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 + 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 − 18·25-s + 220·27-s + 16·29-s + 160·30-s + 12·31-s + 4·32-s + 20·33-s + 48·34-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 + 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 − 3.59·25-s + 42.3·27-s + 2.97·29-s + 29.2·30-s + 2.15·31-s + 0.707·32-s + 3.48·33-s + 8.23·34-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: | \(0\) |
| Selberg data: | \((20,\ 3^{10} \cdot 7^{20} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(7451.957170\) |
| \(L(\frac12)\) | \(\approx\) | \(7451.957170\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 - p^{2} T + 7 p T^{2} - 9 p^{2} T^{3} + 85 T^{4} - 43 p^{2} T^{5} + 163 p T^{6} - 139 p^{2} T^{7} + 907 T^{8} - 343 p^{2} T^{9} + 1005 p T^{10} - 343 p^{3} T^{11} + 907 p^{2} T^{12} - 139 p^{5} T^{13} + 163 p^{5} T^{14} - 43 p^{7} T^{15} + 85 p^{6} T^{16} - 9 p^{9} T^{17} + 7 p^{9} T^{18} - p^{11} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 - 4 T + 34 T^{2} - 108 T^{3} + 556 T^{4} - 1508 T^{5} + 5906 T^{6} - 13924 T^{7} + 45018 T^{8} - 93032 T^{9} + 258022 T^{10} - 93032 p T^{11} + 45018 p^{2} T^{12} - 13924 p^{3} T^{13} + 5906 p^{4} T^{14} - 1508 p^{5} T^{15} + 556 p^{6} T^{16} - 108 p^{7} T^{17} + 34 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 2 T + 53 T^{2} - 58 T^{3} + 1467 T^{4} - 868 T^{5} + 238 p^{2} T^{6} - 9448 T^{7} + 440841 T^{8} - 92042 T^{9} + 5391743 T^{10} - 92042 p T^{11} + 440841 p^{2} T^{12} - 9448 p^{3} T^{13} + 238 p^{6} T^{14} - 868 p^{5} T^{15} + 1467 p^{6} T^{16} - 58 p^{7} T^{17} + 53 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + 6 p T^{2} + 96 T^{3} + 2944 T^{4} + 6152 T^{5} + 75492 T^{6} + 183648 T^{7} + 1463056 T^{8} + 3439112 T^{9} + 21751764 T^{10} + 3439112 p T^{11} + 1463056 p^{2} T^{12} + 183648 p^{3} T^{13} + 75492 p^{4} T^{14} + 6152 p^{5} T^{15} + 2944 p^{6} T^{16} + 96 p^{7} T^{17} + 6 p^{9} T^{18} + p^{10} T^{20} \) | |
| 17 | \( 1 - 12 T + 180 T^{2} - 1576 T^{3} + 13833 T^{4} - 95140 T^{5} + 618512 T^{6} - 3478232 T^{7} + 18137749 T^{8} - 85091020 T^{9} + 367878516 T^{10} - 85091020 p T^{11} + 18137749 p^{2} T^{12} - 3478232 p^{3} T^{13} + 618512 p^{4} T^{14} - 95140 p^{5} T^{15} + 13833 p^{6} T^{16} - 1576 p^{7} T^{17} + 180 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 26 T + 429 T^{2} - 274 p T^{3} + 51371 T^{4} - 22492 p T^{5} + 3088378 T^{6} - 19663876 T^{7} + 111685989 T^{8} - 568937798 T^{9} + 2612078711 T^{10} - 568937798 p T^{11} + 111685989 p^{2} T^{12} - 19663876 p^{3} T^{13} + 3088378 p^{4} T^{14} - 22492 p^{6} T^{15} + 51371 p^{6} T^{16} - 274 p^{8} T^{17} + 429 p^{8} T^{18} - 26 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 16 T + 300 T^{2} - 3064 T^{3} + 33545 T^{4} - 255228 T^{5} + 2079528 T^{6} - 12799412 T^{7} + 86607901 T^{8} - 460942964 T^{9} + 2777917388 T^{10} - 460942964 p T^{11} + 86607901 p^{2} T^{12} - 12799412 p^{3} T^{13} + 2079528 p^{4} T^{14} - 255228 p^{5} T^{15} + 33545 p^{6} T^{16} - 3064 p^{7} T^{17} + 300 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 12 T + 208 T^{2} - 1676 T^{3} + 18395 T^{4} - 121848 T^{5} + 1082170 T^{6} - 6297744 T^{7} + 48334917 T^{8} - 251240140 T^{9} + 1694559822 T^{10} - 251240140 p T^{11} + 48334917 p^{2} T^{12} - 6297744 p^{3} T^{13} + 1082170 p^{4} T^{14} - 121848 p^{5} T^{15} + 18395 p^{6} T^{16} - 1676 p^{7} T^{17} + 208 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 8 T + 258 T^{2} + 1604 T^{3} + 31055 T^{4} + 159736 T^{5} + 2392566 T^{6} + 10515492 T^{7} + 132762765 T^{8} + 507746604 T^{9} + 5591956836 T^{10} + 507746604 p T^{11} + 132762765 p^{2} T^{12} + 10515492 p^{3} T^{13} + 2392566 p^{4} T^{14} + 159736 p^{5} T^{15} + 31055 p^{6} T^{16} + 1604 p^{7} T^{17} + 258 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 10 T + 219 T^{2} - 38 p T^{3} + 20867 T^{4} - 110252 T^{5} + 1223250 T^{6} - 4989184 T^{7} + 54793657 T^{8} - 187156698 T^{9} + 2252707265 T^{10} - 187156698 p T^{11} + 54793657 p^{2} T^{12} - 4989184 p^{3} T^{13} + 1223250 p^{4} T^{14} - 110252 p^{5} T^{15} + 20867 p^{6} T^{16} - 38 p^{8} T^{17} + 219 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 4 T + 182 T^{2} + 872 T^{3} + 18608 T^{4} + 96912 T^{5} + 1399440 T^{6} + 7144384 T^{7} + 83139024 T^{8} + 391881500 T^{9} + 3968877576 T^{10} + 391881500 p T^{11} + 83139024 p^{2} T^{12} + 7144384 p^{3} T^{13} + 1399440 p^{4} T^{14} + 96912 p^{5} T^{15} + 18608 p^{6} T^{16} + 872 p^{7} T^{17} + 182 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 2 T + 165 T^{2} - 58 T^{3} + 15617 T^{4} + 9860 T^{5} + 1034576 T^{6} + 1434836 T^{7} + 57608062 T^{8} + 94245240 T^{9} + 2822859238 T^{10} + 94245240 p T^{11} + 57608062 p^{2} T^{12} + 1434836 p^{3} T^{13} + 1034576 p^{4} T^{14} + 9860 p^{5} T^{15} + 15617 p^{6} T^{16} - 58 p^{7} T^{17} + 165 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 - 14 T + 369 T^{2} - 3682 T^{3} + 59098 T^{4} - 480686 T^{5} + 6072577 T^{6} - 42730682 T^{7} + 460709288 T^{8} - 2869013198 T^{9} + 27334783233 T^{10} - 2869013198 p T^{11} + 460709288 p^{2} T^{12} - 42730682 p^{3} T^{13} + 6072577 p^{4} T^{14} - 480686 p^{5} T^{15} + 59098 p^{6} T^{16} - 3682 p^{7} T^{17} + 369 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 38 T + 1031 T^{2} - 19698 T^{3} + 314638 T^{4} - 4181578 T^{5} + 49326103 T^{6} - 514101730 T^{7} + 4903669192 T^{8} - 42457689022 T^{9} + 340951259991 T^{10} - 42457689022 p T^{11} + 4903669192 p^{2} T^{12} - 514101730 p^{3} T^{13} + 49326103 p^{4} T^{14} - 4181578 p^{5} T^{15} + 314638 p^{6} T^{16} - 19698 p^{7} T^{17} + 1031 p^{8} T^{18} - 38 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 14 T + 559 T^{2} - 6558 T^{3} + 140846 T^{4} - 1408346 T^{5} + 21327609 T^{6} - 183175262 T^{7} + 2170985152 T^{8} - 15994130670 T^{9} + 156288538549 T^{10} - 15994130670 p T^{11} + 2170985152 p^{2} T^{12} - 183175262 p^{3} T^{13} + 21327609 p^{4} T^{14} - 1408346 p^{5} T^{15} + 140846 p^{6} T^{16} - 6558 p^{7} T^{17} + 559 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 6 p T^{2} - 508 T^{3} + 78312 T^{4} - 180408 T^{5} + 10068272 T^{6} - 29521420 T^{7} + 967063496 T^{8} - 2949275496 T^{9} + 72798638602 T^{10} - 2949275496 p T^{11} + 967063496 p^{2} T^{12} - 29521420 p^{3} T^{13} + 10068272 p^{4} T^{14} - 180408 p^{5} T^{15} + 78312 p^{6} T^{16} - 508 p^{7} T^{17} + 6 p^{9} T^{18} + p^{10} T^{20} \) | |
| 71 | \( 1 - 24 T + 524 T^{2} - 6624 T^{3} + 84930 T^{4} - 781148 T^{5} + 8521216 T^{6} - 72982964 T^{7} + 785496516 T^{8} - 6447287672 T^{9} + 62929081596 T^{10} - 6447287672 p T^{11} + 785496516 p^{2} T^{12} - 72982964 p^{3} T^{13} + 8521216 p^{4} T^{14} - 781148 p^{5} T^{15} + 84930 p^{6} T^{16} - 6624 p^{7} T^{17} + 524 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 8 T + 592 T^{2} - 5072 T^{3} + 162515 T^{4} - 1443448 T^{5} + 27512994 T^{6} - 241803056 T^{7} + 3214264741 T^{8} - 26261195000 T^{9} + 273013980742 T^{10} - 26261195000 p T^{11} + 3214264741 p^{2} T^{12} - 241803056 p^{3} T^{13} + 27512994 p^{4} T^{14} - 1443448 p^{5} T^{15} + 162515 p^{6} T^{16} - 5072 p^{7} T^{17} + 592 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 16 T + 618 T^{2} + 6140 T^{3} + 135279 T^{4} + 660264 T^{5} + 12233402 T^{6} - 36646124 T^{7} + 170326597 T^{8} - 13709505108 T^{9} - 34377725056 T^{10} - 13709505108 p T^{11} + 170326597 p^{2} T^{12} - 36646124 p^{3} T^{13} + 12233402 p^{4} T^{14} + 660264 p^{5} T^{15} + 135279 p^{6} T^{16} + 6140 p^{7} T^{17} + 618 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 28 T + 946 T^{2} - 18928 T^{3} + 372113 T^{4} - 5772732 T^{5} + 83636824 T^{6} - 1050612080 T^{7} + 12164172229 T^{8} - 126560579516 T^{9} + 1210530130186 T^{10} - 126560579516 p T^{11} + 12164172229 p^{2} T^{12} - 1050612080 p^{3} T^{13} + 83636824 p^{4} T^{14} - 5772732 p^{5} T^{15} + 372113 p^{6} T^{16} - 18928 p^{7} T^{17} + 946 p^{8} T^{18} - 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 32 T + 1186 T^{2} - 25240 T^{3} + 545746 T^{4} - 8763504 T^{5} + 138653822 T^{6} - 1778280524 T^{7} + 22266573204 T^{8} - 234209505616 T^{9} + 2398708941826 T^{10} - 234209505616 p T^{11} + 22266573204 p^{2} T^{12} - 1778280524 p^{3} T^{13} + 138653822 p^{4} T^{14} - 8763504 p^{5} T^{15} + 545746 p^{6} T^{16} - 25240 p^{7} T^{17} + 1186 p^{8} T^{18} - 32 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 4 T + 510 T^{2} + 804 T^{3} + 116219 T^{4} + 744784 T^{5} + 18609908 T^{6} + 180841560 T^{7} + 2361709013 T^{8} + 26573677556 T^{9} + 246376505122 T^{10} + 26573677556 p T^{11} + 2361709013 p^{2} T^{12} + 180841560 p^{3} T^{13} + 18609908 p^{4} T^{14} + 744784 p^{5} T^{15} + 116219 p^{6} T^{16} + 804 p^{7} T^{17} + 510 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−3.06399701808452138095063257058, −2.93074858041025688904657588692, −2.92501351622503944404100173388, −2.82570435483691589908307410643, −2.53719037585441741855944853521, −2.51495620369713243127499504069, −2.44884730482372453002985728687, −2.39967382973503373758963639606, −2.37394240642366791808495220936, −2.35380704542449241040928377849, −2.02830335015444566955575901363, −1.97112842972265617618999994459, −1.88754636002486190681909779785, −1.78401650124264498128028464061, −1.75362546081299381716595901714, −1.53536512490765243276599083032, −1.50910221415371351899918456478, −1.29219103340161047862692660881, −1.10036700646671570764338202068, −0.968986014427637494419018884979, −0.887764461634161294504265624443, −0.825516397558115688178841952308, −0.68583916633256404655013082876, −0.64154992193545169012728136860, −0.46869460210101907415866383410, 0.46869460210101907415866383410, 0.64154992193545169012728136860, 0.68583916633256404655013082876, 0.825516397558115688178841952308, 0.887764461634161294504265624443, 0.968986014427637494419018884979, 1.10036700646671570764338202068, 1.29219103340161047862692660881, 1.50910221415371351899918456478, 1.53536512490765243276599083032, 1.75362546081299381716595901714, 1.78401650124264498128028464061, 1.88754636002486190681909779785, 1.97112842972265617618999994459, 2.02830335015444566955575901363, 2.35380704542449241040928377849, 2.37394240642366791808495220936, 2.39967382973503373758963639606, 2.44884730482372453002985728687, 2.51495620369713243127499504069, 2.53719037585441741855944853521, 2.82570435483691589908307410643, 2.92501351622503944404100173388, 2.93074858041025688904657588692, 3.06399701808452138095063257058
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.