Dirichlet series
| L(s) = 1 | + 10·3-s + 2·4-s + 6·5-s − 7-s + 55·9-s + 6·11-s + 20·12-s − 13-s + 60·15-s − 16-s + 8·17-s − 5·19-s + 12·20-s − 10·21-s − 7·23-s − 11·25-s + 220·27-s − 2·28-s − 12·29-s − 12·31-s − 32-s + 60·33-s − 6·35-s + 110·36-s − 17·37-s − 10·39-s + 13·41-s + ⋯ |
| L(s) = 1 | + 5.77·3-s + 4-s + 2.68·5-s − 0.377·7-s + 55/3·9-s + 1.80·11-s + 5.77·12-s − 0.277·13-s + 15.4·15-s − 1/4·16-s + 1.94·17-s − 1.14·19-s + 2.68·20-s − 2.18·21-s − 1.45·23-s − 2.19·25-s + 42.3·27-s − 0.377·28-s − 2.22·29-s − 2.15·31-s − 0.176·32-s + 10.4·33-s − 1.01·35-s + 55/3·36-s − 2.79·37-s − 1.60·39-s + 2.03·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 67^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 67^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 67^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(113.430\) |
| Root analytic conductor: | \(1.26688\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{10} \cdot 67^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(50.31865153\) |
| \(L(\frac12)\) | \(\approx\) | \(50.31865153\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{10} \) |
| 67 | \( 1 - 2 T + 108 T^{2} - 663 T^{3} - 609 T^{4} - 70902 T^{5} - 609 p T^{6} - 663 p^{2} T^{7} + 108 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 2 | \( 1 - p T^{2} + 5 T^{4} + T^{5} - p T^{6} - p^{2} T^{7} - 11 T^{8} - T^{9} + 31 T^{10} - p T^{11} - 11 p^{2} T^{12} - p^{5} T^{13} - p^{5} T^{14} + p^{5} T^{15} + 5 p^{6} T^{16} - p^{9} T^{18} + p^{10} T^{20} \) |
| 5 | \( ( 1 - 3 T + 19 T^{2} - 49 T^{3} + 176 T^{4} - 336 T^{5} + 176 p T^{6} - 49 p^{2} T^{7} + 19 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 7 | \( 1 + T - 25 T^{2} - 30 T^{3} + p^{3} T^{4} + 411 T^{5} - 3169 T^{6} - 3189 T^{7} + 23137 T^{8} + 9593 T^{9} - 159214 T^{10} + 9593 p T^{11} + 23137 p^{2} T^{12} - 3189 p^{3} T^{13} - 3169 p^{4} T^{14} + 411 p^{5} T^{15} + p^{9} T^{16} - 30 p^{7} T^{17} - 25 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 6 T - 2 T^{2} - 32 T^{3} + 475 T^{4} + 152 T^{5} - 668 T^{6} - 21446 T^{7} + 9413 T^{8} - 22120 T^{9} + 881362 T^{10} - 22120 p T^{11} + 9413 p^{2} T^{12} - 21446 p^{3} T^{13} - 668 p^{4} T^{14} + 152 p^{5} T^{15} + 475 p^{6} T^{16} - 32 p^{7} T^{17} - 2 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + T - 3 p T^{2} - 4 T^{3} + 873 T^{4} - 297 T^{5} - 9619 T^{6} + 9753 T^{7} + 44855 T^{8} - 59517 T^{9} + 141714 T^{10} - 59517 p T^{11} + 44855 p^{2} T^{12} + 9753 p^{3} T^{13} - 9619 p^{4} T^{14} - 297 p^{5} T^{15} + 873 p^{6} T^{16} - 4 p^{7} T^{17} - 3 p^{9} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 8 T - 5 T^{2} + 38 T^{3} + 942 T^{4} - 1960 T^{5} - 3632 T^{6} - 31122 T^{7} - 17598 T^{8} + 154944 T^{9} + 3171184 T^{10} + 154944 p T^{11} - 17598 p^{2} T^{12} - 31122 p^{3} T^{13} - 3632 p^{4} T^{14} - 1960 p^{5} T^{15} + 942 p^{6} T^{16} + 38 p^{7} T^{17} - 5 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 5 T + 9 T^{2} + 20 T^{3} - 543 T^{4} - 2347 T^{5} - 3137 T^{6} - 25261 T^{7} + 112571 T^{8} + 1056217 T^{9} + 2867534 T^{10} + 1056217 p T^{11} + 112571 p^{2} T^{12} - 25261 p^{3} T^{13} - 3137 p^{4} T^{14} - 2347 p^{5} T^{15} - 543 p^{6} T^{16} + 20 p^{7} T^{17} + 9 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 + 7 T - 28 T^{2} - 169 T^{3} + 490 T^{4} - 2635 T^{5} - 47402 T^{6} - 68213 T^{7} + 1115861 T^{8} + 3074918 T^{9} - 10160692 T^{10} + 3074918 p T^{11} + 1115861 p^{2} T^{12} - 68213 p^{3} T^{13} - 47402 p^{4} T^{14} - 2635 p^{5} T^{15} + 490 p^{6} T^{16} - 169 p^{7} T^{17} - 28 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 12 T + 49 T^{2} - 698 T^{3} - 8162 T^{4} - 29314 T^{5} + 242314 T^{6} + 2648902 T^{7} + 294046 p T^{8} - 51892474 T^{9} - 519226304 T^{10} - 51892474 p T^{11} + 294046 p^{3} T^{12} + 2648902 p^{3} T^{13} + 242314 p^{4} T^{14} - 29314 p^{5} T^{15} - 8162 p^{6} T^{16} - 698 p^{7} T^{17} + 49 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 12 T + 26 T^{2} - 608 T^{3} - 5365 T^{4} - 352 p T^{5} + 4200 p T^{6} + 1149340 T^{7} + 3094221 T^{8} - 19496512 T^{9} - 205293462 T^{10} - 19496512 p T^{11} + 3094221 p^{2} T^{12} + 1149340 p^{3} T^{13} + 4200 p^{5} T^{14} - 352 p^{6} T^{15} - 5365 p^{6} T^{16} - 608 p^{7} T^{17} + 26 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 17 T + 79 T^{2} - 250 T^{3} - 3783 T^{4} - 21237 T^{5} - 177053 T^{6} - 967703 T^{7} + 1810739 T^{8} + 55321381 T^{9} + 416535186 T^{10} + 55321381 p T^{11} + 1810739 p^{2} T^{12} - 967703 p^{3} T^{13} - 177053 p^{4} T^{14} - 21237 p^{5} T^{15} - 3783 p^{6} T^{16} - 250 p^{7} T^{17} + 79 p^{8} T^{18} + 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 13 T - 3 T^{2} + 528 T^{3} - 423 T^{4} + 633 T^{5} - 145243 T^{6} + 325119 T^{7} + 5344351 T^{8} - 15495599 T^{9} - 86063806 T^{10} - 15495599 p T^{11} + 5344351 p^{2} T^{12} + 325119 p^{3} T^{13} - 145243 p^{4} T^{14} + 633 p^{5} T^{15} - 423 p^{6} T^{16} + 528 p^{7} T^{17} - 3 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( ( 1 + 2 T + 136 T^{2} + 213 T^{3} + 9751 T^{4} + 12954 T^{5} + 9751 p T^{6} + 213 p^{2} T^{7} + 136 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 47 | \( 1 + 25 T + 229 T^{2} + 1600 T^{3} + 17233 T^{4} + 107717 T^{5} + 1759 T^{6} - 2747789 T^{7} - 27929365 T^{8} - 392253539 T^{9} - 3590510690 T^{10} - 392253539 p T^{11} - 27929365 p^{2} T^{12} - 2747789 p^{3} T^{13} + 1759 p^{4} T^{14} + 107717 p^{5} T^{15} + 17233 p^{6} T^{16} + 1600 p^{7} T^{17} + 229 p^{8} T^{18} + 25 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( ( 1 + 6 T + 156 T^{2} + 881 T^{3} + 13099 T^{4} + 61642 T^{5} + 13099 p T^{6} + 881 p^{2} T^{7} + 156 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 59 | \( ( 1 + 6 T + 170 T^{2} + 709 T^{3} + 16557 T^{4} + 62562 T^{5} + 16557 p T^{6} + 709 p^{2} T^{7} + 170 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 61 | \( 1 - 9 T - 143 T^{2} + 720 T^{3} + 17731 T^{4} - 41977 T^{5} - 1063593 T^{6} + 191383 T^{7} + 37041985 T^{8} + 39709251 T^{9} - 1285232122 T^{10} + 39709251 p T^{11} + 37041985 p^{2} T^{12} + 191383 p^{3} T^{13} - 1063593 p^{4} T^{14} - 41977 p^{5} T^{15} + 17731 p^{6} T^{16} + 720 p^{7} T^{17} - 143 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 29 T + 172 T^{2} + 227 T^{3} + 42010 T^{4} - 655063 T^{5} + 551678 T^{6} - 7889593 T^{7} + 675212805 T^{8} - 3579881474 T^{9} - 7395575892 T^{10} - 3579881474 p T^{11} + 675212805 p^{2} T^{12} - 7889593 p^{3} T^{13} + 551678 p^{4} T^{14} - 655063 p^{5} T^{15} + 42010 p^{6} T^{16} + 227 p^{7} T^{17} + 172 p^{8} T^{18} - 29 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 12 T - 243 T^{2} + 2166 T^{3} + 50962 T^{4} - 296558 T^{5} - 6805814 T^{6} + 19816122 T^{7} + 747409258 T^{8} - 754426666 T^{9} - 59977062832 T^{10} - 754426666 p T^{11} + 747409258 p^{2} T^{12} + 19816122 p^{3} T^{13} - 6805814 p^{4} T^{14} - 296558 p^{5} T^{15} + 50962 p^{6} T^{16} + 2166 p^{7} T^{17} - 243 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + T - 163 T^{2} - 2432 T^{3} + 17745 T^{4} + 386997 T^{5} + 1447855 T^{6} - 43350169 T^{7} - 351131257 T^{8} + 1211710577 T^{9} + 44054595094 T^{10} + 1211710577 p T^{11} - 351131257 p^{2} T^{12} - 43350169 p^{3} T^{13} + 1447855 p^{4} T^{14} + 386997 p^{5} T^{15} + 17745 p^{6} T^{16} - 2432 p^{7} T^{17} - 163 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 6 T - 333 T^{2} - 1472 T^{3} + 66934 T^{4} + 204696 T^{5} - 9747600 T^{6} - 15450810 T^{7} + 1133101468 T^{8} + 537045640 T^{9} - 104602238660 T^{10} + 537045640 p T^{11} + 1133101468 p^{2} T^{12} - 15450810 p^{3} T^{13} - 9747600 p^{4} T^{14} + 204696 p^{5} T^{15} + 66934 p^{6} T^{16} - 1472 p^{7} T^{17} - 333 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( ( 1 + 2 T + 38 T^{2} - 847 T^{3} + 4877 T^{4} - 52366 T^{5} + 4877 p T^{6} - 847 p^{2} T^{7} + 38 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 97 | \( 1 - 11 T - 377 T^{2} + 2738 T^{3} + 105319 T^{4} - 490863 T^{5} - 19253567 T^{6} + 44453801 T^{7} + 2768038153 T^{8} - 2124828485 T^{9} - 299323562154 T^{10} - 2124828485 p T^{11} + 2768038153 p^{2} T^{12} + 44453801 p^{3} T^{13} - 19253567 p^{4} T^{14} - 490863 p^{5} T^{15} + 105319 p^{6} T^{16} + 2738 p^{7} T^{17} - 377 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.89108159660950148230863896360, −4.65937422092524327315080307155, −4.65646907511242764279650756825, −4.21055193546763403048821059784, −4.17701499046807973345567801097, −3.94187672688033619991623523962, −3.86156171773861345670511935426, −3.80773631165421845400615602069, −3.77670281082303885224820998995, −3.58067226623907146239874404239, −3.52547736420761054589252486495, −3.38352303657382582811153811814, −3.34706000030559883175994745061, −3.04611866380702998196398389298, −2.72520902016900659549312044782, −2.62937691908371709389409993061, −2.43677869433105711311987098506, −2.24747788216582025412230321079, −2.20203839720023169456144547298, −1.93373287850054899980332920601, −1.92197515840039723400502622448, −1.87348836601426198156778167167, −1.69920071051184467238663619252, −1.46335661689208396874154308218, −1.40623896330831541420203795394, 1.40623896330831541420203795394, 1.46335661689208396874154308218, 1.69920071051184467238663619252, 1.87348836601426198156778167167, 1.92197515840039723400502622448, 1.93373287850054899980332920601, 2.20203839720023169456144547298, 2.24747788216582025412230321079, 2.43677869433105711311987098506, 2.62937691908371709389409993061, 2.72520902016900659549312044782, 3.04611866380702998196398389298, 3.34706000030559883175994745061, 3.38352303657382582811153811814, 3.52547736420761054589252486495, 3.58067226623907146239874404239, 3.77670281082303885224820998995, 3.80773631165421845400615602069, 3.86156171773861345670511935426, 3.94187672688033619991623523962, 4.17701499046807973345567801097, 4.21055193546763403048821059784, 4.65646907511242764279650756825, 4.65937422092524327315080307155, 4.89108159660950148230863896360
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.