Dirichlet series
| L(s) = 1 | − 7·5-s + 4·7-s + 4·9-s + 14·11-s + 4·13-s + 17-s + 4·19-s + 12·23-s + 21·25-s − 2·27-s − 4·29-s − 24·31-s − 28·35-s + 10·37-s + 5·41-s − 6·43-s − 28·45-s − 16·47-s + 27·49-s + 14·53-s − 98·55-s − 16·59-s + 12·61-s + 16·63-s − 28·65-s − 9·71-s − 40·73-s + ⋯ |
| L(s) = 1 | − 3.13·5-s + 1.51·7-s + 4/3·9-s + 4.22·11-s + 1.10·13-s + 0.242·17-s + 0.917·19-s + 2.50·23-s + 21/5·25-s − 0.384·27-s − 0.742·29-s − 4.31·31-s − 4.73·35-s + 1.64·37-s + 0.780·41-s − 0.914·43-s − 4.17·45-s − 2.33·47-s + 27/7·49-s + 1.92·53-s − 13.2·55-s − 2.08·59-s + 1.53·61-s + 2.01·63-s − 3.47·65-s − 1.06·71-s − 4.68·73-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{28} \cdot 5^{14} \cdot 37^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.32591\times 10^{10}\) |
| Root analytic conductor: | \(2.43082\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{28} \cdot 5^{14} \cdot 37^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(4.793290500\) |
| \(L(\frac12)\) | \(\approx\) | \(4.793290500\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T + T^{2} )^{7} \) | |
| 37 | \( 1 - 10 T + T^{2} + 302 T^{3} - 1283 T^{4} - 904 T^{5} + 9618 T^{6} + 95439 T^{7} + 9618 p T^{8} - 904 p^{2} T^{9} - 1283 p^{3} T^{10} + 302 p^{4} T^{11} + p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| good | 3 | \( 1 - 4 T^{2} + 2 T^{3} + p^{2} T^{4} + T^{5} - 5 p^{2} T^{6} - 2 p^{2} T^{7} + 13 p^{2} T^{8} + 43 T^{9} - 32 p^{2} T^{10} - 325 T^{11} + 1223 T^{12} + 29 p^{3} T^{13} - 3137 T^{14} + 29 p^{4} T^{15} + 1223 p^{2} T^{16} - 325 p^{3} T^{17} - 32 p^{6} T^{18} + 43 p^{5} T^{19} + 13 p^{8} T^{20} - 2 p^{9} T^{21} - 5 p^{10} T^{22} + p^{9} T^{23} + p^{12} T^{24} + 2 p^{11} T^{25} - 4 p^{12} T^{26} + p^{14} T^{28} \) |
| 7 | \( 1 - 4 T - 11 T^{2} + 88 T^{3} - 81 T^{4} - 12 p^{2} T^{5} + 1546 T^{6} + 619 T^{7} - 4692 T^{8} - 8431 T^{9} + 1460 T^{10} + 249336 T^{11} - 576070 T^{12} - 163026 p T^{13} + 7383895 T^{14} - 163026 p^{2} T^{15} - 576070 p^{2} T^{16} + 249336 p^{3} T^{17} + 1460 p^{4} T^{18} - 8431 p^{5} T^{19} - 4692 p^{6} T^{20} + 619 p^{7} T^{21} + 1546 p^{8} T^{22} - 12 p^{11} T^{23} - 81 p^{10} T^{24} + 88 p^{11} T^{25} - 11 p^{12} T^{26} - 4 p^{13} T^{27} + p^{14} T^{28} \) | |
| 11 | \( ( 1 - 7 T + 68 T^{2} - 298 T^{3} + 1726 T^{4} - 5618 T^{5} + 25615 T^{6} - 69943 T^{7} + 25615 p T^{8} - 5618 p^{2} T^{9} + 1726 p^{3} T^{10} - 298 p^{4} T^{11} + 68 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 13 | \( 1 - 4 T - 29 T^{2} + 8 T^{3} + 881 T^{4} + 1675 T^{5} - 12474 T^{6} - 68589 T^{7} + 5488 p T^{8} + 1335963 T^{9} + 2051137 T^{10} - 17517124 T^{11} - 63610426 T^{12} + 84615029 T^{13} + 1123980365 T^{14} + 84615029 p T^{15} - 63610426 p^{2} T^{16} - 17517124 p^{3} T^{17} + 2051137 p^{4} T^{18} + 1335963 p^{5} T^{19} + 5488 p^{7} T^{20} - 68589 p^{7} T^{21} - 12474 p^{8} T^{22} + 1675 p^{9} T^{23} + 881 p^{10} T^{24} + 8 p^{11} T^{25} - 29 p^{12} T^{26} - 4 p^{13} T^{27} + p^{14} T^{28} \) | |
| 17 | \( 1 - T - 67 T^{2} - 42 T^{3} + 124 p T^{4} + 4102 T^{5} - 48187 T^{6} - 86608 T^{7} + 63357 p T^{8} + 791766 T^{9} - 23887128 T^{10} - 12319299 T^{11} + 476185429 T^{12} + 146652386 T^{13} - 8484743545 T^{14} + 146652386 p T^{15} + 476185429 p^{2} T^{16} - 12319299 p^{3} T^{17} - 23887128 p^{4} T^{18} + 791766 p^{5} T^{19} + 63357 p^{7} T^{20} - 86608 p^{7} T^{21} - 48187 p^{8} T^{22} + 4102 p^{9} T^{23} + 124 p^{11} T^{24} - 42 p^{11} T^{25} - 67 p^{12} T^{26} - p^{13} T^{27} + p^{14} T^{28} \) | |
| 19 | \( 1 - 4 T - 53 T^{2} + 14 p T^{3} + 1286 T^{4} - 7385 T^{5} - 24 p^{2} T^{6} + 44058 T^{7} - 158009 T^{8} + 2645067 T^{9} + 3724963 T^{10} - 82681114 T^{11} + 26727803 T^{12} + 738261632 T^{13} - 1257114511 T^{14} + 738261632 p T^{15} + 26727803 p^{2} T^{16} - 82681114 p^{3} T^{17} + 3724963 p^{4} T^{18} + 2645067 p^{5} T^{19} - 158009 p^{6} T^{20} + 44058 p^{7} T^{21} - 24 p^{10} T^{22} - 7385 p^{9} T^{23} + 1286 p^{10} T^{24} + 14 p^{12} T^{25} - 53 p^{12} T^{26} - 4 p^{13} T^{27} + p^{14} T^{28} \) | |
| 23 | \( ( 1 - 6 T + 94 T^{2} - 196 T^{3} + 2396 T^{4} + 5783 T^{5} + 20082 T^{6} + 309979 T^{7} + 20082 p T^{8} + 5783 p^{2} T^{9} + 2396 p^{3} T^{10} - 196 p^{4} T^{11} + 94 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 29 | \( ( 1 + 2 T + 95 T^{2} + 338 T^{3} + 4351 T^{4} + 23938 T^{5} + 141364 T^{6} + 919883 T^{7} + 141364 p T^{8} + 23938 p^{2} T^{9} + 4351 p^{3} T^{10} + 338 p^{4} T^{11} + 95 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 31 | \( ( 1 + 12 T + 148 T^{2} + 1130 T^{3} + 8007 T^{4} + 1490 p T^{5} + 264795 T^{6} + 1417527 T^{7} + 264795 p T^{8} + 1490 p^{3} T^{9} + 8007 p^{3} T^{10} + 1130 p^{4} T^{11} + 148 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 41 | \( 1 - 5 T - 160 T^{2} + 831 T^{3} + 11765 T^{4} - 56059 T^{5} - 737329 T^{6} + 2277982 T^{7} + 50366202 T^{8} - 93013296 T^{9} - 71174628 p T^{10} + 4007963640 T^{11} + 133581761428 T^{12} - 79099642565 T^{13} - 5513828221933 T^{14} - 79099642565 p T^{15} + 133581761428 p^{2} T^{16} + 4007963640 p^{3} T^{17} - 71174628 p^{5} T^{18} - 93013296 p^{5} T^{19} + 50366202 p^{6} T^{20} + 2277982 p^{7} T^{21} - 737329 p^{8} T^{22} - 56059 p^{9} T^{23} + 11765 p^{10} T^{24} + 831 p^{11} T^{25} - 160 p^{12} T^{26} - 5 p^{13} T^{27} + p^{14} T^{28} \) | |
| 43 | \( ( 1 + 3 T + 154 T^{2} + 523 T^{3} + 12258 T^{4} + 47329 T^{5} + 680981 T^{6} + 2556969 T^{7} + 680981 p T^{8} + 47329 p^{2} T^{9} + 12258 p^{3} T^{10} + 523 p^{4} T^{11} + 154 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 47 | \( ( 1 + 8 T + 152 T^{2} + 969 T^{3} + 14757 T^{4} + 1815 p T^{5} + 950377 T^{6} + 4564577 T^{7} + 950377 p T^{8} + 1815 p^{3} T^{9} + 14757 p^{3} T^{10} + 969 p^{4} T^{11} + 152 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 53 | \( 1 - 14 T - 199 T^{2} + 2672 T^{3} + 34729 T^{4} - 344976 T^{5} - 4305943 T^{6} + 29199407 T^{7} + 438739915 T^{8} - 1835622443 T^{9} - 36092154604 T^{10} + 79265460888 T^{11} + 2474190518163 T^{12} - 1616142026928 T^{13} - 142732435757343 T^{14} - 1616142026928 p T^{15} + 2474190518163 p^{2} T^{16} + 79265460888 p^{3} T^{17} - 36092154604 p^{4} T^{18} - 1835622443 p^{5} T^{19} + 438739915 p^{6} T^{20} + 29199407 p^{7} T^{21} - 4305943 p^{8} T^{22} - 344976 p^{9} T^{23} + 34729 p^{10} T^{24} + 2672 p^{11} T^{25} - 199 p^{12} T^{26} - 14 p^{13} T^{27} + p^{14} T^{28} \) | |
| 59 | \( 1 + 16 T - T^{2} - 2322 T^{3} - 19894 T^{4} + 53279 T^{5} + 2127620 T^{6} + 11823688 T^{7} - 68383785 T^{8} - 1235454999 T^{9} - 3209096835 T^{10} + 53286879990 T^{11} + 433400970991 T^{12} - 919662380654 T^{13} - 27664829074567 T^{14} - 919662380654 p T^{15} + 433400970991 p^{2} T^{16} + 53286879990 p^{3} T^{17} - 3209096835 p^{4} T^{18} - 1235454999 p^{5} T^{19} - 68383785 p^{6} T^{20} + 11823688 p^{7} T^{21} + 2127620 p^{8} T^{22} + 53279 p^{9} T^{23} - 19894 p^{10} T^{24} - 2322 p^{11} T^{25} - p^{12} T^{26} + 16 p^{13} T^{27} + p^{14} T^{28} \) | |
| 61 | \( 1 - 12 T - 241 T^{2} + 2434 T^{3} + 42946 T^{4} - 305005 T^{5} - 5397116 T^{6} + 24178444 T^{7} + 536688389 T^{8} - 1325105735 T^{9} - 43762802327 T^{10} + 50317582658 T^{11} + 50155703041 p T^{12} - 880006674542 T^{13} - 195145922612881 T^{14} - 880006674542 p T^{15} + 50155703041 p^{3} T^{16} + 50317582658 p^{3} T^{17} - 43762802327 p^{4} T^{18} - 1325105735 p^{5} T^{19} + 536688389 p^{6} T^{20} + 24178444 p^{7} T^{21} - 5397116 p^{8} T^{22} - 305005 p^{9} T^{23} + 42946 p^{10} T^{24} + 2434 p^{11} T^{25} - 241 p^{12} T^{26} - 12 p^{13} T^{27} + p^{14} T^{28} \) | |
| 67 | \( 1 + 2 T^{2} + 664 T^{3} - 6221 T^{4} - 30196 T^{5} + 835264 T^{6} - 2222267 T^{7} + 79790 p T^{8} + 321601714 T^{9} - 3243999674 T^{10} - 24933819622 T^{11} + 67681228162 T^{12} - 48781545953 T^{13} + 2242848355949 T^{14} - 48781545953 p T^{15} + 67681228162 p^{2} T^{16} - 24933819622 p^{3} T^{17} - 3243999674 p^{4} T^{18} + 321601714 p^{5} T^{19} + 79790 p^{7} T^{20} - 2222267 p^{7} T^{21} + 835264 p^{8} T^{22} - 30196 p^{9} T^{23} - 6221 p^{10} T^{24} + 664 p^{11} T^{25} + 2 p^{12} T^{26} + p^{14} T^{28} \) | |
| 71 | \( 1 + 9 T - 226 T^{2} - 277 T^{3} + 43511 T^{4} - 121193 T^{5} - 4098547 T^{6} + 35735848 T^{7} + 223868340 T^{8} - 3921645926 T^{9} + 4212566848 T^{10} + 270197092146 T^{11} - 1699888271214 T^{12} - 7525214838709 T^{13} + 167644146182909 T^{14} - 7525214838709 p T^{15} - 1699888271214 p^{2} T^{16} + 270197092146 p^{3} T^{17} + 4212566848 p^{4} T^{18} - 3921645926 p^{5} T^{19} + 223868340 p^{6} T^{20} + 35735848 p^{7} T^{21} - 4098547 p^{8} T^{22} - 121193 p^{9} T^{23} + 43511 p^{10} T^{24} - 277 p^{11} T^{25} - 226 p^{12} T^{26} + 9 p^{13} T^{27} + p^{14} T^{28} \) | |
| 73 | \( ( 1 + 20 T + 599 T^{2} + 8268 T^{3} + 140282 T^{4} + 1456206 T^{5} + 17513556 T^{6} + 140037961 T^{7} + 17513556 p T^{8} + 1456206 p^{2} T^{9} + 140282 p^{3} T^{10} + 8268 p^{4} T^{11} + 599 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 79 | \( 1 - 3 T - 377 T^{2} + 1240 T^{3} + 73519 T^{4} - 236021 T^{5} - 10105910 T^{6} + 26131018 T^{7} + 1133872798 T^{8} - 1822427677 T^{9} - 113401373051 T^{10} + 79613475179 T^{11} + 10407148707095 T^{12} - 1709240900902 T^{13} - 866951759701009 T^{14} - 1709240900902 p T^{15} + 10407148707095 p^{2} T^{16} + 79613475179 p^{3} T^{17} - 113401373051 p^{4} T^{18} - 1822427677 p^{5} T^{19} + 1133872798 p^{6} T^{20} + 26131018 p^{7} T^{21} - 10105910 p^{8} T^{22} - 236021 p^{9} T^{23} + 73519 p^{10} T^{24} + 1240 p^{11} T^{25} - 377 p^{12} T^{26} - 3 p^{13} T^{27} + p^{14} T^{28} \) | |
| 83 | \( 1 - 5 T - 143 T^{2} + 344 T^{3} + 6652 T^{4} + 51426 T^{5} + 394484 T^{6} - 9645556 T^{7} - 73509565 T^{8} + 763848965 T^{9} + 3666354918 T^{10} - 25500527510 T^{11} - 175988359285 T^{12} - 46309986056 T^{13} + 20945335689585 T^{14} - 46309986056 p T^{15} - 175988359285 p^{2} T^{16} - 25500527510 p^{3} T^{17} + 3666354918 p^{4} T^{18} + 763848965 p^{5} T^{19} - 73509565 p^{6} T^{20} - 9645556 p^{7} T^{21} + 394484 p^{8} T^{22} + 51426 p^{9} T^{23} + 6652 p^{10} T^{24} + 344 p^{11} T^{25} - 143 p^{12} T^{26} - 5 p^{13} T^{27} + p^{14} T^{28} \) | |
| 89 | \( 1 - 20 T - 13 T^{2} + 1044 T^{3} + 15236 T^{4} + 18542 T^{5} - 1797541 T^{6} - 21566087 T^{7} + 143484861 T^{8} + 2036088027 T^{9} - 4987306587 T^{10} - 1213512840 T^{11} - 1247156945495 T^{12} + 612449435368 T^{13} + 107918638165217 T^{14} + 612449435368 p T^{15} - 1247156945495 p^{2} T^{16} - 1213512840 p^{3} T^{17} - 4987306587 p^{4} T^{18} + 2036088027 p^{5} T^{19} + 143484861 p^{6} T^{20} - 21566087 p^{7} T^{21} - 1797541 p^{8} T^{22} + 18542 p^{9} T^{23} + 15236 p^{10} T^{24} + 1044 p^{11} T^{25} - 13 p^{12} T^{26} - 20 p^{13} T^{27} + p^{14} T^{28} \) | |
| 97 | \( ( 1 + 29 T + 688 T^{2} + 10824 T^{3} + 159996 T^{4} + 1957914 T^{5} + 23248897 T^{6} + 234806951 T^{7} + 23248897 p T^{8} + 1957914 p^{2} T^{9} + 159996 p^{3} T^{10} + 10824 p^{4} T^{11} + 688 p^{5} T^{12} + 29 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.13131445526065924943991845282, −3.09641358486611664790803875047, −2.79373642081762938904924406502, −2.75303989505794867449750786566, −2.68912114642349115826108276383, −2.68229605331598466557914236334, −2.58978171418149088011686135761, −2.29753871569070874792518872334, −2.20716366866756353003116585056, −2.03319468142698947849209715490, −2.02488443039844408422707875584, −1.92544880431644764003812946226, −1.74525084781031246545391386869, −1.70316615296555571295665007636, −1.68484272330559826601901895246, −1.46880390456496246911726659128, −1.36964507989973209134439364480, −1.17817162413384539693300540459, −1.17111398101975140750488728806, −1.13015570294098831575096864903, −1.07184276885251054154845660716, −0.69553040862571575333832357639, −0.64033160418422637090155482574, −0.48165199457674286313000900505, −0.14171746695346799049878837646, 0.14171746695346799049878837646, 0.48165199457674286313000900505, 0.64033160418422637090155482574, 0.69553040862571575333832357639, 1.07184276885251054154845660716, 1.13015570294098831575096864903, 1.17111398101975140750488728806, 1.17817162413384539693300540459, 1.36964507989973209134439364480, 1.46880390456496246911726659128, 1.68484272330559826601901895246, 1.70316615296555571295665007636, 1.74525084781031246545391386869, 1.92544880431644764003812946226, 2.02488443039844408422707875584, 2.03319468142698947849209715490, 2.20716366866756353003116585056, 2.29753871569070874792518872334, 2.58978171418149088011686135761, 2.68229605331598466557914236334, 2.68912114642349115826108276383, 2.75303989505794867449750786566, 2.79373642081762938904924406502, 3.09641358486611664790803875047, 3.13131445526065924943991845282
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.