Dirichlet series
| L(s) = 1 | − 12·2-s + 5·3-s + 84·4-s − 60·6-s + 42·7-s − 448·8-s − 38·9-s − 49·11-s + 420·12-s + 16·13-s − 504·14-s + 2.01e3·16-s + 175·17-s + 456·18-s − 229·19-s + 210·21-s + 588·22-s + 138·23-s − 2.24e3·24-s − 192·26-s − 162·27-s + 3.52e3·28-s − 182·29-s + 114·31-s − 8.06e3·32-s − 245·33-s − 2.10e3·34-s + ⋯ |
| L(s) = 1 | − 4.24·2-s + 0.962·3-s + 21/2·4-s − 4.08·6-s + 2.26·7-s − 19.7·8-s − 1.40·9-s − 1.34·11-s + 10.1·12-s + 0.341·13-s − 9.62·14-s + 63/2·16-s + 2.49·17-s + 5.97·18-s − 2.76·19-s + 2.18·21-s + 5.69·22-s + 1.25·23-s − 19.0·24-s − 1.44·26-s − 1.15·27-s + 23.8·28-s − 1.16·29-s + 0.660·31-s − 44.5·32-s − 1.29·33-s − 10.5·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 5^{12} \cdot 23^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.75850\times 10^{10}\) |
| Root analytic conductor: | \(8.23724\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{6} \cdot 5^{12} \cdot 23^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(1.800186214\) |
| \(L(\frac12)\) | \(\approx\) | \(1.800186214\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p T )^{6} \) |
| 5 | \( 1 \) | |
| 23 | \( ( 1 - p T )^{6} \) | |
| good | 3 | \( 1 - 5 T + 7 p^{2} T^{2} - 343 T^{3} + 2684 T^{4} - 1678 p^{2} T^{5} + 28516 p T^{6} - 1678 p^{5} T^{7} + 2684 p^{6} T^{8} - 343 p^{9} T^{9} + 7 p^{14} T^{10} - 5 p^{15} T^{11} + p^{18} T^{12} \) |
| 7 | \( 1 - 6 p T + 1587 T^{2} - 37802 T^{3} + 847647 T^{4} - 15031848 T^{5} + 290726906 T^{6} - 15031848 p^{3} T^{7} + 847647 p^{6} T^{8} - 37802 p^{9} T^{9} + 1587 p^{12} T^{10} - 6 p^{16} T^{11} + p^{18} T^{12} \) | |
| 11 | \( 1 + 49 T + 5437 T^{2} + 255910 T^{3} + 13369547 T^{4} + 598134541 T^{5} + 21024562150 T^{6} + 598134541 p^{3} T^{7} + 13369547 p^{6} T^{8} + 255910 p^{9} T^{9} + 5437 p^{12} T^{10} + 49 p^{15} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 - 16 T + 1736 T^{2} - 16525 T^{3} - 1905808 T^{4} + 1014680 p^{2} T^{5} - 10408996660 T^{6} + 1014680 p^{5} T^{7} - 1905808 p^{6} T^{8} - 16525 p^{9} T^{9} + 1736 p^{12} T^{10} - 16 p^{15} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 - 175 T + 1570 p T^{2} - 2965479 T^{3} + 307614775 T^{4} - 25370128666 T^{5} + 1931720001964 T^{6} - 25370128666 p^{3} T^{7} + 307614775 p^{6} T^{8} - 2965479 p^{9} T^{9} + 1570 p^{13} T^{10} - 175 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 + 229 T + 37201 T^{2} + 4698978 T^{3} + 528094007 T^{4} + 53571419393 T^{5} + 4705404038350 T^{6} + 53571419393 p^{3} T^{7} + 528094007 p^{6} T^{8} + 4698978 p^{9} T^{9} + 37201 p^{12} T^{10} + 229 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 + 182 T + 91020 T^{2} + 14857459 T^{3} + 4189177066 T^{4} + 571257794354 T^{5} + 125266691581814 T^{6} + 571257794354 p^{3} T^{7} + 4189177066 p^{6} T^{8} + 14857459 p^{9} T^{9} + 91020 p^{12} T^{10} + 182 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 - 114 T + 107143 T^{2} - 519135 p T^{3} + 5920766387 T^{4} - 860717466447 T^{5} + 216689737211050 T^{6} - 860717466447 p^{3} T^{7} + 5920766387 p^{6} T^{8} - 519135 p^{10} T^{9} + 107143 p^{12} T^{10} - 114 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 - 64 T + 129110 T^{2} + 147736 T^{3} + 7390747047 T^{4} + 905353959816 T^{5} + 339617737815444 T^{6} + 905353959816 p^{3} T^{7} + 7390747047 p^{6} T^{8} + 147736 p^{9} T^{9} + 129110 p^{12} T^{10} - 64 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 - 243 T + 217354 T^{2} - 51111450 T^{3} + 28972990659 T^{4} - 5821706605929 T^{5} + 2392660354692683 T^{6} - 5821706605929 p^{3} T^{7} + 28972990659 p^{6} T^{8} - 51111450 p^{9} T^{9} + 217354 p^{12} T^{10} - 243 p^{15} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 - 182 T + 260711 T^{2} - 78970510 T^{3} + 33096675423 T^{4} - 12627843861384 T^{5} + 2963222307919890 T^{6} - 12627843861384 p^{3} T^{7} + 33096675423 p^{6} T^{8} - 78970510 p^{9} T^{9} + 260711 p^{12} T^{10} - 182 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 - 498 T + 509619 T^{2} - 135877151 T^{3} + 83998064637 T^{4} - 12243416324769 T^{5} + 8566851173627134 T^{6} - 12243416324769 p^{3} T^{7} + 83998064637 p^{6} T^{8} - 135877151 p^{9} T^{9} + 509619 p^{12} T^{10} - 498 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 - 1290 T + 1240218 T^{2} - 836015682 T^{3} + 483217979607 T^{4} - 227680588290828 T^{5} + 95811508569502636 T^{6} - 227680588290828 p^{3} T^{7} + 483217979607 p^{6} T^{8} - 836015682 p^{9} T^{9} + 1240218 p^{12} T^{10} - 1290 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 + 559 T + 505009 T^{2} + 126099326 T^{3} + 122476607499 T^{4} + 40791666593975 T^{5} + 34896017937142934 T^{6} + 40791666593975 p^{3} T^{7} + 122476607499 p^{6} T^{8} + 126099326 p^{9} T^{9} + 505009 p^{12} T^{10} + 559 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 + 688 T + 705982 T^{2} + 453964488 T^{3} + 280924931687 T^{4} + 133731264949352 T^{5} + 77783878544342884 T^{6} + 133731264949352 p^{3} T^{7} + 280924931687 p^{6} T^{8} + 453964488 p^{9} T^{9} + 705982 p^{12} T^{10} + 688 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 + 2069 T + 2400766 T^{2} + 1531587331 T^{3} + 478304331079 T^{4} - 123858083758798 T^{5} - 164138098237517020 T^{6} - 123858083758798 p^{3} T^{7} + 478304331079 p^{6} T^{8} + 1531587331 p^{9} T^{9} + 2400766 p^{12} T^{10} + 2069 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 - 584 T + 1644691 T^{2} - 921288723 T^{3} + 1269926126623 T^{4} - 616747945790643 T^{5} + 578286194429532426 T^{6} - 616747945790643 p^{3} T^{7} + 1269926126623 p^{6} T^{8} - 921288723 p^{9} T^{9} + 1644691 p^{12} T^{10} - 584 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 - 2485 T + 4574426 T^{2} - 5799531842 T^{3} + 5968575565743 T^{4} - 4910784012962487 T^{5} + 3377893957023056247 T^{6} - 4910784012962487 p^{3} T^{7} + 5968575565743 p^{6} T^{8} - 5799531842 p^{9} T^{9} + 4574426 p^{12} T^{10} - 2485 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 + 1432 T + 2108895 T^{2} + 1489738990 T^{3} + 1128186053351 T^{4} + 448670081214670 T^{5} + 358134172709224002 T^{6} + 448670081214670 p^{3} T^{7} + 1128186053351 p^{6} T^{8} + 1489738990 p^{9} T^{9} + 2108895 p^{12} T^{10} + 1432 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 - 2089 T + 2805961 T^{2} - 3233295446 T^{3} + 3234574571091 T^{4} - 2785691913858353 T^{5} + 2180446078776301958 T^{6} - 2785691913858353 p^{3} T^{7} + 3234574571091 p^{6} T^{8} - 3233295446 p^{9} T^{9} + 2805961 p^{12} T^{10} - 2089 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 + 591 T + 2481630 T^{2} + 1931033355 T^{3} + 3220232489463 T^{4} + 2481025134045822 T^{5} + 2760586998440640820 T^{6} + 2481025134045822 p^{3} T^{7} + 3220232489463 p^{6} T^{8} + 1931033355 p^{9} T^{9} + 2481630 p^{12} T^{10} + 591 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 + 968 T + 2732362 T^{2} + 3531908816 T^{3} + 4951320686511 T^{4} + 4930878757316040 T^{5} + 5965760995609978188 T^{6} + 4930878757316040 p^{3} T^{7} + 4951320686511 p^{6} T^{8} + 3531908816 p^{9} T^{9} + 2732362 p^{12} T^{10} + 968 p^{15} T^{11} + p^{18} T^{12} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.84793215596000795566834850041, −4.71986391621217752105552784405, −4.29234122538525818419634575925, −4.19865109645254006597786473984, −4.02118317602473161574000774678, −3.92069723481791779353058669234, −3.66211598251460802698794717116, −3.23488961858563785820968652259, −3.11593373105241193282252090259, −3.07690552595976990549092676841, −2.81643068171012886685584302463, −2.78385303650012838736972539211, −2.70194371641779890499938308996, −2.22906457414017634777173715430, −2.12582358353521947077132908313, −1.99062574972165672126429798982, −1.89881370030052494107551114612, −1.78287682064527137772770715663, −1.40894899581373596402215818236, −1.16824705987822713060943279537, −0.966067361723845895074759816554, −0.842921937877344002769425761715, −0.57969260093478085053477405992, −0.34988410088271391737304969019, −0.28457286783539098265113130393, 0.28457286783539098265113130393, 0.34988410088271391737304969019, 0.57969260093478085053477405992, 0.842921937877344002769425761715, 0.966067361723845895074759816554, 1.16824705987822713060943279537, 1.40894899581373596402215818236, 1.78287682064527137772770715663, 1.89881370030052494107551114612, 1.99062574972165672126429798982, 2.12582358353521947077132908313, 2.22906457414017634777173715430, 2.70194371641779890499938308996, 2.78385303650012838736972539211, 2.81643068171012886685584302463, 3.07690552595976990549092676841, 3.11593373105241193282252090259, 3.23488961858563785820968652259, 3.66211598251460802698794717116, 3.92069723481791779353058669234, 4.02118317602473161574000774678, 4.19865109645254006597786473984, 4.29234122538525818419634575925, 4.71986391621217752105552784405, 4.84793215596000795566834850041
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.