Dirichlet series
| L(s) = 1 | + 14·3-s − 20·4-s − 4·7-s + 49·9-s − 50·11-s − 280·12-s + 240·16-s − 56·21-s + 275·25-s − 72·27-s + 80·28-s − 700·33-s − 980·36-s + 82·37-s − 1.19e3·41-s + 1.00e3·44-s + 464·47-s + 3.36e3·48-s − 516·49-s − 692·53-s − 196·63-s − 2.24e3·64-s + 1.11e3·67-s − 1.46e3·71-s + 2.08e3·73-s + 3.85e3·75-s + 200·77-s + ⋯ |
| L(s) = 1 | + 2.69·3-s − 5/2·4-s − 0.215·7-s + 1.81·9-s − 1.37·11-s − 6.73·12-s + 15/4·16-s − 0.581·21-s + 11/5·25-s − 0.513·27-s + 0.539·28-s − 3.69·33-s − 4.53·36-s + 0.364·37-s − 4.54·41-s + 3.42·44-s + 1.44·47-s + 10.1·48-s − 1.50·49-s − 1.79·53-s − 0.391·63-s − 4.37·64-s + 2.03·67-s − 2.44·71-s + 3.33·73-s + 5.92·75-s + 0.296·77-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 37^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.51754\times 10^{6}\) |
| Root analytic conductor: | \(2.08953\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 37^{10} ,\ ( \ : [3/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(3.648603621\) |
| \(L(\frac12)\) | \(\approx\) | \(3.648603621\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{2} T^{2} )^{5} \) |
| 37 | \( 1 - 82 T + 35677 T^{2} - 26624 p T^{3} - 2456290 p^{2} T^{4} + 1665940 p^{3} T^{5} - 2456290 p^{5} T^{6} - 26624 p^{7} T^{7} + 35677 p^{9} T^{8} - 82 p^{12} T^{9} + p^{15} T^{10} \) | |
| good | 3 | \( ( 1 - 7 T + 49 T^{2} - 307 T^{3} + 1507 T^{4} - 6184 T^{5} + 1507 p^{3} T^{6} - 307 p^{6} T^{7} + 49 p^{9} T^{8} - 7 p^{12} T^{9} + p^{15} T^{10} )^{2} \) |
| 5 | \( 1 - 11 p^{2} T^{2} + 67662 T^{4} - 2279794 p T^{6} + 1808160817 T^{8} - 224752350054 T^{10} + 1808160817 p^{6} T^{12} - 2279794 p^{13} T^{14} + 67662 p^{18} T^{16} - 11 p^{26} T^{18} + p^{30} T^{20} \) | |
| 7 | \( ( 1 + 2 T + 264 T^{2} - 3604 T^{3} + 169191 T^{4} + 4548 T^{5} + 169191 p^{3} T^{6} - 3604 p^{6} T^{7} + 264 p^{9} T^{8} + 2 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 11 | \( ( 1 + 25 T + 2847 T^{2} + 8851 p T^{3} + 6135187 T^{4} + 136603824 T^{5} + 6135187 p^{3} T^{6} + 8851 p^{7} T^{7} + 2847 p^{9} T^{8} + 25 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 13 | \( 1 - 10419 T^{2} + 56318750 T^{4} - 209723295466 T^{6} + 608400827558625 T^{8} - 1458884220713830342 T^{10} + 608400827558625 p^{6} T^{12} - 209723295466 p^{12} T^{14} + 56318750 p^{18} T^{16} - 10419 p^{24} T^{18} + p^{30} T^{20} \) | |
| 17 | \( 1 - 22598 T^{2} + 277866733 T^{4} - 2427385958856 T^{6} + 16385233008912498 T^{8} - 89054145725313909988 T^{10} + 16385233008912498 p^{6} T^{12} - 2427385958856 p^{12} T^{14} + 277866733 p^{18} T^{16} - 22598 p^{24} T^{18} + p^{30} T^{20} \) | |
| 19 | \( 1 - 45222 T^{2} + 1013534357 T^{4} - 14831019553768 T^{6} + 156081171751601634 T^{8} - \)\(12\!\cdots\!80\)\( T^{10} + 156081171751601634 p^{6} T^{12} - 14831019553768 p^{12} T^{14} + 1013534357 p^{18} T^{16} - 45222 p^{24} T^{18} + p^{30} T^{20} \) | |
| 23 | \( 1 - 86731 T^{2} + 3706201722 T^{4} - 101315505902914 T^{6} + 1950315774181446117 T^{8} - \)\(27\!\cdots\!14\)\( T^{10} + 1950315774181446117 p^{6} T^{12} - 101315505902914 p^{12} T^{14} + 3706201722 p^{18} T^{16} - 86731 p^{24} T^{18} + p^{30} T^{20} \) | |
| 29 | \( 1 - 2727 p T^{2} + 3549779414 T^{4} - 135664138885042 T^{6} + 4165815967332158905 T^{8} - \)\(10\!\cdots\!94\)\( T^{10} + 4165815967332158905 p^{6} T^{12} - 135664138885042 p^{12} T^{14} + 3549779414 p^{18} T^{16} - 2727 p^{25} T^{18} + p^{30} T^{20} \) | |
| 31 | \( 1 - 210199 T^{2} + 20699407914 T^{4} - 1282099129269986 T^{6} + 56678703318818266997 T^{8} - \)\(19\!\cdots\!58\)\( T^{10} + 56678703318818266997 p^{6} T^{12} - 1282099129269986 p^{12} T^{14} + 20699407914 p^{18} T^{16} - 210199 p^{24} T^{18} + p^{30} T^{20} \) | |
| 41 | \( ( 1 + 597 T + 389761 T^{2} + 150827227 T^{3} + 57037169035 T^{4} + 15118030173814 T^{5} + 57037169035 p^{3} T^{6} + 150827227 p^{6} T^{7} + 389761 p^{9} T^{8} + 597 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 43 | \( 1 - 194666 T^{2} + 29103524917 T^{4} - 3071077966516536 T^{6} + \)\(30\!\cdots\!62\)\( T^{8} - \)\(24\!\cdots\!80\)\( T^{10} + \)\(30\!\cdots\!62\)\( p^{6} T^{12} - 3071077966516536 p^{12} T^{14} + 29103524917 p^{18} T^{16} - 194666 p^{24} T^{18} + p^{30} T^{20} \) | |
| 47 | \( ( 1 - 232 T + 440052 T^{2} - 86857942 T^{3} + 85409509171 T^{4} - 12964195734404 T^{5} + 85409509171 p^{3} T^{6} - 86857942 p^{6} T^{7} + 440052 p^{9} T^{8} - 232 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 53 | \( ( 1 + 346 T + 596314 T^{2} + 141947412 T^{3} + 152218022257 T^{4} + 27422016390844 T^{5} + 152218022257 p^{3} T^{6} + 141947412 p^{6} T^{7} + 596314 p^{9} T^{8} + 346 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 59 | \( 1 - 1487102 T^{2} + 1018136989333 T^{4} - 431689438223827944 T^{6} + \)\(12\!\cdots\!82\)\( T^{8} - \)\(29\!\cdots\!92\)\( T^{10} + \)\(12\!\cdots\!82\)\( p^{6} T^{12} - 431689438223827944 p^{12} T^{14} + 1018136989333 p^{18} T^{16} - 1487102 p^{24} T^{18} + p^{30} T^{20} \) | |
| 61 | \( 1 - 1430195 T^{2} + 1045586842462 T^{4} - 502380973012996578 T^{6} + \)\(17\!\cdots\!01\)\( T^{8} - \)\(45\!\cdots\!38\)\( T^{10} + \)\(17\!\cdots\!01\)\( p^{6} T^{12} - 502380973012996578 p^{12} T^{14} + 1045586842462 p^{18} T^{16} - 1430195 p^{24} T^{18} + p^{30} T^{20} \) | |
| 67 | \( ( 1 - 557 T + 602988 T^{2} - 371477200 T^{3} + 267242587103 T^{4} - 106061161368918 T^{5} + 267242587103 p^{3} T^{6} - 371477200 p^{6} T^{7} + 602988 p^{9} T^{8} - 557 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 71 | \( ( 1 + 730 T + 1843496 T^{2} + 994047024 T^{3} + 1327124697487 T^{4} + 523802364342796 T^{5} + 1327124697487 p^{3} T^{6} + 994047024 p^{6} T^{7} + 1843496 p^{9} T^{8} + 730 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 73 | \( ( 1 - 1041 T + 1426205 T^{2} - 1293938387 T^{3} + 976789040679 T^{4} - 713004204100466 T^{5} + 976789040679 p^{3} T^{6} - 1293938387 p^{6} T^{7} + 1426205 p^{9} T^{8} - 1041 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 79 | \( 1 - 2923579 T^{2} + 4187599615482 T^{4} - 3948434818693177370 T^{6} + \)\(27\!\cdots\!65\)\( T^{8} - \)\(15\!\cdots\!90\)\( T^{10} + \)\(27\!\cdots\!65\)\( p^{6} T^{12} - 3948434818693177370 p^{12} T^{14} + 4187599615482 p^{18} T^{16} - 2923579 p^{24} T^{18} + p^{30} T^{20} \) | |
| 83 | \( ( 1 + 682 T + 892880 T^{2} + 254405640 T^{3} + 700678328659 T^{4} + 310081841027260 T^{5} + 700678328659 p^{3} T^{6} + 254405640 p^{6} T^{7} + 892880 p^{9} T^{8} + 682 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 89 | \( 1 - 3075818 T^{2} + 4748831371549 T^{4} - 5125962483144715704 T^{6} + \)\(45\!\cdots\!62\)\( T^{8} - \)\(34\!\cdots\!08\)\( T^{10} + \)\(45\!\cdots\!62\)\( p^{6} T^{12} - 5125962483144715704 p^{12} T^{14} + 4748831371549 p^{18} T^{16} - 3075818 p^{24} T^{18} + p^{30} T^{20} \) | |
| 97 | \( 1 - 6553662 T^{2} + 20374979496349 T^{4} - 40116269824070150024 T^{6} + \)\(56\!\cdots\!22\)\( T^{8} - \)\(58\!\cdots\!12\)\( T^{10} + \)\(56\!\cdots\!22\)\( p^{6} T^{12} - 40116269824070150024 p^{12} T^{14} + 20374979496349 p^{18} T^{16} - 6553662 p^{24} T^{18} + p^{30} T^{20} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.29942635609025221030273285925, −5.29078506098441040173064101678, −5.16923778750482219107148364834, −5.05156997286459255184733962116, −4.75048798126884956137107331155, −4.71240654897176763884341170451, −4.46526820581421215375895934330, −4.30461050133311448316020976486, −4.19489515334973722539682901661, −3.74107680751875433621709311278, −3.70312467222885868317765993402, −3.68834931538718750428768007776, −3.33445681914912380303200567535, −3.13727408686809356813724977389, −3.08140333921697894308466505557, −2.84243861556447914107317339178, −2.74557956493813177053342022186, −2.70241858725120767396868639989, −2.29930446899146428342066576605, −1.83431965271329396847907830165, −1.65766942906267841588348876238, −1.52035539451378723934098670060, −0.78150981503872105537388907560, −0.49359829192791579967273283785, −0.35307995440977658816849724499, 0.35307995440977658816849724499, 0.49359829192791579967273283785, 0.78150981503872105537388907560, 1.52035539451378723934098670060, 1.65766942906267841588348876238, 1.83431965271329396847907830165, 2.29930446899146428342066576605, 2.70241858725120767396868639989, 2.74557956493813177053342022186, 2.84243861556447914107317339178, 3.08140333921697894308466505557, 3.13727408686809356813724977389, 3.33445681914912380303200567535, 3.68834931538718750428768007776, 3.70312467222885868317765993402, 3.74107680751875433621709311278, 4.19489515334973722539682901661, 4.30461050133311448316020976486, 4.46526820581421215375895934330, 4.71240654897176763884341170451, 4.75048798126884956137107331155, 5.05156997286459255184733962116, 5.16923778750482219107148364834, 5.29078506098441040173064101678, 5.29942635609025221030273285925
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.