Dirichlet series
| L(s) = 1 | − 4·2-s + 10·3-s + 2·4-s − 4·5-s − 40·6-s + 12·8-s + 55·9-s + 16·10-s − 2·11-s + 20·12-s − 16·13-s − 40·15-s − 17·16-s − 12·17-s − 220·18-s − 26·19-s − 8·20-s + 8·22-s + 10·23-s + 120·24-s − 10·25-s + 64·26-s + 220·27-s − 16·29-s + 160·30-s − 20·31-s − 4·32-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5.77·3-s + 4-s − 1.78·5-s − 16.3·6-s + 4.24·8-s + 55/3·9-s + 5.05·10-s − 0.603·11-s + 5.77·12-s − 4.43·13-s − 10.3·15-s − 4.25·16-s − 2.91·17-s − 51.8·18-s − 5.96·19-s − 1.78·20-s + 1.70·22-s + 2.08·23-s + 24.4·24-s − 2·25-s + 12.5·26-s + 42.3·27-s − 2.97·29-s + 29.2·30-s − 3.59·31-s − 0.707·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 7^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 7^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 7^{20} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.05694\times 10^{14}\) |
| Root analytic conductor: | \(5.19590\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(10\) |
| Selberg data: | \((20,\ 3^{10} \cdot 7^{20} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{10} \) |
| 7 | \( 1 \) | |
| 23 | \( ( 1 - T )^{10} \) | |
| good | 2 | \( 1 + p^{2} T + 7 p T^{2} + 9 p^{2} T^{3} + 85 T^{4} + 43 p^{2} T^{5} + 163 p T^{6} + 141 p^{2} T^{7} + 923 T^{8} + 353 p^{2} T^{9} + 1029 p T^{10} + 353 p^{3} T^{11} + 923 p^{2} T^{12} + 141 p^{5} T^{13} + 163 p^{5} T^{14} + 43 p^{7} T^{15} + 85 p^{6} T^{16} + 9 p^{9} T^{17} + 7 p^{9} T^{18} + p^{11} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 + 4 T + 26 T^{2} + 68 T^{3} + 52 p T^{4} + 388 T^{5} + 946 T^{6} - 804 T^{7} - 2926 T^{8} - 22048 T^{9} - 38458 T^{10} - 22048 p T^{11} - 2926 p^{2} T^{12} - 804 p^{3} T^{13} + 946 p^{4} T^{14} + 388 p^{5} T^{15} + 52 p^{7} T^{16} + 68 p^{7} T^{17} + 26 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 + 2 T + 45 T^{2} + 130 T^{3} + 1027 T^{4} + 364 p T^{5} + 17814 T^{6} + 76640 T^{7} + 267081 T^{8} + 1057026 T^{9} + 3300639 T^{10} + 1057026 p T^{11} + 267081 p^{2} T^{12} + 76640 p^{3} T^{13} + 17814 p^{4} T^{14} + 364 p^{6} T^{15} + 1027 p^{6} T^{16} + 130 p^{7} T^{17} + 45 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + 16 T + 174 T^{2} + 1408 T^{3} + 9640 T^{4} + 56736 T^{5} + 299836 T^{6} + 110048 p T^{7} + 6284464 T^{8} + 25383560 T^{9} + 95214372 T^{10} + 25383560 p T^{11} + 6284464 p^{2} T^{12} + 110048 p^{4} T^{13} + 299836 p^{4} T^{14} + 56736 p^{5} T^{15} + 9640 p^{6} T^{16} + 1408 p^{7} T^{17} + 174 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 12 T + 140 T^{2} + 1064 T^{3} + 8121 T^{4} + 49452 T^{5} + 299368 T^{6} + 1536704 T^{7} + 7834469 T^{8} + 2048756 p T^{9} + 153403828 T^{10} + 2048756 p^{2} T^{11} + 7834469 p^{2} T^{12} + 1536704 p^{3} T^{13} + 299368 p^{4} T^{14} + 49452 p^{5} T^{15} + 8121 p^{6} T^{16} + 1064 p^{7} T^{17} + 140 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 26 T + 445 T^{2} + 5494 T^{3} + 55403 T^{4} + 466964 T^{5} + 3420282 T^{6} + 21950948 T^{7} + 6610343 p T^{8} + 641704422 T^{9} + 2952231687 T^{10} + 641704422 p T^{11} + 6610343 p^{3} T^{12} + 21950948 p^{3} T^{13} + 3420282 p^{4} T^{14} + 466964 p^{5} T^{15} + 55403 p^{6} T^{16} + 5494 p^{7} T^{17} + 445 p^{8} T^{18} + 26 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 16 T + 284 T^{2} + 3072 T^{3} + 33593 T^{4} + 282628 T^{5} + 2351880 T^{6} + 16317652 T^{7} + 111371709 T^{8} + 654706724 T^{9} + 3781129948 T^{10} + 654706724 p T^{11} + 111371709 p^{2} T^{12} + 16317652 p^{3} T^{13} + 2351880 p^{4} T^{14} + 282628 p^{5} T^{15} + 33593 p^{6} T^{16} + 3072 p^{7} T^{17} + 284 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 20 T + 312 T^{2} + 108 p T^{3} + 30387 T^{4} + 218696 T^{5} + 1410 p^{2} T^{6} + 6771872 T^{7} + 29688117 T^{8} + 111033780 T^{9} + 529179102 T^{10} + 111033780 p T^{11} + 29688117 p^{2} T^{12} + 6771872 p^{3} T^{13} + 1410 p^{6} T^{14} + 218696 p^{5} T^{15} + 30387 p^{6} T^{16} + 108 p^{8} T^{17} + 312 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 8 T + 186 T^{2} - 1428 T^{3} + 475 p T^{4} - 128080 T^{5} + 1166702 T^{6} - 7651324 T^{7} + 60027389 T^{8} - 346094116 T^{9} + 2471030340 T^{10} - 346094116 p T^{11} + 60027389 p^{2} T^{12} - 7651324 p^{3} T^{13} + 1166702 p^{4} T^{14} - 128080 p^{5} T^{15} + 475 p^{7} T^{16} - 1428 p^{7} T^{17} + 186 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 22 T + 363 T^{2} + 3882 T^{3} + 36947 T^{4} + 6500 p T^{5} + 1777650 T^{6} + 8495984 T^{7} + 39359993 T^{8} + 103934710 T^{9} + 640472689 T^{10} + 103934710 p T^{11} + 39359993 p^{2} T^{12} + 8495984 p^{3} T^{13} + 1777650 p^{4} T^{14} + 6500 p^{6} T^{15} + 36947 p^{6} T^{16} + 3882 p^{7} T^{17} + 363 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 4 T + 214 T^{2} + 1464 T^{3} + 25536 T^{4} + 197368 T^{5} + 2258920 T^{6} + 16123368 T^{7} + 148278232 T^{8} + 944834252 T^{9} + 7290599032 T^{10} + 944834252 p T^{11} + 148278232 p^{2} T^{12} + 16123368 p^{3} T^{13} + 2258920 p^{4} T^{14} + 197368 p^{5} T^{15} + 25536 p^{6} T^{16} + 1464 p^{7} T^{17} + 214 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 + 6 T + 221 T^{2} + 1774 T^{3} + 28113 T^{4} + 239124 T^{5} + 2610736 T^{6} + 20340772 T^{7} + 185643566 T^{8} + 1246420184 T^{9} + 10044377014 T^{10} + 1246420184 p T^{11} + 185643566 p^{2} T^{12} + 20340772 p^{3} T^{13} + 2610736 p^{4} T^{14} + 239124 p^{5} T^{15} + 28113 p^{6} T^{16} + 1774 p^{7} T^{17} + 221 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 30 T + 641 T^{2} + 9610 T^{3} + 123946 T^{4} + 1358950 T^{5} + 13846889 T^{6} + 127809962 T^{7} + 1117553064 T^{8} + 8960384934 T^{9} + 68056322929 T^{10} + 8960384934 p T^{11} + 1117553064 p^{2} T^{12} + 127809962 p^{3} T^{13} + 13846889 p^{4} T^{14} + 1358950 p^{5} T^{15} + 123946 p^{6} T^{16} + 9610 p^{7} T^{17} + 641 p^{8} T^{18} + 30 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 42 T + 1159 T^{2} + 23006 T^{3} + 371158 T^{4} + 5005854 T^{5} + 58798975 T^{6} + 611415166 T^{7} + 5756361672 T^{8} + 49592950034 T^{9} + 395689035399 T^{10} + 49592950034 p T^{11} + 5756361672 p^{2} T^{12} + 611415166 p^{3} T^{13} + 58798975 p^{4} T^{14} + 5005854 p^{5} T^{15} + 371158 p^{6} T^{16} + 23006 p^{7} T^{17} + 1159 p^{8} T^{18} + 42 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 14 T + 247 T^{2} + 54 p T^{3} + 40478 T^{4} + 415610 T^{5} + 4386777 T^{6} + 40901182 T^{7} + 368230920 T^{8} + 3061116430 T^{9} + 25525101093 T^{10} + 3061116430 p T^{11} + 368230920 p^{2} T^{12} + 40901182 p^{3} T^{13} + 4386777 p^{4} T^{14} + 415610 p^{5} T^{15} + 40478 p^{6} T^{16} + 54 p^{8} T^{17} + 247 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 138 T^{2} - 340 T^{3} + 256 p T^{4} - 70360 T^{5} + 1518528 T^{6} - 7466580 T^{7} + 137555528 T^{8} - 578947680 T^{9} + 9641276026 T^{10} - 578947680 p T^{11} + 137555528 p^{2} T^{12} - 7466580 p^{3} T^{13} + 1518528 p^{4} T^{14} - 70360 p^{5} T^{15} + 256 p^{7} T^{16} - 340 p^{7} T^{17} + 138 p^{8} T^{18} + p^{10} T^{20} \) | |
| 71 | \( 1 - 8 T + 492 T^{2} - 4280 T^{3} + 119042 T^{4} - 1051460 T^{5} + 18649728 T^{6} - 156919980 T^{7} + 2080536932 T^{8} - 15817853464 T^{9} + 171224592828 T^{10} - 15817853464 p T^{11} + 2080536932 p^{2} T^{12} - 156919980 p^{3} T^{13} + 18649728 p^{4} T^{14} - 1051460 p^{5} T^{15} + 119042 p^{6} T^{16} - 4280 p^{7} T^{17} + 492 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 + 24 T + 8 p T^{2} + 9152 T^{3} + 142347 T^{4} + 1810120 T^{5} + 22284378 T^{6} + 241618128 T^{7} + 2495752597 T^{8} + 23462140056 T^{9} + 209097796598 T^{10} + 23462140056 p T^{11} + 2495752597 p^{2} T^{12} + 241618128 p^{3} T^{13} + 22284378 p^{4} T^{14} + 1810120 p^{5} T^{15} + 142347 p^{6} T^{16} + 9152 p^{7} T^{17} + 8 p^{9} T^{18} + 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 32 T + 818 T^{2} - 14028 T^{3} + 218615 T^{4} - 2821792 T^{5} + 35008002 T^{6} - 382750140 T^{7} + 4056045269 T^{8} - 38640584356 T^{9} + 360053260576 T^{10} - 38640584356 p T^{11} + 4056045269 p^{2} T^{12} - 382750140 p^{3} T^{13} + 35008002 p^{4} T^{14} - 2821792 p^{5} T^{15} + 218615 p^{6} T^{16} - 14028 p^{7} T^{17} + 818 p^{8} T^{18} - 32 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 28 T + 666 T^{2} + 11072 T^{3} + 176305 T^{4} + 2353556 T^{5} + 30234112 T^{6} + 342959416 T^{7} + 3760459605 T^{8} + 37214634052 T^{9} + 355901361770 T^{10} + 37214634052 p T^{11} + 3760459605 p^{2} T^{12} + 342959416 p^{3} T^{13} + 30234112 p^{4} T^{14} + 2353556 p^{5} T^{15} + 176305 p^{6} T^{16} + 11072 p^{7} T^{17} + 666 p^{8} T^{18} + 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 242 T^{2} - 1040 T^{3} + 38506 T^{4} - 251352 T^{5} + 4918134 T^{6} - 33810308 T^{7} + 600757708 T^{8} - 3191084544 T^{9} + 59690451634 T^{10} - 3191084544 p T^{11} + 600757708 p^{2} T^{12} - 33810308 p^{3} T^{13} + 4918134 p^{4} T^{14} - 251352 p^{5} T^{15} + 38506 p^{6} T^{16} - 1040 p^{7} T^{17} + 242 p^{8} T^{18} + p^{10} T^{20} \) | |
| 97 | \( 1 - 12 T + 782 T^{2} - 8644 T^{3} + 284139 T^{4} - 2897360 T^{5} + 63905988 T^{6} - 594793256 T^{7} + 9940480725 T^{8} - 82441637460 T^{9} + 1123237206882 T^{10} - 82441637460 p T^{11} + 9940480725 p^{2} T^{12} - 594793256 p^{3} T^{13} + 63905988 p^{4} T^{14} - 2897360 p^{5} T^{15} + 284139 p^{6} T^{16} - 8644 p^{7} T^{17} + 782 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−3.50433591118086768680096856838, −3.50363367256697089484738486546, −3.17361315139694884711708146518, −3.14902353674175131361057190940, −3.09838141166331010005107641267, −2.79715851705411588400788885968, −2.72956104812659916080441465036, −2.71872806474941342819360105742, −2.65345160357094621511664168725, −2.63739500271458801460355135928, −2.33533841334892703190646642089, −2.31806723616174552464593405283, −2.28623849422151230136641676540, −2.26121647272640605631100316673, −2.23126354513778769811494602465, −2.19925870703484000566330538766, −1.99026214723177515932425598026, −1.62620851574321565815562537612, −1.57786710814096676856157246516, −1.54876122945227292144329065789, −1.49203700209929031292463344720, −1.44913947015696894686454100870, −1.30162299942761007516123903862, −1.25032164717767978971388368609, −1.15741131812739791443645662712, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.15741131812739791443645662712, 1.25032164717767978971388368609, 1.30162299942761007516123903862, 1.44913947015696894686454100870, 1.49203700209929031292463344720, 1.54876122945227292144329065789, 1.57786710814096676856157246516, 1.62620851574321565815562537612, 1.99026214723177515932425598026, 2.19925870703484000566330538766, 2.23126354513778769811494602465, 2.26121647272640605631100316673, 2.28623849422151230136641676540, 2.31806723616174552464593405283, 2.33533841334892703190646642089, 2.63739500271458801460355135928, 2.65345160357094621511664168725, 2.71872806474941342819360105742, 2.72956104812659916080441465036, 2.79715851705411588400788885968, 3.09838141166331010005107641267, 3.14902353674175131361057190940, 3.17361315139694884711708146518, 3.50363367256697089484738486546, 3.50433591118086768680096856838
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.