Dirichlet series
| L(s) = 1 | + 6·2-s − 6·3-s + 7·4-s + 8·5-s − 36·6-s + 53·7-s − 34·8-s − 63·9-s + 48·10-s − 66·11-s − 42·12-s + 318·14-s − 48·15-s − 153·16-s − 117·17-s − 378·18-s + 67·19-s + 56·20-s − 318·21-s − 396·22-s − 158·23-s + 204·24-s − 460·25-s + 327·27-s + 371·28-s − 145·29-s − 288·30-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.15·3-s + 7/8·4-s + 0.715·5-s − 2.44·6-s + 2.86·7-s − 1.50·8-s − 7/3·9-s + 1.51·10-s − 1.80·11-s − 1.01·12-s + 6.07·14-s − 0.826·15-s − 2.39·16-s − 1.66·17-s − 4.94·18-s + 0.808·19-s + 0.626·20-s − 3.30·21-s − 3.83·22-s − 1.43·23-s + 1.73·24-s − 3.67·25-s + 2.33·27-s + 2.50·28-s − 0.928·29-s − 1.75·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(11^{6} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.74129\times 10^{12}\) |
| Root analytic conductor: | \(10.4730\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 11^{6} \cdot 13^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.07677491777\) |
| \(L(\frac12)\) | \(\approx\) | \(0.07677491777\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 11 | \( ( 1 + p T )^{6} \) |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - 3 p T + 29 T^{2} - 49 p T^{3} + 167 p T^{4} - 249 p^{2} T^{5} + 365 p^{3} T^{6} - 249 p^{5} T^{7} + 167 p^{7} T^{8} - 49 p^{10} T^{9} + 29 p^{12} T^{10} - 3 p^{16} T^{11} + p^{18} T^{12} \) |
| 3 | \( 1 + 2 p T + 11 p^{2} T^{2} + 215 p T^{3} + 1793 p T^{4} + 9830 p T^{5} + 185564 T^{6} + 9830 p^{4} T^{7} + 1793 p^{7} T^{8} + 215 p^{10} T^{9} + 11 p^{14} T^{10} + 2 p^{16} T^{11} + p^{18} T^{12} \) | |
| 5 | \( 1 - 8 T + 524 T^{2} - 748 p T^{3} + 126936 T^{4} - 160136 p T^{5} + 19280306 T^{6} - 160136 p^{4} T^{7} + 126936 p^{6} T^{8} - 748 p^{10} T^{9} + 524 p^{12} T^{10} - 8 p^{15} T^{11} + p^{18} T^{12} \) | |
| 7 | \( 1 - 53 T + 2188 T^{2} - 63724 T^{3} + 1619606 T^{4} - 34855843 T^{5} + 689625078 T^{6} - 34855843 p^{3} T^{7} + 1619606 p^{6} T^{8} - 63724 p^{9} T^{9} + 2188 p^{12} T^{10} - 53 p^{15} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 + 117 T + 24314 T^{2} + 2079401 T^{3} + 238132471 T^{4} + 16370187938 T^{5} + 1403367857084 T^{6} + 16370187938 p^{3} T^{7} + 238132471 p^{6} T^{8} + 2079401 p^{9} T^{9} + 24314 p^{12} T^{10} + 117 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 - 67 T + 19555 T^{2} - 1430237 T^{3} + 189694295 T^{4} - 14259310118 T^{5} + 1358301471498 T^{6} - 14259310118 p^{3} T^{7} + 189694295 p^{6} T^{8} - 1430237 p^{9} T^{9} + 19555 p^{12} T^{10} - 67 p^{15} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 + 158 T + 60143 T^{2} + 9801237 T^{3} + 1597069421 T^{4} + 10467599190 p T^{5} + 24658906276302 T^{6} + 10467599190 p^{4} T^{7} + 1597069421 p^{6} T^{8} + 9801237 p^{9} T^{9} + 60143 p^{12} T^{10} + 158 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 + 5 p T + 51110 T^{2} + 6949165 T^{3} + 1910142423 T^{4} + 196277478998 T^{5} + 46668691986932 T^{6} + 196277478998 p^{3} T^{7} + 1910142423 p^{6} T^{8} + 6949165 p^{9} T^{9} + 51110 p^{12} T^{10} + 5 p^{16} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 - 58 T + 80939 T^{2} + 2470446 T^{3} + 2548903859 T^{4} + 391912538240 T^{5} + 58746638942978 T^{6} + 391912538240 p^{3} T^{7} + 2548903859 p^{6} T^{8} + 2470446 p^{9} T^{9} + 80939 p^{12} T^{10} - 58 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 - 753 T + 459983 T^{2} - 192457488 T^{3} + 68461040657 T^{4} - 19481753226715 T^{5} + 4833581242805878 T^{6} - 19481753226715 p^{3} T^{7} + 68461040657 p^{6} T^{8} - 192457488 p^{9} T^{9} + 459983 p^{12} T^{10} - 753 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 - 232 T + 293174 T^{2} - 50891293 T^{3} + 41369329299 T^{4} - 5922498625163 T^{5} + 3591496531232636 T^{6} - 5922498625163 p^{3} T^{7} + 41369329299 p^{6} T^{8} - 50891293 p^{9} T^{9} + 293174 p^{12} T^{10} - 232 p^{15} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 + 390 T + 331377 T^{2} + 111021008 T^{3} + 52213632111 T^{4} + 14785338458754 T^{5} + 5113492517171742 T^{6} + 14785338458754 p^{3} T^{7} + 52213632111 p^{6} T^{8} + 111021008 p^{9} T^{9} + 331377 p^{12} T^{10} + 390 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 - 205 T + 376182 T^{2} - 60248011 T^{3} + 76098985567 T^{4} - 9841839554830 T^{5} + 9570220744928788 T^{6} - 9841839554830 p^{3} T^{7} + 76098985567 p^{6} T^{8} - 60248011 p^{9} T^{9} + 376182 p^{12} T^{10} - 205 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 + 65 T + 376413 T^{2} + 37485577 T^{3} + 89225598299 T^{4} + 9289836579170 T^{5} + 16447221425778766 T^{6} + 9289836579170 p^{3} T^{7} + 89225598299 p^{6} T^{8} + 37485577 p^{9} T^{9} + 376413 p^{12} T^{10} + 65 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 + 1735 T + 2213729 T^{2} + 1908715930 T^{3} + 1378264924419 T^{4} + 790628633816423 T^{5} + 395281859448122702 T^{6} + 790628633816423 p^{3} T^{7} + 1378264924419 p^{6} T^{8} + 1908715930 p^{9} T^{9} + 2213729 p^{12} T^{10} + 1735 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 - 421 T + 494172 T^{2} - 339729533 T^{3} + 203199465943 T^{4} - 98739474575410 T^{5} + 65417171328743016 T^{6} - 98739474575410 p^{3} T^{7} + 203199465943 p^{6} T^{8} - 339729533 p^{9} T^{9} + 494172 p^{12} T^{10} - 421 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 - 703 T + 956391 T^{2} - 416394566 T^{3} + 454601379157 T^{4} - 175163240959519 T^{5} + 162473364652255542 T^{6} - 175163240959519 p^{3} T^{7} + 454601379157 p^{6} T^{8} - 416394566 p^{9} T^{9} + 956391 p^{12} T^{10} - 703 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 + 445 T + 756355 T^{2} - 70046778 T^{3} + 316878902777 T^{4} + 26286080897545 T^{5} + 198270129155701318 T^{6} + 26286080897545 p^{3} T^{7} + 316878902777 p^{6} T^{8} - 70046778 p^{9} T^{9} + 756355 p^{12} T^{10} + 445 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 - 2340 T + 4178150 T^{2} - 5127086007 T^{3} + 5198236125083 T^{4} - 4213395057987785 T^{5} + 2897553162992121292 T^{6} - 4213395057987785 p^{3} T^{7} + 5198236125083 p^{6} T^{8} - 5127086007 p^{9} T^{9} + 4178150 p^{12} T^{10} - 2340 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 + 1234 T + 1066126 T^{2} + 993800414 T^{3} + 1043674824191 T^{4} + 708556670766956 T^{5} + 442068787705598244 T^{6} + 708556670766956 p^{3} T^{7} + 1043674824191 p^{6} T^{8} + 993800414 p^{9} T^{9} + 1066126 p^{12} T^{10} + 1234 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 - 1601 T + 3317865 T^{2} - 3575728359 T^{3} + 4430716234155 T^{4} - 3613002332703914 T^{5} + 3295285147131882454 T^{6} - 3613002332703914 p^{3} T^{7} + 4430716234155 p^{6} T^{8} - 3575728359 p^{9} T^{9} + 3317865 p^{12} T^{10} - 1601 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 - 442 T + 2403055 T^{2} - 620978590 T^{3} + 3152593252419 T^{4} - 696193209089576 T^{5} + 2749929345322336810 T^{6} - 696193209089576 p^{3} T^{7} + 3152593252419 p^{6} T^{8} - 620978590 p^{9} T^{9} + 2403055 p^{12} T^{10} - 442 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 - 2682 T + 6059759 T^{2} - 9052497762 T^{3} + 12463006983923 T^{4} - 13577755911419396 T^{5} + 14221317897418765834 T^{6} - 13577755911419396 p^{3} T^{7} + 12463006983923 p^{6} T^{8} - 9052497762 p^{9} T^{9} + 6059759 p^{12} T^{10} - 2682 p^{15} T^{11} + p^{18} T^{12} \) | |
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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
−4.53343547868207098111597713644, −4.46092342985913895210156169719, −4.13349598283920173389931687192, −4.11787904254384056809979881724, −4.01122966296749370938154140889, −3.82574909392174705895060949322, −3.60375733346209966716815757148, −3.23718286267961238507358967402, −3.23345047427566223381153866475, −3.19392784112542003655088681738, −2.92412473433842462979713264374, −2.51916346476042013176835811703, −2.50457683112176169446606312327, −2.36133474410920990440970023830, −2.18027186774178930252586353122, −1.91505660752925297658899760239, −1.89900096781902568071605911874, −1.86949220965668716462536973191, −1.54486634712596821598820976550, −1.30369761435294513500963316375, −0.77189231034219707065057613496, −0.65274502321674712922044159675, −0.47295498888752633896356475846, −0.32873441361858327823708630967, −0.02323120483489540459764369528, 0.02323120483489540459764369528, 0.32873441361858327823708630967, 0.47295498888752633896356475846, 0.65274502321674712922044159675, 0.77189231034219707065057613496, 1.30369761435294513500963316375, 1.54486634712596821598820976550, 1.86949220965668716462536973191, 1.89900096781902568071605911874, 1.91505660752925297658899760239, 2.18027186774178930252586353122, 2.36133474410920990440970023830, 2.50457683112176169446606312327, 2.51916346476042013176835811703, 2.92412473433842462979713264374, 3.19392784112542003655088681738, 3.23345047427566223381153866475, 3.23718286267961238507358967402, 3.60375733346209966716815757148, 3.82574909392174705895060949322, 4.01122966296749370938154140889, 4.11787904254384056809979881724, 4.13349598283920173389931687192, 4.46092342985913895210156169719, 4.53343547868207098111597713644
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.