Dirichlet series
| L(s) = 1 | + 2-s + 9·3-s − 3·4-s + 9·6-s + 5·7-s + 45·9-s + 3·11-s − 27·12-s + 5·13-s + 5·14-s + 16-s + 16·17-s + 45·18-s − 19-s + 45·21-s + 3·22-s − 10·23-s − 18·25-s + 5·26-s + 165·27-s − 15·28-s − 4·31-s − 11·32-s + 27·33-s + 16·34-s − 135·36-s + 25·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 5.19·3-s − 3/2·4-s + 3.67·6-s + 1.88·7-s + 15·9-s + 0.904·11-s − 7.79·12-s + 1.38·13-s + 1.33·14-s + 1/4·16-s + 3.88·17-s + 10.6·18-s − 0.229·19-s + 9.81·21-s + 0.639·22-s − 2.08·23-s − 3.59·25-s + 0.980·26-s + 31.7·27-s − 2.83·28-s − 0.718·31-s − 1.94·32-s + 4.70·33-s + 2.74·34-s − 22.5·36-s + 4.10·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 29^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 29^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(3^{9} \cdot 29^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.46700\times 10^{11}\) |
| Root analytic conductor: | \(4.48845\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 3^{9} \cdot 29^{18} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(784.4245991\) |
| \(L(\frac12)\) | \(\approx\) | \(784.4245991\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{9} \) |
| 29 | \( 1 \) | |
| good | 2 | \( 1 - T + p^{2} T^{2} - 7 T^{3} + 9 p T^{4} - 27 T^{5} + 55 T^{6} - 39 p T^{7} + 67 p T^{8} - 183 T^{9} + 67 p^{2} T^{10} - 39 p^{3} T^{11} + 55 p^{3} T^{12} - 27 p^{4} T^{13} + 9 p^{6} T^{14} - 7 p^{6} T^{15} + p^{9} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) |
| 5 | \( 1 + 18 T^{2} + 24 T^{3} + 177 T^{4} + 337 T^{5} + 1492 T^{6} + 2633 T^{7} + 9457 T^{8} + 15649 T^{9} + 9457 p T^{10} + 2633 p^{2} T^{11} + 1492 p^{3} T^{12} + 337 p^{4} T^{13} + 177 p^{5} T^{14} + 24 p^{6} T^{15} + 18 p^{7} T^{16} + p^{9} T^{18} \) | |
| 7 | \( 1 - 5 T + 31 T^{2} - 128 T^{3} + 463 T^{4} - 1405 T^{5} + 4100 T^{6} - 9956 T^{7} + 25827 T^{8} - 65596 T^{9} + 25827 p T^{10} - 9956 p^{2} T^{11} + 4100 p^{3} T^{12} - 1405 p^{4} T^{13} + 463 p^{5} T^{14} - 128 p^{6} T^{15} + 31 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 - 3 T + 49 T^{2} - 52 T^{3} + 991 T^{4} + 513 T^{5} + 13154 T^{6} + 27866 T^{7} + 1147 p^{2} T^{8} + 445824 T^{9} + 1147 p^{3} T^{10} + 27866 p^{2} T^{11} + 13154 p^{3} T^{12} + 513 p^{4} T^{13} + 991 p^{5} T^{14} - 52 p^{6} T^{15} + 49 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 - 5 T + 102 T^{2} - 33 p T^{3} + 4788 T^{4} - 17120 T^{5} + 135677 T^{6} - 412188 T^{7} + 2557299 T^{8} - 6530585 T^{9} + 2557299 p T^{10} - 412188 p^{2} T^{11} + 135677 p^{3} T^{12} - 17120 p^{4} T^{13} + 4788 p^{5} T^{14} - 33 p^{7} T^{15} + 102 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 - 16 T + 190 T^{2} - 1544 T^{3} + 11104 T^{4} - 66663 T^{5} + 376150 T^{6} - 1868520 T^{7} + 8820919 T^{8} - 37225411 T^{9} + 8820919 p T^{10} - 1868520 p^{2} T^{11} + 376150 p^{3} T^{12} - 66663 p^{4} T^{13} + 11104 p^{5} T^{14} - 1544 p^{6} T^{15} + 190 p^{7} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 + T + 144 T^{2} + 154 T^{3} + 9521 T^{4} + 10329 T^{5} + 382344 T^{6} + 394400 T^{7} + 10349226 T^{8} + 9381584 T^{9} + 10349226 p T^{10} + 394400 p^{2} T^{11} + 382344 p^{3} T^{12} + 10329 p^{4} T^{13} + 9521 p^{5} T^{14} + 154 p^{6} T^{15} + 144 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 10 T + 8 p T^{2} + 1416 T^{3} + 15023 T^{4} + 94178 T^{5} + 734858 T^{6} + 3853590 T^{7} + 24114842 T^{8} + 106498388 T^{9} + 24114842 p T^{10} + 3853590 p^{2} T^{11} + 734858 p^{3} T^{12} + 94178 p^{4} T^{13} + 15023 p^{5} T^{14} + 1416 p^{6} T^{15} + 8 p^{8} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 + 4 T + 95 T^{2} + 604 T^{3} + 4724 T^{4} + 31293 T^{5} + 171357 T^{6} + 961972 T^{7} + 4798255 T^{8} + 27630886 T^{9} + 4798255 p T^{10} + 961972 p^{2} T^{11} + 171357 p^{3} T^{12} + 31293 p^{4} T^{13} + 4724 p^{5} T^{14} + 604 p^{6} T^{15} + 95 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - 25 T + 476 T^{2} - 6436 T^{3} + 74730 T^{4} - 726039 T^{5} + 6338716 T^{6} - 48776974 T^{7} + 344118462 T^{8} - 2179787403 T^{9} + 344118462 p T^{10} - 48776974 p^{2} T^{11} + 6338716 p^{3} T^{12} - 726039 p^{4} T^{13} + 74730 p^{5} T^{14} - 6436 p^{6} T^{15} + 476 p^{7} T^{16} - 25 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 - 34 T + 788 T^{2} - 12970 T^{3} + 175640 T^{4} - 1971845 T^{5} + 19301782 T^{6} - 164588816 T^{7} + 1253221727 T^{8} - 8463182695 T^{9} + 1253221727 p T^{10} - 164588816 p^{2} T^{11} + 19301782 p^{3} T^{12} - 1971845 p^{4} T^{13} + 175640 p^{5} T^{14} - 12970 p^{6} T^{15} + 788 p^{7} T^{16} - 34 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 - 12 T + 217 T^{2} - 2702 T^{3} + 28804 T^{4} - 281927 T^{5} + 2520869 T^{6} - 19935336 T^{7} + 147994305 T^{8} - 1019469718 T^{9} + 147994305 p T^{10} - 19935336 p^{2} T^{11} + 2520869 p^{3} T^{12} - 281927 p^{4} T^{13} + 28804 p^{5} T^{14} - 2702 p^{6} T^{15} + 217 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 8 T + 268 T^{2} - 2089 T^{3} + 38781 T^{4} - 267015 T^{5} + 3604314 T^{6} - 21845867 T^{7} + 235040160 T^{8} - 1222532802 T^{9} + 235040160 p T^{10} - 21845867 p^{2} T^{11} + 3604314 p^{3} T^{12} - 267015 p^{4} T^{13} + 38781 p^{5} T^{14} - 2089 p^{6} T^{15} + 268 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 + 32 T + 777 T^{2} + 13556 T^{3} + 199931 T^{4} + 2475132 T^{5} + 27002767 T^{6} + 258706391 T^{7} + 2223873503 T^{8} + 17047663403 T^{9} + 2223873503 p T^{10} + 258706391 p^{2} T^{11} + 27002767 p^{3} T^{12} + 2475132 p^{4} T^{13} + 199931 p^{5} T^{14} + 13556 p^{6} T^{15} + 777 p^{7} T^{16} + 32 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 - 10 T + 263 T^{2} - 28 p T^{3} + 30369 T^{4} - 110304 T^{5} + 2000488 T^{6} - 1128580 T^{7} + 96017121 T^{8} + 142377228 T^{9} + 96017121 p T^{10} - 1128580 p^{2} T^{11} + 2000488 p^{3} T^{12} - 110304 p^{4} T^{13} + 30369 p^{5} T^{14} - 28 p^{7} T^{15} + 263 p^{7} T^{16} - 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 51 T + 1613 T^{2} - 36289 T^{3} + 10652 p T^{4} - 9576025 T^{5} + 1971498 p T^{6} - 1302879380 T^{7} + 12352053601 T^{8} - 102748700237 T^{9} + 12352053601 p T^{10} - 1302879380 p^{2} T^{11} + 1971498 p^{4} T^{12} - 9576025 p^{4} T^{13} + 10652 p^{6} T^{14} - 36289 p^{6} T^{15} + 1613 p^{7} T^{16} - 51 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 - 7 T + 271 T^{2} - 2196 T^{3} + 43720 T^{4} - 328746 T^{5} + 4849035 T^{6} - 33932684 T^{7} + 408787801 T^{8} - 2583256998 T^{9} + 408787801 p T^{10} - 33932684 p^{2} T^{11} + 4849035 p^{3} T^{12} - 328746 p^{4} T^{13} + 43720 p^{5} T^{14} - 2196 p^{6} T^{15} + 271 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 - 7 T + 403 T^{2} - 2578 T^{3} + 79996 T^{4} - 463984 T^{5} + 10354421 T^{6} - 54242872 T^{7} + 972589151 T^{8} - 4513565462 T^{9} + 972589151 p T^{10} - 54242872 p^{2} T^{11} + 10354421 p^{3} T^{12} - 463984 p^{4} T^{13} + 79996 p^{5} T^{14} - 2578 p^{6} T^{15} + 403 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 - 17 T + 401 T^{2} - 6426 T^{3} + 92991 T^{4} - 1179384 T^{5} + 13953041 T^{6} - 144174105 T^{7} + 1416502918 T^{8} - 12512366475 T^{9} + 1416502918 p T^{10} - 144174105 p^{2} T^{11} + 13953041 p^{3} T^{12} - 1179384 p^{4} T^{13} + 92991 p^{5} T^{14} - 6426 p^{6} T^{15} + 401 p^{7} T^{16} - 17 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 - 13 T + 543 T^{2} - 6692 T^{3} + 144736 T^{4} - 1578690 T^{5} + 24184789 T^{6} - 226132206 T^{7} + 2738544831 T^{8} - 21609843142 T^{9} + 2738544831 p T^{10} - 226132206 p^{2} T^{11} + 24184789 p^{3} T^{12} - 1578690 p^{4} T^{13} + 144736 p^{5} T^{14} - 6692 p^{6} T^{15} + 543 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 + 31 T + 1023 T^{2} + 20178 T^{3} + 391292 T^{4} + 5722828 T^{5} + 81071705 T^{6} + 933533148 T^{7} + 10375704955 T^{8} + 96245293542 T^{9} + 10375704955 p T^{10} + 933533148 p^{2} T^{11} + 81071705 p^{3} T^{12} + 5722828 p^{4} T^{13} + 391292 p^{5} T^{14} + 20178 p^{6} T^{15} + 1023 p^{7} T^{16} + 31 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 32 T + 894 T^{2} + 16539 T^{3} + 282952 T^{4} + 3946842 T^{5} + 51974585 T^{6} + 593765484 T^{7} + 6445461922 T^{8} + 62230806243 T^{9} + 6445461922 p T^{10} + 593765484 p^{2} T^{11} + 51974585 p^{3} T^{12} + 3946842 p^{4} T^{13} + 282952 p^{5} T^{14} + 16539 p^{6} T^{15} + 894 p^{7} T^{16} + 32 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 - 16 T + 446 T^{2} - 5487 T^{3} + 94470 T^{4} - 931906 T^{5} + 13217605 T^{6} - 110931128 T^{7} + 1428628652 T^{8} - 11184892691 T^{9} + 1428628652 p T^{10} - 110931128 p^{2} T^{11} + 13217605 p^{3} T^{12} - 931906 p^{4} T^{13} + 94470 p^{5} T^{14} - 5487 p^{6} T^{15} + 446 p^{7} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.53269985047208945955943788342, −3.35728049878546155691485862163, −3.31301686238696893621401657194, −3.14290691604254052858212129524, −2.83153064707314165880836621751, −2.80586452443707872763580330191, −2.74718352081279154359818833385, −2.61124287712323453283774993431, −2.54721740341077306209339293743, −2.48018483363078393814937966698, −2.11288301615040538975344383317, −2.08690429227919172411047336554, −2.07395145273429544543306432019, −2.00507173398224234399566099927, −2.00387167805507805308711083837, −1.75658108574976970475325510291, −1.73961659817377672090806975657, −1.36823080135923419487002372820, −1.22037220261402304154742936630, −1.20156259228613038576132505057, −1.06555775352255085648153426782, −0.848958585214715597394713242811, −0.806989254111012787227852243433, −0.65618362649692908041785953673, −0.41215820423216672000067788353, 0.41215820423216672000067788353, 0.65618362649692908041785953673, 0.806989254111012787227852243433, 0.848958585214715597394713242811, 1.06555775352255085648153426782, 1.20156259228613038576132505057, 1.22037220261402304154742936630, 1.36823080135923419487002372820, 1.73961659817377672090806975657, 1.75658108574976970475325510291, 2.00387167805507805308711083837, 2.00507173398224234399566099927, 2.07395145273429544543306432019, 2.08690429227919172411047336554, 2.11288301615040538975344383317, 2.48018483363078393814937966698, 2.54721740341077306209339293743, 2.61124287712323453283774993431, 2.74718352081279154359818833385, 2.80586452443707872763580330191, 2.83153064707314165880836621751, 3.14290691604254052858212129524, 3.31301686238696893621401657194, 3.35728049878546155691485862163, 3.53269985047208945955943788342
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.