Dirichlet series
| L(s) = 1 | − 4·2-s + 3-s + 13·4-s + 5·5-s − 4·6-s − 8·7-s − 33·8-s − 20·10-s + 7·11-s + 13·12-s − 30·13-s + 32·14-s + 5·15-s + 77·16-s − 2·17-s + 10·19-s + 65·20-s − 8·21-s − 28·22-s − 23-s − 33·24-s + 27·25-s + 120·26-s − 104·28-s − 14·29-s − 20·30-s − 28·31-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 0.577·3-s + 13/2·4-s + 2.23·5-s − 1.63·6-s − 3.02·7-s − 11.6·8-s − 6.32·10-s + 2.11·11-s + 3.75·12-s − 8.32·13-s + 8.55·14-s + 1.29·15-s + 77/4·16-s − 0.485·17-s + 2.29·19-s + 14.5·20-s − 1.74·21-s − 5.96·22-s − 0.208·23-s − 6.73·24-s + 27/5·25-s + 23.5·26-s − 19.6·28-s − 2.59·29-s − 3.65·30-s − 5.02·31-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.00257787\) |
| Root analytic conductor: | \(0.742272\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{10} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1661917358\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1661917358\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 23 | \( 1 + T + T^{2} - 21 T^{3} - 219 T^{4} + 1365 T^{5} - 219 p T^{6} - 21 p^{2} T^{7} + p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 2 | \( 1 + p^{2} T + 3 T^{2} - 7 T^{3} - 3 p^{2} T^{4} - 3 p^{2} T^{5} - 35 T^{6} - 39 T^{7} + 23 p T^{8} + 119 T^{9} + 131 T^{10} + 119 p T^{11} + 23 p^{3} T^{12} - 39 p^{3} T^{13} - 35 p^{4} T^{14} - 3 p^{7} T^{15} - 3 p^{8} T^{16} - 7 p^{7} T^{17} + 3 p^{8} T^{18} + p^{11} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 - p T - 2 T^{2} + 57 T^{3} - 11 p T^{4} - 18 p^{2} T^{5} + 1051 T^{6} + 1813 T^{7} - 7896 T^{8} - 4323 T^{9} + 47631 T^{10} - 4323 p T^{11} - 7896 p^{2} T^{12} + 1813 p^{3} T^{13} + 1051 p^{4} T^{14} - 18 p^{7} T^{15} - 11 p^{7} T^{16} + 57 p^{7} T^{17} - 2 p^{8} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) | |
| 7 | \( 1 + 8 T + 5 p T^{2} + 10 p T^{3} + 18 T^{4} - 390 T^{5} - 892 T^{6} - 930 T^{7} + 212 T^{8} - 9504 T^{9} - 33571 T^{10} - 9504 p T^{11} + 212 p^{2} T^{12} - 930 p^{3} T^{13} - 892 p^{4} T^{14} - 390 p^{5} T^{15} + 18 p^{6} T^{16} + 10 p^{8} T^{17} + 5 p^{9} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 7 T + 5 T^{2} + 64 T^{3} - 239 T^{4} + 243 T^{5} + 4448 T^{6} - 26384 T^{7} + 38597 T^{8} + 113996 T^{9} - 669129 T^{10} + 113996 p T^{11} + 38597 p^{2} T^{12} - 26384 p^{3} T^{13} + 4448 p^{4} T^{14} + 243 p^{5} T^{15} - 239 p^{6} T^{16} + 64 p^{7} T^{17} + 5 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + 30 T + 425 T^{2} + 3802 T^{3} + 24319 T^{4} + 713 p^{2} T^{5} + 500055 T^{6} + 1914482 T^{7} + 7374031 T^{8} + 28798957 T^{9} + 107783391 T^{10} + 28798957 p T^{11} + 7374031 p^{2} T^{12} + 1914482 p^{3} T^{13} + 500055 p^{4} T^{14} + 713 p^{7} T^{15} + 24319 p^{6} T^{16} + 3802 p^{7} T^{17} + 425 p^{8} T^{18} + 30 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 2 T - 35 T^{2} - 214 T^{3} + 431 T^{4} + 5688 T^{5} + 9373 T^{6} - 83714 T^{7} - 326439 T^{8} + 397822 T^{9} + 6715059 T^{10} + 397822 p T^{11} - 326439 p^{2} T^{12} - 83714 p^{3} T^{13} + 9373 p^{4} T^{14} + 5688 p^{5} T^{15} + 431 p^{6} T^{16} - 214 p^{7} T^{17} - 35 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 10 T + 37 T^{2} - 224 T^{3} + 1515 T^{4} - 3535 T^{5} + 3837 T^{6} - 16998 T^{7} - 265439 T^{8} + 2145367 T^{9} - 8068919 T^{10} + 2145367 p T^{11} - 265439 p^{2} T^{12} - 16998 p^{3} T^{13} + 3837 p^{4} T^{14} - 3535 p^{5} T^{15} + 1515 p^{6} T^{16} - 224 p^{7} T^{17} + 37 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 14 T + 79 T^{2} + 282 T^{3} + 205 T^{4} - 9642 T^{5} - 75 p^{2} T^{6} - 88676 T^{7} + 844649 T^{8} + 14330294 T^{9} + 114680611 T^{10} + 14330294 p T^{11} + 844649 p^{2} T^{12} - 88676 p^{3} T^{13} - 75 p^{6} T^{14} - 9642 p^{5} T^{15} + 205 p^{6} T^{16} + 282 p^{7} T^{17} + 79 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 28 T + 269 T^{2} + 119 T^{3} - 17987 T^{4} - 131312 T^{5} - 6733 T^{6} + 4623801 T^{7} + 683578 p T^{8} - 45481198 T^{9} - 771383701 T^{10} - 45481198 p T^{11} + 683578 p^{3} T^{12} + 4623801 p^{3} T^{13} - 6733 p^{4} T^{14} - 131312 p^{5} T^{15} - 17987 p^{6} T^{16} + 119 p^{7} T^{17} + 269 p^{8} T^{18} + 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 19 T + 115 T^{2} + 520 T^{3} - 13101 T^{4} + 63491 T^{5} + 359070 T^{6} - 6076284 T^{7} + 23241523 T^{8} + 120898426 T^{9} - 1625899439 T^{10} + 120898426 p T^{11} + 23241523 p^{2} T^{12} - 6076284 p^{3} T^{13} + 359070 p^{4} T^{14} + 63491 p^{5} T^{15} - 13101 p^{6} T^{16} + 520 p^{7} T^{17} + 115 p^{8} T^{18} - 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 19 T + 155 T^{2} - 1143 T^{3} + 10918 T^{4} - 91631 T^{5} + 600318 T^{6} - 3717353 T^{7} + 26862105 T^{8} - 183442874 T^{9} + 1136819815 T^{10} - 183442874 p T^{11} + 26862105 p^{2} T^{12} - 3717353 p^{3} T^{13} + 600318 p^{4} T^{14} - 91631 p^{5} T^{15} + 10918 p^{6} T^{16} - 1143 p^{7} T^{17} + 155 p^{8} T^{18} - 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 24 T + 6 p T^{2} + 1772 T^{3} + 13900 T^{4} + 150088 T^{5} + 1402735 T^{6} + 10050602 T^{7} + 66969414 T^{8} + 483886438 T^{9} + 3400444453 T^{10} + 483886438 p T^{11} + 66969414 p^{2} T^{12} + 10050602 p^{3} T^{13} + 1402735 p^{4} T^{14} + 150088 p^{5} T^{15} + 13900 p^{6} T^{16} + 1772 p^{7} T^{17} + 6 p^{9} T^{18} + 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 - 13 T + 208 T^{2} - 1958 T^{3} + 18034 T^{4} - 129125 T^{5} + 18034 p T^{6} - 1958 p^{2} T^{7} + 208 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( 1 + T - 107 T^{2} + 445 T^{3} + 6457 T^{4} - 70093 T^{5} + 56506 T^{6} + 5497335 T^{7} - 26132342 T^{8} - 128397962 T^{9} + 2205948207 T^{10} - 128397962 p T^{11} - 26132342 p^{2} T^{12} + 5497335 p^{3} T^{13} + 56506 p^{4} T^{14} - 70093 p^{5} T^{15} + 6457 p^{6} T^{16} + 445 p^{7} T^{17} - 107 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 2 T - 22 T^{2} - 289 T^{3} + 3163 T^{4} - 43780 T^{5} + 236872 T^{6} - 1038099 T^{7} + 4999500 T^{8} - 123257282 T^{9} + 2028123943 T^{10} - 123257282 p T^{11} + 4999500 p^{2} T^{12} - 1038099 p^{3} T^{13} + 236872 p^{4} T^{14} - 43780 p^{5} T^{15} + 3163 p^{6} T^{16} - 289 p^{7} T^{17} - 22 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 30 T + 520 T^{2} - 6411 T^{3} + 61137 T^{4} - 460332 T^{5} + 2375048 T^{6} - 1406971 T^{7} - 147896090 T^{8} + 2165902942 T^{9} - 19886912607 T^{10} + 2165902942 p T^{11} - 147896090 p^{2} T^{12} - 1406971 p^{3} T^{13} + 2375048 p^{4} T^{14} - 460332 p^{5} T^{15} + 61137 p^{6} T^{16} - 6411 p^{7} T^{17} + 520 p^{8} T^{18} - 30 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 4 T - 51 T^{2} + 1902 T^{3} - 6171 T^{4} - 78902 T^{5} + 1471697 T^{6} - 3428762 T^{7} - 27566353 T^{8} + 396448910 T^{9} - 1474341681 T^{10} + 396448910 p T^{11} - 27566353 p^{2} T^{12} - 3428762 p^{3} T^{13} + 1471697 p^{4} T^{14} - 78902 p^{5} T^{15} - 6171 p^{6} T^{16} + 1902 p^{7} T^{17} - 51 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 14 T + 59 T^{2} + 888 T^{3} + 13006 T^{4} + 37174 T^{5} - 294838 T^{6} - 1123008 T^{7} - 29023298 T^{8} - 578340936 T^{9} - 5350091175 T^{10} - 578340936 p T^{11} - 29023298 p^{2} T^{12} - 1123008 p^{3} T^{13} - 294838 p^{4} T^{14} + 37174 p^{5} T^{15} + 13006 p^{6} T^{16} + 888 p^{7} T^{17} + 59 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 + 26 T + 229 T^{2} - 916 T^{3} - 41006 T^{4} - 400668 T^{5} - 969020 T^{6} + 17362110 T^{7} + 191304780 T^{8} + 474080036 T^{9} - 2635024327 T^{10} + 474080036 p T^{11} + 191304780 p^{2} T^{12} + 17362110 p^{3} T^{13} - 969020 p^{4} T^{14} - 400668 p^{5} T^{15} - 41006 p^{6} T^{16} - 916 p^{7} T^{17} + 229 p^{8} T^{18} + 26 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 20 T + 244 T^{2} - 2662 T^{3} + 28442 T^{4} - 265746 T^{5} + 2590151 T^{6} - 26295368 T^{7} + 256238658 T^{8} - 2511930928 T^{9} + 24180646357 T^{10} - 2511930928 p T^{11} + 256238658 p^{2} T^{12} - 26295368 p^{3} T^{13} + 2590151 p^{4} T^{14} - 265746 p^{5} T^{15} + 28442 p^{6} T^{16} - 2662 p^{7} T^{17} + 244 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 18 T + 109 T^{2} - 72 T^{3} - 10248 T^{4} + 168066 T^{5} - 1355830 T^{6} + 6119988 T^{7} + 37604796 T^{8} - 775038374 T^{9} + 6144118025 T^{10} - 775038374 p T^{11} + 37604796 p^{2} T^{12} + 6119988 p^{3} T^{13} - 1355830 p^{4} T^{14} + 168066 p^{5} T^{15} - 10248 p^{6} T^{16} - 72 p^{7} T^{17} + 109 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 5 T - 108 T^{2} + 1061 T^{3} + 9362 T^{4} - 308869 T^{5} + 52007 T^{6} + 26546636 T^{7} - 244910129 T^{8} - 887314323 T^{9} + 35250962075 T^{10} - 887314323 p T^{11} - 244910129 p^{2} T^{12} + 26546636 p^{3} T^{13} + 52007 p^{4} T^{14} - 308869 p^{5} T^{15} + 9362 p^{6} T^{16} + 1061 p^{7} T^{17} - 108 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 15 T + 282 T^{2} - 1939 T^{3} + 18231 T^{4} + 44616 T^{5} + 131865 T^{6} + 10371571 T^{7} + 194532548 T^{8} - 2033558595 T^{9} + 42781964191 T^{10} - 2033558595 p T^{11} + 194532548 p^{2} T^{12} + 10371571 p^{3} T^{13} + 131865 p^{4} T^{14} + 44616 p^{5} T^{15} + 18231 p^{6} T^{16} - 1939 p^{7} T^{17} + 282 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−6.46079870070064049352366716025, −6.43712617853384580873324239530, −5.96729268497104026043874092327, −5.87852267674029089422716125515, −5.76369502463848315320219951541, −5.52877607906957651857956608863, −5.40046090275397100460822961372, −5.35306336464161833200999969309, −5.04042543215895933950633249381, −4.99675200914753326960356661409, −4.93317686616211836923991637321, −4.52500105086563036338449452214, −4.21836561575609531150688297118, −3.79690007635598834504112983659, −3.69176065791913204470937826730, −3.55432385581394752105215415927, −3.29938586733823977250338832436, −2.94217272145115175623566715862, −2.66114685994115016580552222729, −2.65808088526771181268458438759, −2.40726528784854194911626272339, −2.39490186825623008111632867174, −2.26496102753342117966133774533, −1.98356146236179602396328766504, −1.22816000656139972458694353222, 1.22816000656139972458694353222, 1.98356146236179602396328766504, 2.26496102753342117966133774533, 2.39490186825623008111632867174, 2.40726528784854194911626272339, 2.65808088526771181268458438759, 2.66114685994115016580552222729, 2.94217272145115175623566715862, 3.29938586733823977250338832436, 3.55432385581394752105215415927, 3.69176065791913204470937826730, 3.79690007635598834504112983659, 4.21836561575609531150688297118, 4.52500105086563036338449452214, 4.93317686616211836923991637321, 4.99675200914753326960356661409, 5.04042543215895933950633249381, 5.35306336464161833200999969309, 5.40046090275397100460822961372, 5.52877607906957651857956608863, 5.76369502463848315320219951541, 5.87852267674029089422716125515, 5.96729268497104026043874092327, 6.43712617853384580873324239530, 6.46079870070064049352366716025
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.