Dirichlet series
| L(s) = 1 | − 5·2-s + 10·4-s − 5·5-s − 5·8-s + 3·9-s + 25·10-s − 5·11-s − 2·13-s − 20·16-s − 15·18-s − 4·19-s − 50·20-s + 25·22-s − 7·23-s + 10·25-s + 10·26-s − 6·27-s + 8·29-s − 6·31-s + 49·32-s + 30·36-s − 16·37-s + 20·38-s + 25·40-s − 2·41-s + 26·43-s − 50·44-s + ⋯ |
| L(s) = 1 | − 3.53·2-s + 5·4-s − 2.23·5-s − 1.76·8-s + 9-s + 7.90·10-s − 1.50·11-s − 0.554·13-s − 5·16-s − 3.53·18-s − 0.917·19-s − 11.1·20-s + 5.33·22-s − 1.45·23-s + 2·25-s + 1.96·26-s − 1.15·27-s + 1.48·29-s − 1.07·31-s + 8.66·32-s + 5·36-s − 2.63·37-s + 3.24·38-s + 3.95·40-s − 0.312·41-s + 3.96·43-s − 7.53·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 5^{10} \cdot 7^{10} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 5^{10} \cdot 7^{10} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 5^{10} \cdot 7^{10} \cdot 11^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.72108\times 10^{7}\) |
| Root analytic conductor: | \(2.47961\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 5^{10} \cdot 7^{10} \cdot 11^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.07295702804\) |
| \(L(\frac12)\) | \(\approx\) | \(0.07295702804\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{5} \) |
| 5 | \( ( 1 + T + T^{2} )^{5} \) | |
| 7 | \( 1 - p T^{2} - 25 T^{3} + p^{2} T^{4} + 139 T^{5} + p^{3} T^{6} - 25 p^{2} T^{7} - p^{4} T^{8} + p^{5} T^{10} \) | |
| 11 | \( ( 1 + T + T^{2} )^{5} \) | |
| good | 3 | \( 1 - p T^{2} + 2 p T^{3} - 4 T^{4} - 20 T^{5} + 16 p T^{6} - 62 T^{8} + 10 T^{9} - 224 T^{10} + 10 p T^{11} - 62 p^{2} T^{12} + 16 p^{5} T^{14} - 20 p^{5} T^{15} - 4 p^{6} T^{16} + 2 p^{8} T^{17} - p^{9} T^{18} + p^{10} T^{20} \) |
| 13 | \( ( 1 + T + 37 T^{2} + 32 T^{3} + 766 T^{4} + 638 T^{5} + 766 p T^{6} + 32 p^{2} T^{7} + 37 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 17 | \( 1 - 42 T^{2} + 150 T^{3} + 895 T^{4} - 5476 T^{5} - 1327 T^{6} + 108534 T^{7} - 246101 T^{8} - 715694 T^{9} + 6860121 T^{10} - 715694 p T^{11} - 246101 p^{2} T^{12} + 108534 p^{3} T^{13} - 1327 p^{4} T^{14} - 5476 p^{5} T^{15} + 895 p^{6} T^{16} + 150 p^{7} T^{17} - 42 p^{8} T^{18} + p^{10} T^{20} \) | |
| 19 | \( 1 + 4 T - 58 T^{2} - 162 T^{3} + 2057 T^{4} + 160 p T^{5} - 62139 T^{6} - 59380 T^{7} + 1498555 T^{8} + 33550 p T^{9} - 1577785 p T^{10} + 33550 p^{2} T^{11} + 1498555 p^{2} T^{12} - 59380 p^{3} T^{13} - 62139 p^{4} T^{14} + 160 p^{6} T^{15} + 2057 p^{6} T^{16} - 162 p^{7} T^{17} - 58 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 + 7 T - 50 T^{2} - 219 T^{3} + 2735 T^{4} + 4430 T^{5} - 92040 T^{6} - 45874 T^{7} + 95363 p T^{8} - 481663 T^{9} - 53985826 T^{10} - 481663 p T^{11} + 95363 p^{3} T^{12} - 45874 p^{3} T^{13} - 92040 p^{4} T^{14} + 4430 p^{5} T^{15} + 2735 p^{6} T^{16} - 219 p^{7} T^{17} - 50 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( ( 1 - 4 T + 71 T^{2} - 265 T^{3} + 111 p T^{4} - 8959 T^{5} + 111 p^{2} T^{6} - 265 p^{2} T^{7} + 71 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 31 | \( 1 + 6 T - 106 T^{2} - 502 T^{3} + 7611 T^{4} + 24660 T^{5} - 403459 T^{6} - 660382 T^{7} + 17703555 T^{8} + 8562458 T^{9} - 610673415 T^{10} + 8562458 p T^{11} + 17703555 p^{2} T^{12} - 660382 p^{3} T^{13} - 403459 p^{4} T^{14} + 24660 p^{5} T^{15} + 7611 p^{6} T^{16} - 502 p^{7} T^{17} - 106 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 16 T + 76 T^{2} - 210 T^{3} - 4681 T^{4} - 36562 T^{5} - 217481 T^{6} - 589838 T^{7} + 4060831 T^{8} + 53274548 T^{9} + 355150225 T^{10} + 53274548 p T^{11} + 4060831 p^{2} T^{12} - 589838 p^{3} T^{13} - 217481 p^{4} T^{14} - 36562 p^{5} T^{15} - 4681 p^{6} T^{16} - 210 p^{7} T^{17} + 76 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( ( 1 + T + 45 T^{2} - 208 T^{3} + 450 T^{4} - 24002 T^{5} + 450 p T^{6} - 208 p^{2} T^{7} + 45 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 43 | \( ( 1 - 13 T + 247 T^{2} - 2200 T^{3} + 22530 T^{4} - 141142 T^{5} + 22530 p T^{6} - 2200 p^{2} T^{7} + 247 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 47 | \( 1 + 8 T - 87 T^{2} - 600 T^{3} + 3883 T^{4} + 8888 T^{5} - 323766 T^{6} - 971704 T^{7} + 16239317 T^{8} + 49532048 T^{9} - 507170701 T^{10} + 49532048 p T^{11} + 16239317 p^{2} T^{12} - 971704 p^{3} T^{13} - 323766 p^{4} T^{14} + 8888 p^{5} T^{15} + 3883 p^{6} T^{16} - 600 p^{7} T^{17} - 87 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 4 T - 76 T^{2} + 258 T^{3} + 285 T^{4} - 38010 T^{5} + 32043 T^{6} - 1511220 T^{7} + 10943763 T^{8} + 123446776 T^{9} - 965629247 T^{10} + 123446776 p T^{11} + 10943763 p^{2} T^{12} - 1511220 p^{3} T^{13} + 32043 p^{4} T^{14} - 38010 p^{5} T^{15} + 285 p^{6} T^{16} + 258 p^{7} T^{17} - 76 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 9 T - 108 T^{2} - 9 p T^{3} + 8317 T^{4} - 15502 T^{5} - 761590 T^{6} - 51696 T^{7} + 42861673 T^{8} + 73257385 T^{9} - 1734691842 T^{10} + 73257385 p T^{11} + 42861673 p^{2} T^{12} - 51696 p^{3} T^{13} - 761590 p^{4} T^{14} - 15502 p^{5} T^{15} + 8317 p^{6} T^{16} - 9 p^{8} T^{17} - 108 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 2 T - 163 T^{2} - 324 T^{3} + 12638 T^{4} + 26858 T^{5} - 609018 T^{6} - 2310206 T^{7} + 21323344 T^{8} + 81115550 T^{9} - 715190428 T^{10} + 81115550 p T^{11} + 21323344 p^{2} T^{12} - 2310206 p^{3} T^{13} - 609018 p^{4} T^{14} + 26858 p^{5} T^{15} + 12638 p^{6} T^{16} - 324 p^{7} T^{17} - 163 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 5 T - 272 T^{2} - 879 T^{3} + 44738 T^{4} + 89591 T^{5} - 5296054 T^{6} - 5517283 T^{7} + 492213997 T^{8} + 159267050 T^{9} - 36561572532 T^{10} + 159267050 p T^{11} + 492213997 p^{2} T^{12} - 5517283 p^{3} T^{13} - 5296054 p^{4} T^{14} + 89591 p^{5} T^{15} + 44738 p^{6} T^{16} - 879 p^{7} T^{17} - 272 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( ( 1 - 28 T + 542 T^{2} - 7453 T^{3} + 83509 T^{4} - 763486 T^{5} + 83509 p T^{6} - 7453 p^{2} T^{7} + 542 p^{3} T^{8} - 28 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 73 | \( 1 - T - 93 T^{2} - 896 T^{3} + 11646 T^{4} + 96664 T^{5} + 99321 T^{6} - 10645367 T^{7} - 44086605 T^{8} + 2855784 p T^{9} + 7561599492 T^{10} + 2855784 p^{2} T^{11} - 44086605 p^{2} T^{12} - 10645367 p^{3} T^{13} + 99321 p^{4} T^{14} + 96664 p^{5} T^{15} + 11646 p^{6} T^{16} - 896 p^{7} T^{17} - 93 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 23 T + 140 T^{2} - 525 T^{3} - 14755 T^{4} - 245350 T^{5} - 3218034 T^{6} - 16969592 T^{7} + 114620497 T^{8} + 2280041639 T^{9} + 20390960354 T^{10} + 2280041639 p T^{11} + 114620497 p^{2} T^{12} - 16969592 p^{3} T^{13} - 3218034 p^{4} T^{14} - 245350 p^{5} T^{15} - 14755 p^{6} T^{16} - 525 p^{7} T^{17} + 140 p^{8} T^{18} + 23 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( ( 1 + 23 T + 509 T^{2} + 6764 T^{3} + 88456 T^{4} + 818602 T^{5} + 88456 p T^{6} + 6764 p^{2} T^{7} + 509 p^{3} T^{8} + 23 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 89 | \( 1 - 9 T - 231 T^{2} + 640 T^{3} + 39212 T^{4} + 51604 T^{5} - 4336732 T^{6} - 10441300 T^{7} + 352441008 T^{8} + 398422554 T^{9} - 27054430762 T^{10} + 398422554 p T^{11} + 352441008 p^{2} T^{12} - 10441300 p^{3} T^{13} - 4336732 p^{4} T^{14} + 51604 p^{5} T^{15} + 39212 p^{6} T^{16} + 640 p^{7} T^{17} - 231 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( ( 1 + 13 T + 247 T^{2} + 1724 T^{3} + 31000 T^{4} + 210014 T^{5} + 31000 p T^{6} + 1724 p^{2} T^{7} + 247 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
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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
−3.76358851857842878296958497737, −3.68325808786897709051777374179, −3.59219601509572126803160446504, −3.42213953609200855325507197110, −3.38838748930564284716243066781, −3.37092749255674851597666226186, −3.21429221262610674295723559414, −2.77923126943930636878063480524, −2.62200247349415929642191656418, −2.61770210167950801205196542829, −2.52237931212648941637985975169, −2.44719979750550486223249067719, −2.44460085450796860450684865574, −2.02037818854087896694741839382, −1.99720126832606614787343787633, −1.87118063572578099330866715146, −1.53414731940096683594148227850, −1.51720719381139141444660881168, −1.37038229232794234625686086347, −1.22154299213764863391281248332, −1.09529631868568785232808973671, −0.56517257477657132758852502137, −0.46638765305618305728479887086, −0.32015030768246258855828400863, −0.24755680280847952080129005369, 0.24755680280847952080129005369, 0.32015030768246258855828400863, 0.46638765305618305728479887086, 0.56517257477657132758852502137, 1.09529631868568785232808973671, 1.22154299213764863391281248332, 1.37038229232794234625686086347, 1.51720719381139141444660881168, 1.53414731940096683594148227850, 1.87118063572578099330866715146, 1.99720126832606614787343787633, 2.02037818854087896694741839382, 2.44460085450796860450684865574, 2.44719979750550486223249067719, 2.52237931212648941637985975169, 2.61770210167950801205196542829, 2.62200247349415929642191656418, 2.77923126943930636878063480524, 3.21429221262610674295723559414, 3.37092749255674851597666226186, 3.38838748930564284716243066781, 3.42213953609200855325507197110, 3.59219601509572126803160446504, 3.68325808786897709051777374179, 3.76358851857842878296958497737
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.