Dirichlet series
| L(s) = 1 | + 2·2-s − 2·4-s + 2·7-s − 6·8-s + 7·11-s + 3·13-s + 4·14-s − 16-s + 7·17-s − 7·19-s + 14·22-s + 18·23-s − 20·25-s + 6·26-s − 4·28-s + 14·29-s − 8·31-s + 4·32-s + 14·34-s − 2·37-s − 14·38-s + 18·41-s − 4·43-s − 14·44-s + 36·46-s + 10·47-s − 28·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 4-s + 0.755·7-s − 2.12·8-s + 2.11·11-s + 0.832·13-s + 1.06·14-s − 1/4·16-s + 1.69·17-s − 1.60·19-s + 2.98·22-s + 3.75·23-s − 4·25-s + 1.17·26-s − 0.755·28-s + 2.59·29-s − 1.43·31-s + 0.707·32-s + 2.40·34-s − 0.328·37-s − 2.27·38-s + 2.81·41-s − 0.609·43-s − 2.11·44-s + 5.30·46-s + 1.45·47-s − 4·49-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 11^{7} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 11^{7} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{14} \cdot 11^{7} \cdot 19^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.72449\times 10^{8}\) |
| Root analytic conductor: | \(3.87554\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 3^{14} \cdot 11^{7} \cdot 19^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(26.11347717\) |
| \(L(\frac12)\) | \(\approx\) | \(26.11347717\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 11 | \( ( 1 - T )^{7} \) | |
| 19 | \( ( 1 + T )^{7} \) | |
| good | 2 | \( 1 - p T + 3 p T^{2} - 5 p T^{3} + 21 T^{4} - p^{5} T^{5} + 55 T^{6} - 39 p T^{7} + 55 p T^{8} - p^{7} T^{9} + 21 p^{3} T^{10} - 5 p^{5} T^{11} + 3 p^{6} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) |
| 5 | \( 1 + 4 p T^{2} + 8 T^{3} + 42 p T^{4} + 104 T^{5} + 1496 T^{6} + 662 T^{7} + 1496 p T^{8} + 104 p^{2} T^{9} + 42 p^{4} T^{10} + 8 p^{4} T^{11} + 4 p^{6} T^{12} + p^{7} T^{14} \) | |
| 7 | \( 1 - 2 T + 32 T^{2} - 62 T^{3} + 534 T^{4} - 902 T^{5} + 5580 T^{6} - 7976 T^{7} + 5580 p T^{8} - 902 p^{2} T^{9} + 534 p^{3} T^{10} - 62 p^{4} T^{11} + 32 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 3 T + 4 p T^{2} - 167 T^{3} + 1464 T^{4} - 4317 T^{5} + 27662 T^{6} - 68781 T^{7} + 27662 p T^{8} - 4317 p^{2} T^{9} + 1464 p^{3} T^{10} - 167 p^{4} T^{11} + 4 p^{6} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 7 T + 75 T^{2} - 292 T^{3} + 1692 T^{4} - 3046 T^{5} + 16149 T^{6} - 5591 T^{7} + 16149 p T^{8} - 3046 p^{2} T^{9} + 1692 p^{3} T^{10} - 292 p^{4} T^{11} + 75 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 18 T + 250 T^{2} - 2456 T^{3} + 20475 T^{4} - 139784 T^{5} + 837561 T^{6} - 4269484 T^{7} + 837561 p T^{8} - 139784 p^{2} T^{9} + 20475 p^{3} T^{10} - 2456 p^{4} T^{11} + 250 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 14 T + 207 T^{2} - 1678 T^{3} + 13600 T^{4} - 76898 T^{5} + 469825 T^{6} - 2314678 T^{7} + 469825 p T^{8} - 76898 p^{2} T^{9} + 13600 p^{3} T^{10} - 1678 p^{4} T^{11} + 207 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 8 T + 127 T^{2} + 546 T^{3} + 6732 T^{4} + 24318 T^{5} + 295517 T^{6} + 968768 T^{7} + 295517 p T^{8} + 24318 p^{2} T^{9} + 6732 p^{3} T^{10} + 546 p^{4} T^{11} + 127 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 2 T + 168 T^{2} + 88 T^{3} + 12985 T^{4} - 6784 T^{5} + 643605 T^{6} - 553156 T^{7} + 643605 p T^{8} - 6784 p^{2} T^{9} + 12985 p^{3} T^{10} + 88 p^{4} T^{11} + 168 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 18 T + 330 T^{2} - 3598 T^{3} + 38232 T^{4} - 305214 T^{5} + 2408210 T^{6} - 15423692 T^{7} + 2408210 p T^{8} - 305214 p^{2} T^{9} + 38232 p^{3} T^{10} - 3598 p^{4} T^{11} + 330 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 4 T + 121 T^{2} + 138 T^{3} + 6656 T^{4} - 1626 T^{5} + 349013 T^{6} - 19992 T^{7} + 349013 p T^{8} - 1626 p^{2} T^{9} + 6656 p^{3} T^{10} + 138 p^{4} T^{11} + 121 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 10 T + 169 T^{2} - 498 T^{3} + 6964 T^{4} + 13746 T^{5} + 308671 T^{6} + 722962 T^{7} + 308671 p T^{8} + 13746 p^{2} T^{9} + 6964 p^{3} T^{10} - 498 p^{4} T^{11} + 169 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 23 T + 399 T^{2} - 4428 T^{3} + 43616 T^{4} - 334702 T^{5} + 2604471 T^{6} - 17887421 T^{7} + 2604471 p T^{8} - 334702 p^{2} T^{9} + 43616 p^{3} T^{10} - 4428 p^{4} T^{11} + 399 p^{5} T^{12} - 23 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 15 T + 376 T^{2} - 3565 T^{3} + 51380 T^{4} - 348761 T^{5} + 3987278 T^{6} - 22528211 T^{7} + 3987278 p T^{8} - 348761 p^{2} T^{9} + 51380 p^{3} T^{10} - 3565 p^{4} T^{11} + 376 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 12 T + 362 T^{2} - 2844 T^{3} + 50298 T^{4} - 275584 T^{5} + 4043948 T^{6} - 17899330 T^{7} + 4043948 p T^{8} - 275584 p^{2} T^{9} + 50298 p^{3} T^{10} - 2844 p^{4} T^{11} + 362 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 6 T + 188 T^{2} + 2642 T^{3} + 23796 T^{4} + 306650 T^{5} + 3040264 T^{6} + 20691976 T^{7} + 3040264 p T^{8} + 306650 p^{2} T^{9} + 23796 p^{3} T^{10} + 2642 p^{4} T^{11} + 188 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 39 T + 1033 T^{2} - 19420 T^{3} + 297470 T^{4} - 3723376 T^{5} + 39822981 T^{6} - 360875937 T^{7} + 39822981 p T^{8} - 3723376 p^{2} T^{9} + 297470 p^{3} T^{10} - 19420 p^{4} T^{11} + 1033 p^{5} T^{12} - 39 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 4 T + 433 T^{2} + 1378 T^{3} + 84678 T^{4} + 217302 T^{5} + 9725129 T^{6} + 20117620 T^{7} + 9725129 p T^{8} + 217302 p^{2} T^{9} + 84678 p^{3} T^{10} + 1378 p^{4} T^{11} + 433 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 21 T + 567 T^{2} - 106 p T^{3} + 133576 T^{4} - 1497888 T^{5} + 17422579 T^{6} - 152791853 T^{7} + 17422579 p T^{8} - 1497888 p^{2} T^{9} + 133576 p^{3} T^{10} - 106 p^{5} T^{11} + 567 p^{5} T^{12} - 21 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 15 T + 414 T^{2} - 4607 T^{3} + 68030 T^{4} - 624005 T^{5} + 6744800 T^{6} - 57163121 T^{7} + 6744800 p T^{8} - 624005 p^{2} T^{9} + 68030 p^{3} T^{10} - 4607 p^{4} T^{11} + 414 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 13 T + 401 T^{2} - 5034 T^{3} + 85044 T^{4} - 902320 T^{5} + 11588685 T^{6} - 98948575 T^{7} + 11588685 p T^{8} - 902320 p^{2} T^{9} + 85044 p^{3} T^{10} - 5034 p^{4} T^{11} + 401 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 4 T + 209 T^{2} + 1362 T^{3} + 35936 T^{4} + 254438 T^{5} + 4487773 T^{6} + 27053728 T^{7} + 4487773 p T^{8} + 254438 p^{2} T^{9} + 35936 p^{3} T^{10} + 1362 p^{4} T^{11} + 209 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−4.32423457717477854872680787337, −4.23772515332942656249178921838, −4.09361894890067103364869562717, −3.97373644405628991896590119856, −3.65929456737270177859443667004, −3.63802659008398983633233615074, −3.59478924863030054221177332663, −3.42387014926094513277114877021, −3.42374169207616093391104773936, −3.31671748581748399603682858719, −2.89803824266721195106099238268, −2.84785314088186182278795014423, −2.46821678307035284204214865241, −2.45539372791089766333096787716, −2.38183794725398940136413427448, −1.94095112426756435184080673079, −1.90717743731700716161288814707, −1.90107182878510573559145157483, −1.70683293323315154287678161540, −1.25048820984180224348414809615, −1.08336537325345904003292173584, −0.881111248182183080275837920457, −0.820000338745602561813494757279, −0.63940104413889996952447650411, −0.41359203633662299368586027131, 0.41359203633662299368586027131, 0.63940104413889996952447650411, 0.820000338745602561813494757279, 0.881111248182183080275837920457, 1.08336537325345904003292173584, 1.25048820984180224348414809615, 1.70683293323315154287678161540, 1.90107182878510573559145157483, 1.90717743731700716161288814707, 1.94095112426756435184080673079, 2.38183794725398940136413427448, 2.45539372791089766333096787716, 2.46821678307035284204214865241, 2.84785314088186182278795014423, 2.89803824266721195106099238268, 3.31671748581748399603682858719, 3.42374169207616093391104773936, 3.42387014926094513277114877021, 3.59478924863030054221177332663, 3.63802659008398983633233615074, 3.65929456737270177859443667004, 3.97373644405628991896590119856, 4.09361894890067103364869562717, 4.23772515332942656249178921838, 4.32423457717477854872680787337
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.