Dirichlet series
| L(s) = 1 | + 3·3-s + 9·7-s + 9-s + 9·11-s − 9·13-s + 5·17-s + 10·19-s + 27·21-s + 8·23-s − 21·25-s + 14·29-s + 11·31-s + 27·33-s − 27·39-s + 14·41-s + 8·43-s + 10·47-s + 45·49-s + 15·51-s + 21·53-s + 30·57-s + 23·59-s + 34·61-s + 9·63-s + 10·67-s + 24·69-s + 4·71-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 3.40·7-s + 1/3·9-s + 2.71·11-s − 2.49·13-s + 1.21·17-s + 2.29·19-s + 5.89·21-s + 1.66·23-s − 4.19·25-s + 2.59·29-s + 1.97·31-s + 4.70·33-s − 4.32·39-s + 2.18·41-s + 1.21·43-s + 1.45·47-s + 45/7·49-s + 2.10·51-s + 2.88·53-s + 3.97·57-s + 2.99·59-s + 4.35·61-s + 1.13·63-s + 1.22·67-s + 2.88·69-s + 0.474·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{18} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.49092\times 10^{13}\) |
| Root analytic conductor: | \(5.65438\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 2^{18} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(506.7124374\) |
| \(L(\frac12)\) | \(\approx\) | \(506.7124374\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 - T )^{9} \) | |
| 11 | \( ( 1 - T )^{9} \) | |
| 13 | \( ( 1 + T )^{9} \) | |
| good | 3 | \( 1 - p T + 8 T^{2} - 7 p T^{3} + 41 T^{4} - 85 T^{5} + 56 p T^{6} - 109 p T^{7} + 71 p^{2} T^{8} - 1094 T^{9} + 71 p^{3} T^{10} - 109 p^{3} T^{11} + 56 p^{4} T^{12} - 85 p^{4} T^{13} + 41 p^{5} T^{14} - 7 p^{7} T^{15} + 8 p^{7} T^{16} - p^{9} T^{17} + p^{9} T^{18} \) |
| 5 | \( 1 + 21 T^{2} + 4 T^{3} + 231 T^{4} + p^{3} T^{5} + 1723 T^{6} + 59 p^{2} T^{7} + 10031 T^{8} + 9583 T^{9} + 10031 p T^{10} + 59 p^{4} T^{11} + 1723 p^{3} T^{12} + p^{7} T^{13} + 231 p^{5} T^{14} + 4 p^{6} T^{15} + 21 p^{7} T^{16} + p^{9} T^{18} \) | |
| 17 | \( 1 - 5 T + 58 T^{2} - 269 T^{3} + 1851 T^{4} - 6863 T^{5} + 35928 T^{6} - 115557 T^{7} + 557691 T^{8} - 1709086 T^{9} + 557691 p T^{10} - 115557 p^{2} T^{11} + 35928 p^{3} T^{12} - 6863 p^{4} T^{13} + 1851 p^{5} T^{14} - 269 p^{6} T^{15} + 58 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 - 10 T + 129 T^{2} - 858 T^{3} + 6799 T^{4} - 36687 T^{5} + 230863 T^{6} - 57587 p T^{7} + 5856445 T^{8} - 24273225 T^{9} + 5856445 p T^{10} - 57587 p^{3} T^{11} + 230863 p^{3} T^{12} - 36687 p^{4} T^{13} + 6799 p^{5} T^{14} - 858 p^{6} T^{15} + 129 p^{7} T^{16} - 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 - 8 T + 146 T^{2} - 896 T^{3} + 9956 T^{4} - 51621 T^{5} + 438850 T^{6} - 1963688 T^{7} + 13741055 T^{8} - 53051974 T^{9} + 13741055 p T^{10} - 1963688 p^{2} T^{11} + 438850 p^{3} T^{12} - 51621 p^{4} T^{13} + 9956 p^{5} T^{14} - 896 p^{6} T^{15} + 146 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 - 14 T + 214 T^{2} - 1810 T^{3} + 15904 T^{4} - 98781 T^{5} + 676330 T^{6} - 3606286 T^{7} + 22533647 T^{8} - 111646538 T^{9} + 22533647 p T^{10} - 3606286 p^{2} T^{11} + 676330 p^{3} T^{12} - 98781 p^{4} T^{13} + 15904 p^{5} T^{14} - 1810 p^{6} T^{15} + 214 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 - 11 T + 3 p T^{2} - 474 T^{3} + 4027 T^{4} - 25667 T^{5} + 195325 T^{6} - 1011266 T^{7} + 7205150 T^{8} - 36845532 T^{9} + 7205150 p T^{10} - 1011266 p^{2} T^{11} + 195325 p^{3} T^{12} - 25667 p^{4} T^{13} + 4027 p^{5} T^{14} - 474 p^{6} T^{15} + 3 p^{8} T^{16} - 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 + 145 T^{2} - 175 T^{3} + 9615 T^{4} - 26440 T^{5} + 384903 T^{6} - 1904977 T^{7} + 12041144 T^{8} - 2329344 p T^{9} + 12041144 p T^{10} - 1904977 p^{2} T^{11} + 384903 p^{3} T^{12} - 26440 p^{4} T^{13} + 9615 p^{5} T^{14} - 175 p^{6} T^{15} + 145 p^{7} T^{16} + p^{9} T^{18} \) | |
| 41 | \( 1 - 14 T + 283 T^{2} - 3147 T^{3} + 38143 T^{4} - 8432 p T^{5} + 3189999 T^{6} - 24091885 T^{7} + 183199126 T^{8} - 1168958292 T^{9} + 183199126 p T^{10} - 24091885 p^{2} T^{11} + 3189999 p^{3} T^{12} - 8432 p^{5} T^{13} + 38143 p^{5} T^{14} - 3147 p^{6} T^{15} + 283 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 - 8 T + 176 T^{2} - 22 p T^{3} + 13274 T^{4} - 55927 T^{5} + 667442 T^{6} - 2392933 T^{7} + 27280255 T^{8} - 92929925 T^{9} + 27280255 p T^{10} - 2392933 p^{2} T^{11} + 667442 p^{3} T^{12} - 55927 p^{4} T^{13} + 13274 p^{5} T^{14} - 22 p^{7} T^{15} + 176 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 10 T + 182 T^{2} - 1176 T^{3} + 16608 T^{4} - 94125 T^{5} + 1162654 T^{6} - 5625600 T^{7} + 1332557 p T^{8} - 270886898 T^{9} + 1332557 p^{2} T^{10} - 5625600 p^{2} T^{11} + 1162654 p^{3} T^{12} - 94125 p^{4} T^{13} + 16608 p^{5} T^{14} - 1176 p^{6} T^{15} + 182 p^{7} T^{16} - 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 - 21 T + 277 T^{2} - 56 p T^{3} + 30947 T^{4} - 288185 T^{5} + 2342249 T^{6} - 18000300 T^{7} + 137833141 T^{8} - 1040546905 T^{9} + 137833141 p T^{10} - 18000300 p^{2} T^{11} + 2342249 p^{3} T^{12} - 288185 p^{4} T^{13} + 30947 p^{5} T^{14} - 56 p^{7} T^{15} + 277 p^{7} T^{16} - 21 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 - 23 T + 518 T^{2} - 8163 T^{3} + 115526 T^{4} - 1388946 T^{5} + 15076132 T^{6} - 146537445 T^{7} + 21924049 p T^{8} - 10417971342 T^{9} + 21924049 p^{2} T^{10} - 146537445 p^{2} T^{11} + 15076132 p^{3} T^{12} - 1388946 p^{4} T^{13} + 115526 p^{5} T^{14} - 8163 p^{6} T^{15} + 518 p^{7} T^{16} - 23 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 34 T + 860 T^{2} - 15381 T^{3} + 233681 T^{4} - 2948067 T^{5} + 33192584 T^{6} - 328091222 T^{7} + 2968578671 T^{8} - 24128224246 T^{9} + 2968578671 p T^{10} - 328091222 p^{2} T^{11} + 33192584 p^{3} T^{12} - 2948067 p^{4} T^{13} + 233681 p^{5} T^{14} - 15381 p^{6} T^{15} + 860 p^{7} T^{16} - 34 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 - 10 T + 350 T^{2} - 3424 T^{3} + 56851 T^{4} - 534274 T^{5} + 5808712 T^{6} - 52969882 T^{7} + 447475317 T^{8} - 3957619294 T^{9} + 447475317 p T^{10} - 52969882 p^{2} T^{11} + 5808712 p^{3} T^{12} - 534274 p^{4} T^{13} + 56851 p^{5} T^{14} - 3424 p^{6} T^{15} + 350 p^{7} T^{16} - 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 - 4 T + 386 T^{2} - 1012 T^{3} + 74320 T^{4} - 132554 T^{5} + 9430158 T^{6} - 11810548 T^{7} + 876200831 T^{8} - 881477364 T^{9} + 876200831 p T^{10} - 11810548 p^{2} T^{11} + 9430158 p^{3} T^{12} - 132554 p^{4} T^{13} + 74320 p^{5} T^{14} - 1012 p^{6} T^{15} + 386 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 - 9 T + 193 T^{2} - 663 T^{3} + 12610 T^{4} + 34115 T^{5} + 857410 T^{6} - 120365 T^{7} + 134510578 T^{8} - 427210620 T^{9} + 134510578 p T^{10} - 120365 p^{2} T^{11} + 857410 p^{3} T^{12} + 34115 p^{4} T^{13} + 12610 p^{5} T^{14} - 663 p^{6} T^{15} + 193 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 + 34 T + 954 T^{2} + 18742 T^{3} + 323460 T^{4} + 4647733 T^{5} + 60285514 T^{6} + 685620581 T^{7} + 7129338937 T^{8} + 66150889905 T^{9} + 7129338937 p T^{10} + 685620581 p^{2} T^{11} + 60285514 p^{3} T^{12} + 4647733 p^{4} T^{13} + 323460 p^{5} T^{14} + 18742 p^{6} T^{15} + 954 p^{7} T^{16} + 34 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 - 15 T + 486 T^{2} - 7454 T^{3} + 18 p^{2} T^{4} - 1678655 T^{5} + 21032850 T^{6} - 233678476 T^{7} + 2484495939 T^{8} - 22761826039 T^{9} + 2484495939 p T^{10} - 233678476 p^{2} T^{11} + 21032850 p^{3} T^{12} - 1678655 p^{4} T^{13} + 18 p^{7} T^{14} - 7454 p^{6} T^{15} + 486 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 263 T^{2} - 80 T^{3} + 42883 T^{4} - 977 p T^{5} + 5161687 T^{6} - 14370001 T^{7} + 526563995 T^{8} - 1611468791 T^{9} + 526563995 p T^{10} - 14370001 p^{2} T^{11} + 5161687 p^{3} T^{12} - 977 p^{5} T^{13} + 42883 p^{5} T^{14} - 80 p^{6} T^{15} + 263 p^{7} T^{16} + p^{9} T^{18} \) | |
| 97 | \( 1 - 15 T + 783 T^{2} - 9737 T^{3} + 277122 T^{4} - 2896117 T^{5} + 58631534 T^{6} - 517666299 T^{7} + 8231099808 T^{8} - 61045311040 T^{9} + 8231099808 p T^{10} - 517666299 p^{2} T^{11} + 58631534 p^{3} T^{12} - 2896117 p^{4} T^{13} + 277122 p^{5} T^{14} - 9737 p^{6} T^{15} + 783 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.04195272199534648285815800684, −2.97483367150130843695231480684, −2.91402410924748651064377223770, −2.89796323006007384461581430595, −2.79816978116135269120008604209, −2.63701849096992149493706604659, −2.55564536526413268330600091848, −2.20502514087907679249739438768, −2.20260720843351760001205810270, −2.17551736353612423281819182216, −2.12211351824731037292838855785, −2.11856274965806966026566009313, −2.11281512662127715082794187653, −1.87865606246870239840851066363, −1.67462712621828566996455289811, −1.52639234232923929506575501988, −1.40584260163237271505371678574, −1.11084746127321322104108706742, −1.00045760151462902103448476490, −0.907078862581200667301849797417, −0.893294190739507390801075173165, −0.789509665780532702853107503053, −0.61368916717975188123239548961, −0.60909132930974332063683414057, −0.54992757691179297810048036171, 0.54992757691179297810048036171, 0.60909132930974332063683414057, 0.61368916717975188123239548961, 0.789509665780532702853107503053, 0.893294190739507390801075173165, 0.907078862581200667301849797417, 1.00045760151462902103448476490, 1.11084746127321322104108706742, 1.40584260163237271505371678574, 1.52639234232923929506575501988, 1.67462712621828566996455289811, 1.87865606246870239840851066363, 2.11281512662127715082794187653, 2.11856274965806966026566009313, 2.12211351824731037292838855785, 2.17551736353612423281819182216, 2.20260720843351760001205810270, 2.20502514087907679249739438768, 2.55564536526413268330600091848, 2.63701849096992149493706604659, 2.79816978116135269120008604209, 2.89796323006007384461581430595, 2.91402410924748651064377223770, 2.97483367150130843695231480684, 3.04195272199534648285815800684
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.