Dirichlet series
| L(s) = 1 | + 9·3-s + 5·5-s − 28·7-s − 11·9-s + 77·11-s − 43·13-s + 45·15-s − 6·17-s + 133·19-s − 252·21-s + 159·23-s − 403·25-s − 266·27-s + 69·29-s − 90·31-s + 693·33-s − 140·35-s − 164·37-s − 387·39-s + 214·41-s + 1.16e3·43-s − 55·45-s − 909·47-s − 510·49-s − 54·51-s + 51·53-s + 385·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s − 1.51·7-s − 0.407·9-s + 2.11·11-s − 0.917·13-s + 0.774·15-s − 0.0856·17-s + 1.60·19-s − 2.61·21-s + 1.44·23-s − 3.22·25-s − 1.89·27-s + 0.441·29-s − 0.521·31-s + 3.65·33-s − 0.676·35-s − 0.728·37-s − 1.58·39-s + 0.815·41-s + 4.13·43-s − 0.182·45-s − 2.82·47-s − 1.48·49-s − 0.148·51-s + 0.132·53-s + 0.943·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{42} \cdot 19^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.78582\times 10^{12}\) |
| Root analytic conductor: | \(8.47032\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{42} \cdot 19^{7} ,\ ( \ : [3/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(21.97758462\) |
| \(L(\frac12)\) | \(\approx\) | \(21.97758462\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 19 | \( ( 1 - p T )^{7} \) | |
| good | 3 | \( 1 - p^{2} T + 92 T^{2} - 661 T^{3} + 4730 T^{4} - 27763 T^{5} + 159427 T^{6} - 843550 T^{7} + 159427 p^{3} T^{8} - 27763 p^{6} T^{9} + 4730 p^{9} T^{10} - 661 p^{12} T^{11} + 92 p^{15} T^{12} - p^{20} T^{13} + p^{21} T^{14} \) |
| 5 | \( 1 - p T + 428 T^{2} - 1213 T^{3} + 94984 T^{4} - 3879 p^{2} T^{5} + 14752191 T^{6} - 5798222 T^{7} + 14752191 p^{3} T^{8} - 3879 p^{8} T^{9} + 94984 p^{9} T^{10} - 1213 p^{12} T^{11} + 428 p^{15} T^{12} - p^{19} T^{13} + p^{21} T^{14} \) | |
| 7 | \( 1 + 4 p T + 1294 T^{2} + 35540 T^{3} + 1013399 T^{4} + 22295920 T^{5} + 491192117 T^{6} + 9459986700 T^{7} + 491192117 p^{3} T^{8} + 22295920 p^{6} T^{9} + 1013399 p^{9} T^{10} + 35540 p^{12} T^{11} + 1294 p^{15} T^{12} + 4 p^{19} T^{13} + p^{21} T^{14} \) | |
| 11 | \( 1 - 7 p T + 662 p T^{2} - 383981 T^{3} + 21957654 T^{4} - 938563355 T^{5} + 41881205505 T^{6} - 1510001644054 T^{7} + 41881205505 p^{3} T^{8} - 938563355 p^{6} T^{9} + 21957654 p^{9} T^{10} - 383981 p^{12} T^{11} + 662 p^{16} T^{12} - 7 p^{19} T^{13} + p^{21} T^{14} \) | |
| 13 | \( 1 + 43 T + 10668 T^{2} + 294323 T^{3} + 48033104 T^{4} + 790053241 T^{5} + 133508107855 T^{6} + 1538227036354 T^{7} + 133508107855 p^{3} T^{8} + 790053241 p^{6} T^{9} + 48033104 p^{9} T^{10} + 294323 p^{12} T^{11} + 10668 p^{15} T^{12} + 43 p^{18} T^{13} + p^{21} T^{14} \) | |
| 17 | \( 1 + 6 T + 14052 T^{2} + 61834 T^{3} + 100370665 T^{4} - 1356754060 T^{5} + 538031723837 T^{6} - 13621310102350 T^{7} + 538031723837 p^{3} T^{8} - 1356754060 p^{6} T^{9} + 100370665 p^{9} T^{10} + 61834 p^{12} T^{11} + 14052 p^{15} T^{12} + 6 p^{18} T^{13} + p^{21} T^{14} \) | |
| 23 | \( 1 - 159 T + 40584 T^{2} - 5019067 T^{3} + 930016750 T^{4} - 100961903373 T^{5} + 15694037671231 T^{6} - 1500226296973106 T^{7} + 15694037671231 p^{3} T^{8} - 100961903373 p^{6} T^{9} + 930016750 p^{9} T^{10} - 5019067 p^{12} T^{11} + 40584 p^{15} T^{12} - 159 p^{18} T^{13} + p^{21} T^{14} \) | |
| 29 | \( 1 - 69 T + 146992 T^{2} - 9193221 T^{3} + 9706770288 T^{4} - 528572244175 T^{5} + 374503322355627 T^{6} - 16848051832914238 T^{7} + 374503322355627 p^{3} T^{8} - 528572244175 p^{6} T^{9} + 9706770288 p^{9} T^{10} - 9193221 p^{12} T^{11} + 146992 p^{15} T^{12} - 69 p^{18} T^{13} + p^{21} T^{14} \) | |
| 31 | \( 1 + 90 T + 104805 T^{2} + 1133500 T^{3} + 4449013977 T^{4} - 410087453594 T^{5} + 110983717425973 T^{6} - 21998916488024056 T^{7} + 110983717425973 p^{3} T^{8} - 410087453594 p^{6} T^{9} + 4449013977 p^{9} T^{10} + 1133500 p^{12} T^{11} + 104805 p^{15} T^{12} + 90 p^{18} T^{13} + p^{21} T^{14} \) | |
| 37 | \( 1 + 164 T + 266139 T^{2} + 43940168 T^{3} + 33240364477 T^{4} + 5128980819900 T^{5} + 2546318195316639 T^{6} + 336745175297269744 T^{7} + 2546318195316639 p^{3} T^{8} + 5128980819900 p^{6} T^{9} + 33240364477 p^{9} T^{10} + 43940168 p^{12} T^{11} + 266139 p^{15} T^{12} + 164 p^{18} T^{13} + p^{21} T^{14} \) | |
| 41 | \( 1 - 214 T + 294679 T^{2} - 66316932 T^{3} + 40313902765 T^{4} - 10045000323306 T^{5} + 3669755939349739 T^{6} - 895177731873405240 T^{7} + 3669755939349739 p^{3} T^{8} - 10045000323306 p^{6} T^{9} + 40313902765 p^{9} T^{10} - 66316932 p^{12} T^{11} + 294679 p^{15} T^{12} - 214 p^{18} T^{13} + p^{21} T^{14} \) | |
| 43 | \( 1 - 1167 T + 908260 T^{2} - 514930563 T^{3} + 237703149090 T^{4} - 92225948378885 T^{5} + 31277568237263691 T^{6} - 9328514519042664210 T^{7} + 31277568237263691 p^{3} T^{8} - 92225948378885 p^{6} T^{9} + 237703149090 p^{9} T^{10} - 514930563 p^{12} T^{11} + 908260 p^{15} T^{12} - 1167 p^{18} T^{13} + p^{21} T^{14} \) | |
| 47 | \( 1 + 909 T + 989432 T^{2} + 584967945 T^{3} + 360159852342 T^{4} + 154438826080343 T^{5} + 66690527601259183 T^{6} + 21493021855340770342 T^{7} + 66690527601259183 p^{3} T^{8} + 154438826080343 p^{6} T^{9} + 360159852342 p^{9} T^{10} + 584967945 p^{12} T^{11} + 989432 p^{15} T^{12} + 909 p^{18} T^{13} + p^{21} T^{14} \) | |
| 53 | \( 1 - 51 T + 351086 T^{2} - 3371643 T^{3} + 65402679824 T^{4} - 3307860219329 T^{5} + 10385305632393565 T^{6} - 901449802745835490 T^{7} + 10385305632393565 p^{3} T^{8} - 3307860219329 p^{6} T^{9} + 65402679824 p^{9} T^{10} - 3371643 p^{12} T^{11} + 351086 p^{15} T^{12} - 51 p^{18} T^{13} + p^{21} T^{14} \) | |
| 59 | \( 1 - 1663 T + 2156334 T^{2} - 1975805567 T^{3} + 1510538723774 T^{4} - 965999929142921 T^{5} + 534365303940862077 T^{6} - \)\(25\!\cdots\!62\)\( T^{7} + 534365303940862077 p^{3} T^{8} - 965999929142921 p^{6} T^{9} + 1510538723774 p^{9} T^{10} - 1975805567 p^{12} T^{11} + 2156334 p^{15} T^{12} - 1663 p^{18} T^{13} + p^{21} T^{14} \) | |
| 61 | \( 1 - 463 T + 375042 T^{2} - 251783479 T^{3} + 205763339032 T^{4} - 84717244400109 T^{5} + 54696686746847097 T^{6} - 25840918170514252250 T^{7} + 54696686746847097 p^{3} T^{8} - 84717244400109 p^{6} T^{9} + 205763339032 p^{9} T^{10} - 251783479 p^{12} T^{11} + 375042 p^{15} T^{12} - 463 p^{18} T^{13} + p^{21} T^{14} \) | |
| 67 | \( 1 - 2203 T + 2879222 T^{2} - 2578944899 T^{3} + 1940615797998 T^{4} - 1298858141290669 T^{5} + 829844292151904141 T^{6} - \)\(47\!\cdots\!90\)\( T^{7} + 829844292151904141 p^{3} T^{8} - 1298858141290669 p^{6} T^{9} + 1940615797998 p^{9} T^{10} - 2578944899 p^{12} T^{11} + 2879222 p^{15} T^{12} - 2203 p^{18} T^{13} + p^{21} T^{14} \) | |
| 71 | \( 1 + 404 T + 1129757 T^{2} + 368139240 T^{3} + 723747214473 T^{4} + 229637382608428 T^{5} + 345081241962962749 T^{6} + \)\(10\!\cdots\!08\)\( T^{7} + 345081241962962749 p^{3} T^{8} + 229637382608428 p^{6} T^{9} + 723747214473 p^{9} T^{10} + 368139240 p^{12} T^{11} + 1129757 p^{15} T^{12} + 404 p^{18} T^{13} + p^{21} T^{14} \) | |
| 73 | \( 1 + 308 T + 2031390 T^{2} + 469483150 T^{3} + 1944705763799 T^{4} + 360624193528268 T^{5} + 1147366115516232151 T^{6} + \)\(17\!\cdots\!66\)\( T^{7} + 1147366115516232151 p^{3} T^{8} + 360624193528268 p^{6} T^{9} + 1944705763799 p^{9} T^{10} + 469483150 p^{12} T^{11} + 2031390 p^{15} T^{12} + 308 p^{18} T^{13} + p^{21} T^{14} \) | |
| 79 | \( 1 + 596 T + 2782693 T^{2} + 1485824680 T^{3} + 3574980624857 T^{4} + 1651481617803692 T^{5} + 2737704370839254293 T^{6} + \)\(10\!\cdots\!60\)\( T^{7} + 2737704370839254293 p^{3} T^{8} + 1651481617803692 p^{6} T^{9} + 3574980624857 p^{9} T^{10} + 1485824680 p^{12} T^{11} + 2782693 p^{15} T^{12} + 596 p^{18} T^{13} + p^{21} T^{14} \) | |
| 83 | \( 1 - 2398 T + 4079729 T^{2} - 5417828836 T^{3} + 6427831910089 T^{4} - 6499659942857810 T^{5} + 5843247350765914177 T^{6} - \)\(46\!\cdots\!68\)\( T^{7} + 5843247350765914177 p^{3} T^{8} - 6499659942857810 p^{6} T^{9} + 6427831910089 p^{9} T^{10} - 5417828836 p^{12} T^{11} + 4079729 p^{15} T^{12} - 2398 p^{18} T^{13} + p^{21} T^{14} \) | |
| 89 | \( 1 + 176 T + 2624303 T^{2} + 215821456 T^{3} + 3430695540677 T^{4} - 33150536428592 T^{5} + 3162911514463425499 T^{6} - \)\(14\!\cdots\!96\)\( T^{7} + 3162911514463425499 p^{3} T^{8} - 33150536428592 p^{6} T^{9} + 3430695540677 p^{9} T^{10} + 215821456 p^{12} T^{11} + 2624303 p^{15} T^{12} + 176 p^{18} T^{13} + p^{21} T^{14} \) | |
| 97 | \( 1 - 1444 T + 4003103 T^{2} - 3584391832 T^{3} + 5735650002365 T^{4} - 2775248425986140 T^{5} + 4343170895680018275 T^{6} - \)\(12\!\cdots\!24\)\( T^{7} + 4343170895680018275 p^{3} T^{8} - 2775248425986140 p^{6} T^{9} + 5735650002365 p^{9} T^{10} - 3584391832 p^{12} T^{11} + 4003103 p^{15} T^{12} - 1444 p^{18} T^{13} + p^{21} T^{14} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.99829436080285744706769516092, −3.92424219259267608474237510734, −3.89367844016949114732764617606, −3.73352640791077620636744677606, −3.36067532057901490638191831686, −3.33731553440186170580602639926, −3.32853377814071528929816701540, −3.19397948564209565975714462287, −2.98386692403858892165840218991, −2.93231363027415544532395084422, −2.77434104094170023045184183667, −2.49036246620004796898921990148, −2.29648510495669281948961572439, −2.18429913712771753206883656607, −2.14359579074855441153440107654, −1.94382410686539140202309918724, −1.76221579642716316942824521858, −1.70798381361036475998275562206, −1.29978487379015465396833371561, −1.03615524517535880935849441317, −0.831519056349182815956726965792, −0.67634476647857537229120684312, −0.66451442990479569244669684573, −0.43115429723847000823710233320, −0.20744729023882552803903774969, 0.20744729023882552803903774969, 0.43115429723847000823710233320, 0.66451442990479569244669684573, 0.67634476647857537229120684312, 0.831519056349182815956726965792, 1.03615524517535880935849441317, 1.29978487379015465396833371561, 1.70798381361036475998275562206, 1.76221579642716316942824521858, 1.94382410686539140202309918724, 2.14359579074855441153440107654, 2.18429913712771753206883656607, 2.29648510495669281948961572439, 2.49036246620004796898921990148, 2.77434104094170023045184183667, 2.93231363027415544532395084422, 2.98386692403858892165840218991, 3.19397948564209565975714462287, 3.32853377814071528929816701540, 3.33731553440186170580602639926, 3.36067532057901490638191831686, 3.73352640791077620636744677606, 3.89367844016949114732764617606, 3.92424219259267608474237510734, 3.99829436080285744706769516092
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.