{10} - 3642 p^{4} T^{11} + 404 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 13 T + 4 p T^{2} + 3704 T^{3} + 69071 T^{4} + 565404 T^{5} + 8897921 T^{6} + 64277646 T^{7} + 8897921 p T^{8} + 565404 p^{2} T^{9} + 69071 p^{3} T^{10} + 3704 p^{4} T^{11} + 4 p^{6} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.45198030044210460894640093518, −3.31991884101084332601411198935, −3.23078855802124475306226431247, −3.16813942082485361092378814106, −2.98739554738749479394941468043, −2.84350757071351873550363417147, −2.76064973780287042290941075583, −2.75624896556480366131334213516, −2.64780957267899985759443723181, −2.49115101018975378095105274438, −2.47727663257257431154858621512, −2.13789277956346894997400320308, −2.00200071044010274228059060750, −1.74171990735197728761624661627, −1.66313313364508983070485539940, −1.60449582026964125998963965947, −1.55800097406577375954724829360, −1.06436430187765771414876850459, −1.03876500856507879466165709721, −0.964762793471881548466605779987, −0.795364234252009560588220836194, −0.68311726096880806586257083466, −0.67030270995009675227129658247, −0.44959341818561389998374778731, −0.35842584368134353738288913281, 0.35842584368134353738288913281, 0.44959341818561389998374778731, 0.67030270995009675227129658247, 0.68311726096880806586257083466, 0.795364234252009560588220836194, 0.964762793471881548466605779987, 1.03876500856507879466165709721, 1.06436430187765771414876850459, 1.55800097406577375954724829360, 1.60449582026964125998963965947, 1.66313313364508983070485539940, 1.74171990735197728761624661627, 2.00200071044010274228059060750, 2.13789277956346894997400320308, 2.47727663257257431154858621512, 2.49115101018975378095105274438, 2.64780957267899985759443723181, 2.75624896556480366131334213516, 2.76064973780287042290941075583, 2.84350757071351873550363417147, 2.98739554738749479394941468043, 3.16813942082485361092378814106, 3.23078855802124475306226431247, 3.31991884101084332601411198935, 3.45198030044210460894640093518
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.