Dirichlet series
| L(s) = 1 | + 2·2-s + 4·3-s − 2·4-s − 10·5-s + 8·6-s + 10·7-s − 6·8-s − 4·9-s − 20·10-s − 8·12-s + 26·13-s + 20·14-s − 40·15-s + 16-s + 4·17-s − 8·18-s + 6·19-s + 20·20-s + 40·21-s − 8·23-s − 24·24-s + 55·25-s + 52·26-s − 38·27-s − 20·28-s − 4·29-s − 80·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 2.30·3-s − 4-s − 4.47·5-s + 3.26·6-s + 3.77·7-s − 2.12·8-s − 4/3·9-s − 6.32·10-s − 2.30·12-s + 7.21·13-s + 5.34·14-s − 10.3·15-s + 1/4·16-s + 0.970·17-s − 1.88·18-s + 1.37·19-s + 4.47·20-s + 8.72·21-s − 1.66·23-s − 4.89·24-s + 11·25-s + 10.1·26-s − 7.31·27-s − 3.77·28-s − 0.742·29-s − 14.6·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 7^{10} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 7^{10} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(5^{10} \cdot 7^{10} \cdot 11^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.95571\times 10^{15}\) |
| Root analytic conductor: | \(5.81520\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 5^{10} \cdot 7^{10} \cdot 11^{20} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(193.6456249\) |
| \(L(\frac12)\) | \(\approx\) | \(193.6456249\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + T )^{10} \) |
| 7 | \( ( 1 - T )^{10} \) | |
| 11 | \( 1 \) | |
| good | 2 | \( 1 - p T + 3 p T^{2} - 5 p T^{3} + 19 T^{4} - 15 p T^{5} + 13 p^{2} T^{6} - 5 p^{4} T^{7} + 69 p T^{8} - 95 p T^{9} + 301 T^{10} - 95 p^{2} T^{11} + 69 p^{3} T^{12} - 5 p^{7} T^{13} + 13 p^{6} T^{14} - 15 p^{6} T^{15} + 19 p^{6} T^{16} - 5 p^{8} T^{17} + 3 p^{9} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) |
| 3 | \( 1 - 4 T + 20 T^{2} - 58 T^{3} + 178 T^{4} - 412 T^{5} + 37 p^{3} T^{6} - 664 p T^{7} + 1394 p T^{8} - 7448 T^{9} + 13969 T^{10} - 7448 p T^{11} + 1394 p^{3} T^{12} - 664 p^{4} T^{13} + 37 p^{7} T^{14} - 412 p^{5} T^{15} + 178 p^{6} T^{16} - 58 p^{7} T^{17} + 20 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 2 p T + 376 T^{2} - 3938 T^{3} + 32983 T^{4} - 231826 T^{5} + 108167 p T^{6} - 7491882 T^{7} + 35459780 T^{8} - 150137712 T^{9} + 570875139 T^{10} - 150137712 p T^{11} + 35459780 p^{2} T^{12} - 7491882 p^{3} T^{13} + 108167 p^{5} T^{14} - 231826 p^{5} T^{15} + 32983 p^{6} T^{16} - 3938 p^{7} T^{17} + 376 p^{8} T^{18} - 2 p^{10} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 4 T + 88 T^{2} - 342 T^{3} + 3813 T^{4} - 15066 T^{5} + 107997 T^{6} - 448782 T^{7} + 2334822 T^{8} - 9951724 T^{9} + 42429315 T^{10} - 9951724 p T^{11} + 2334822 p^{2} T^{12} - 448782 p^{3} T^{13} + 107997 p^{4} T^{14} - 15066 p^{5} T^{15} + 3813 p^{6} T^{16} - 342 p^{7} T^{17} + 88 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 6 T + 111 T^{2} - 690 T^{3} + 6506 T^{4} - 37648 T^{5} + 255176 T^{6} - 1324842 T^{7} + 7263766 T^{8} - 33630196 T^{9} + 156978173 T^{10} - 33630196 p T^{11} + 7263766 p^{2} T^{12} - 1324842 p^{3} T^{13} + 255176 p^{4} T^{14} - 37648 p^{5} T^{15} + 6506 p^{6} T^{16} - 690 p^{7} T^{17} + 111 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 + 8 T + 165 T^{2} + 44 p T^{3} + 12145 T^{4} + 60334 T^{5} + 552581 T^{6} + 2316570 T^{7} + 18068419 T^{8} + 66302664 T^{9} + 462596279 T^{10} + 66302664 p T^{11} + 18068419 p^{2} T^{12} + 2316570 p^{3} T^{13} + 552581 p^{4} T^{14} + 60334 p^{5} T^{15} + 12145 p^{6} T^{16} + 44 p^{8} T^{17} + 165 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 4 T + 175 T^{2} + 972 T^{3} + 15441 T^{4} + 98052 T^{5} + 937299 T^{6} + 5770656 T^{7} + 42447957 T^{8} + 230193832 T^{9} + 1432095303 T^{10} + 230193832 p T^{11} + 42447957 p^{2} T^{12} + 5770656 p^{3} T^{13} + 937299 p^{4} T^{14} + 98052 p^{5} T^{15} + 15441 p^{6} T^{16} + 972 p^{7} T^{17} + 175 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 18 T + 280 T^{2} - 3108 T^{3} + 32250 T^{4} - 279200 T^{5} + 2286609 T^{6} - 16467316 T^{7} + 112788486 T^{8} - 693061076 T^{9} + 4062485173 T^{10} - 693061076 p T^{11} + 112788486 p^{2} T^{12} - 16467316 p^{3} T^{13} + 2286609 p^{4} T^{14} - 279200 p^{5} T^{15} + 32250 p^{6} T^{16} - 3108 p^{7} T^{17} + 280 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 16 T + 342 T^{2} + 3826 T^{3} + 49096 T^{4} + 433906 T^{5} + 4213787 T^{6} + 31094278 T^{7} + 248210900 T^{8} + 1568496998 T^{9} + 10660834969 T^{10} + 1568496998 p T^{11} + 248210900 p^{2} T^{12} + 31094278 p^{3} T^{13} + 4213787 p^{4} T^{14} + 433906 p^{5} T^{15} + 49096 p^{6} T^{16} + 3826 p^{7} T^{17} + 342 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 30 T + 673 T^{2} + 10784 T^{3} + 147197 T^{4} + 1686378 T^{5} + 17204279 T^{6} + 154852480 T^{7} + 1265651769 T^{8} + 9308164996 T^{9} + 62659913547 T^{10} + 9308164996 p T^{11} + 1265651769 p^{2} T^{12} + 154852480 p^{3} T^{13} + 17204279 p^{4} T^{14} + 1686378 p^{5} T^{15} + 147197 p^{6} T^{16} + 10784 p^{7} T^{17} + 673 p^{8} T^{18} + 30 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 22 T + 408 T^{2} - 5642 T^{3} + 69286 T^{4} - 725812 T^{5} + 7002509 T^{6} - 60496134 T^{7} + 485843768 T^{8} - 3560204044 T^{9} + 24408114293 T^{10} - 3560204044 p T^{11} + 485843768 p^{2} T^{12} - 60496134 p^{3} T^{13} + 7002509 p^{4} T^{14} - 725812 p^{5} T^{15} + 69286 p^{6} T^{16} - 5642 p^{7} T^{17} + 408 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 14 T + 454 T^{2} - 4958 T^{3} + 90420 T^{4} - 808538 T^{5} + 10690780 T^{6} - 1707598 p T^{7} + 844122391 T^{8} - 5374235994 T^{9} + 46918909069 T^{10} - 5374235994 p T^{11} + 844122391 p^{2} T^{12} - 1707598 p^{4} T^{13} + 10690780 p^{4} T^{14} - 808538 p^{5} T^{15} + 90420 p^{6} T^{16} - 4958 p^{7} T^{17} + 454 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 10 T + 260 T^{2} + 1906 T^{3} + 29952 T^{4} + 177306 T^{5} + 2243675 T^{6} + 11391928 T^{7} + 131097296 T^{8} + 605066540 T^{9} + 6956647477 T^{10} + 605066540 p T^{11} + 131097296 p^{2} T^{12} + 11391928 p^{3} T^{13} + 2243675 p^{4} T^{14} + 177306 p^{5} T^{15} + 29952 p^{6} T^{16} + 1906 p^{7} T^{17} + 260 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 22 T + 646 T^{2} - 10704 T^{3} + 180891 T^{4} - 2365002 T^{5} + 29344155 T^{6} - 311547738 T^{7} + 3077950896 T^{8} - 26970499972 T^{9} + 218980839399 T^{10} - 26970499972 p T^{11} + 3077950896 p^{2} T^{12} - 311547738 p^{3} T^{13} + 29344155 p^{4} T^{14} - 2365002 p^{5} T^{15} + 180891 p^{6} T^{16} - 10704 p^{7} T^{17} + 646 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 12 T + 330 T^{2} - 2854 T^{3} + 48580 T^{4} - 5456 p T^{5} + 4827146 T^{6} - 29712354 T^{7} + 399498407 T^{8} - 2294645898 T^{9} + 27415915441 T^{10} - 2294645898 p T^{11} + 399498407 p^{2} T^{12} - 29712354 p^{3} T^{13} + 4827146 p^{4} T^{14} - 5456 p^{6} T^{15} + 48580 p^{6} T^{16} - 2854 p^{7} T^{17} + 330 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 14 T + 557 T^{2} + 7024 T^{3} + 146228 T^{4} + 1638124 T^{5} + 23719536 T^{6} + 233367328 T^{7} + 2626451572 T^{8} + 22388086782 T^{9} + 207201675905 T^{10} + 22388086782 p T^{11} + 2626451572 p^{2} T^{12} + 233367328 p^{3} T^{13} + 23719536 p^{4} T^{14} + 1638124 p^{5} T^{15} + 146228 p^{6} T^{16} + 7024 p^{7} T^{17} + 557 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 8 T + 416 T^{2} + 2750 T^{3} + 83604 T^{4} + 418194 T^{5} + 10565057 T^{6} + 38459282 T^{7} + 973762874 T^{8} + 2677597648 T^{9} + 74082485569 T^{10} + 2677597648 p T^{11} + 973762874 p^{2} T^{12} + 38459282 p^{3} T^{13} + 10565057 p^{4} T^{14} + 418194 p^{5} T^{15} + 83604 p^{6} T^{16} + 2750 p^{7} T^{17} + 416 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 30 T + 821 T^{2} - 14670 T^{3} + 241420 T^{4} - 3194744 T^{5} + 39822570 T^{6} - 429862700 T^{7} + 4448923394 T^{8} - 41251518108 T^{9} + 370409243969 T^{10} - 41251518108 p T^{11} + 4448923394 p^{2} T^{12} - 429862700 p^{3} T^{13} + 39822570 p^{4} T^{14} - 3194744 p^{5} T^{15} + 241420 p^{6} T^{16} - 14670 p^{7} T^{17} + 821 p^{8} T^{18} - 30 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 8 T + 441 T^{2} - 3512 T^{3} + 94240 T^{4} - 703608 T^{5} + 13147446 T^{6} - 87746244 T^{7} + 1378910660 T^{8} - 8157965776 T^{9} + 118462021641 T^{10} - 8157965776 p T^{11} + 1378910660 p^{2} T^{12} - 87746244 p^{3} T^{13} + 13147446 p^{4} T^{14} - 703608 p^{5} T^{15} + 94240 p^{6} T^{16} - 3512 p^{7} T^{17} + 441 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 14 T + 254 T^{2} + 890 T^{3} + 19233 T^{4} + 7320 T^{5} + 2675219 T^{6} + 2937158 T^{7} + 237998930 T^{8} - 887116970 T^{9} + 12647182171 T^{10} - 887116970 p T^{11} + 237998930 p^{2} T^{12} + 2937158 p^{3} T^{13} + 2675219 p^{4} T^{14} + 7320 p^{5} T^{15} + 19233 p^{6} T^{16} + 890 p^{7} T^{17} + 254 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 6 T + 371 T^{2} - 710 T^{3} + 48696 T^{4} - 629490 T^{5} + 5214703 T^{6} - 99655928 T^{7} + 881406613 T^{8} - 8498462752 T^{9} + 108272614401 T^{10} - 8498462752 p T^{11} + 881406613 p^{2} T^{12} - 99655928 p^{3} T^{13} + 5214703 p^{4} T^{14} - 629490 p^{5} T^{15} + 48696 p^{6} T^{16} - 710 p^{7} T^{17} + 371 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 20 T + 709 T^{2} - 11702 T^{3} + 236826 T^{4} - 3382804 T^{5} + 50193737 T^{6} - 630846688 T^{7} + 7541714553 T^{8} - 83399334306 T^{9} + 842737798505 T^{10} - 83399334306 p T^{11} + 7541714553 p^{2} T^{12} - 630846688 p^{3} T^{13} + 50193737 p^{4} T^{14} - 3382804 p^{5} T^{15} + 236826 p^{6} T^{16} - 11702 p^{7} T^{17} + 709 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−2.98333758489127559594962946326, −2.96852650180932388395407422222, −2.93492358937988770904968052043, −2.74883976525658666514377621724, −2.55259181724287692734521927461, −2.50699408051433320583801139168, −2.39328336598775937674251831488, −2.39034465556301703644960959211, −2.22958390394446801828494319476, −1.90604770621501887547685506611, −1.76370961733339695974179418142, −1.71347165564865721890808193080, −1.67763621142628704183540647523, −1.65355855643117409325222347730, −1.64515909937606493461695423566, −1.47888726743735762614509903287, −1.15063305062422092564151444753, −1.06931384510940210284666866534, −0.839993532939205090817854811733, −0.833929858553142429266822670416, −0.807631661386841966517022149276, −0.55707509551698384075036704194, −0.48620824723451976600370813515, −0.44350679513256187957244364847, −0.44262364938780428290496637483, 0.44262364938780428290496637483, 0.44350679513256187957244364847, 0.48620824723451976600370813515, 0.55707509551698384075036704194, 0.807631661386841966517022149276, 0.833929858553142429266822670416, 0.839993532939205090817854811733, 1.06931384510940210284666866534, 1.15063305062422092564151444753, 1.47888726743735762614509903287, 1.64515909937606493461695423566, 1.65355855643117409325222347730, 1.67763621142628704183540647523, 1.71347165564865721890808193080, 1.76370961733339695974179418142, 1.90604770621501887547685506611, 2.22958390394446801828494319476, 2.39034465556301703644960959211, 2.39328336598775937674251831488, 2.50699408051433320583801139168, 2.55259181724287692734521927461, 2.74883976525658666514377621724, 2.93492358937988770904968052043, 2.96852650180932388395407422222, 2.98333758489127559594962946326
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.