Dirichlet series
| L(s) = 1 | − 54·5-s − 210·7-s + 6.57e3·11-s + 1.00e4·13-s + 2.97e4·17-s + 1.37e5·19-s − 3.96e4·23-s + 1.14e5·25-s − 2.39e5·29-s − 1.45e5·31-s + 1.13e4·35-s − 7.20e5·37-s − 1.08e6·41-s + 2.99e5·43-s + 1.31e5·47-s + 1.49e6·49-s − 1.90e6·53-s − 3.55e5·55-s + 2.50e6·59-s + 7.30e6·61-s − 5.44e5·65-s − 3.43e6·67-s + 2.26e6·71-s − 1.59e7·73-s − 1.38e6·77-s − 7.07e6·79-s − 1.09e7·83-s + ⋯ |
| L(s) = 1 | − 0.193·5-s − 0.231·7-s + 1.49·11-s + 1.27·13-s + 1.47·17-s + 4.59·19-s − 0.679·23-s + 1.46·25-s − 1.82·29-s − 0.878·31-s + 0.0447·35-s − 2.33·37-s − 2.46·41-s + 0.575·43-s + 0.184·47-s + 1.81·49-s − 1.76·53-s − 0.287·55-s + 1.58·59-s + 4.12·61-s − 0.246·65-s − 1.39·67-s + 0.752·71-s − 4.78·73-s − 0.344·77-s − 1.61·79-s − 2.09·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{24} \cdot 3^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.04009\times 10^{12}\) |
| Root analytic conductor: | \(11.6168\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{24} \cdot 3^{18} ,\ ( \ : [7/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(0.007366487230\) |
| \(L(\frac12)\) | \(\approx\) | \(0.007366487230\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 + 54 T - 111768 T^{2} - 11498544 T^{3} + 3715062396 T^{4} + 75281687982 p T^{5} - 536368810346 p^{2} T^{6} + 75281687982 p^{8} T^{7} + 3715062396 p^{14} T^{8} - 11498544 p^{21} T^{9} - 111768 p^{28} T^{10} + 54 p^{35} T^{11} + p^{42} T^{12} \) |
| 7 | \( 1 + 30 p T - 1454352 T^{2} + 402968668 T^{3} + 1020589584408 T^{4} - 442021781623878 T^{5} - 760018152629486082 T^{6} - 442021781623878 p^{7} T^{7} + 1020589584408 p^{14} T^{8} + 402968668 p^{21} T^{9} - 1454352 p^{28} T^{10} + 30 p^{36} T^{11} + p^{42} T^{12} \) | |
| 11 | \( 1 - 6579 T - 12133131 T^{2} + 110744884458 T^{3} + 427470578543355 T^{4} - 132677298636830109 p T^{5} - \)\(49\!\cdots\!94\)\( T^{6} - 132677298636830109 p^{8} T^{7} + 427470578543355 p^{14} T^{8} + 110744884458 p^{21} T^{9} - 12133131 p^{28} T^{10} - 6579 p^{35} T^{11} + p^{42} T^{12} \) | |
| 13 | \( 1 - 10092 T + 52529592 T^{2} - 683259054484 T^{3} + 273864566459520 T^{4} + 27302280412632596388 T^{5} - \)\(50\!\cdots\!98\)\( T^{6} + 27302280412632596388 p^{7} T^{7} + 273864566459520 p^{14} T^{8} - 683259054484 p^{21} T^{9} + 52529592 p^{28} T^{10} - 10092 p^{35} T^{11} + p^{42} T^{12} \) | |
| 17 | \( ( 1 - 14895 T + 563457651 T^{2} - 14438679480378 T^{3} + 563457651 p^{7} T^{4} - 14895 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 19 | \( ( 1 - 68745 T + 3236583129 T^{2} - 100823453324278 T^{3} + 3236583129 p^{7} T^{4} - 68745 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 23 | \( 1 + 39654 T - 6602837784 T^{2} - 241826211089532 T^{3} + 27777515531286961536 T^{4} + \)\(56\!\cdots\!86\)\( T^{5} - \)\(86\!\cdots\!42\)\( T^{6} + \)\(56\!\cdots\!86\)\( p^{7} T^{7} + 27777515531286961536 p^{14} T^{8} - 241826211089532 p^{21} T^{9} - 6602837784 p^{28} T^{10} + 39654 p^{35} T^{11} + p^{42} T^{12} \) | |
| 29 | \( 1 + 239832 T + 12450064224 T^{2} - 2745766217115540 T^{3} - \)\(37\!\cdots\!40\)\( T^{4} + \)\(39\!\cdots\!92\)\( T^{5} + \)\(40\!\cdots\!14\)\( T^{6} + \)\(39\!\cdots\!92\)\( p^{7} T^{7} - \)\(37\!\cdots\!40\)\( p^{14} T^{8} - 2745766217115540 p^{21} T^{9} + 12450064224 p^{28} T^{10} + 239832 p^{35} T^{11} + p^{42} T^{12} \) | |
| 31 | \( 1 + 145704 T - 41125724916 T^{2} - 1234277316213368 T^{3} + \)\(14\!\cdots\!80\)\( T^{4} - \)\(67\!\cdots\!96\)\( T^{5} - \)\(54\!\cdots\!34\)\( T^{6} - \)\(67\!\cdots\!96\)\( p^{7} T^{7} + \)\(14\!\cdots\!80\)\( p^{14} T^{8} - 1234277316213368 p^{21} T^{9} - 41125724916 p^{28} T^{10} + 145704 p^{35} T^{11} + p^{42} T^{12} \) | |
| 37 | \( ( 1 + 360192 T + 180376649547 T^{2} + 60270325951343056 T^{3} + 180376649547 p^{7} T^{4} + 360192 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 41 | \( 1 + 1086993 T + 224205566583 T^{2} + 112685370539499144 T^{3} + \)\(22\!\cdots\!89\)\( T^{4} + \)\(80\!\cdots\!19\)\( T^{5} + \)\(67\!\cdots\!66\)\( T^{6} + \)\(80\!\cdots\!19\)\( p^{7} T^{7} + \)\(22\!\cdots\!89\)\( p^{14} T^{8} + 112685370539499144 p^{21} T^{9} + 224205566583 p^{28} T^{10} + 1086993 p^{35} T^{11} + p^{42} T^{12} \) | |
| 43 | \( 1 - 299967 T - 432077871363 T^{2} + 244584006146300866 T^{3} + \)\(70\!\cdots\!15\)\( T^{4} - \)\(41\!\cdots\!07\)\( T^{5} - \)\(32\!\cdots\!78\)\( T^{6} - \)\(41\!\cdots\!07\)\( p^{7} T^{7} + \)\(70\!\cdots\!15\)\( p^{14} T^{8} + 244584006146300866 p^{21} T^{9} - 432077871363 p^{28} T^{10} - 299967 p^{35} T^{11} + p^{42} T^{12} \) | |
| 47 | \( 1 - 131634 T - 397707348768 T^{2} - 619153327988243244 T^{3} - \)\(85\!\cdots\!60\)\( T^{4} + \)\(13\!\cdots\!06\)\( T^{5} + \)\(24\!\cdots\!38\)\( T^{6} + \)\(13\!\cdots\!06\)\( p^{7} T^{7} - \)\(85\!\cdots\!60\)\( p^{14} T^{8} - 619153327988243244 p^{21} T^{9} - 397707348768 p^{28} T^{10} - 131634 p^{35} T^{11} + p^{42} T^{12} \) | |
| 53 | \( ( 1 + 954576 T + 456802017531 T^{2} - 1124477191430832816 T^{3} + 456802017531 p^{7} T^{4} + 954576 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 59 | \( 1 - 2504853 T + 430627930509 T^{2} + 4140256476775368510 T^{3} - \)\(39\!\cdots\!65\)\( T^{4} + \)\(24\!\cdots\!67\)\( T^{5} - \)\(48\!\cdots\!06\)\( T^{6} + \)\(24\!\cdots\!67\)\( p^{7} T^{7} - \)\(39\!\cdots\!65\)\( p^{14} T^{8} + 4140256476775368510 p^{21} T^{9} + 430627930509 p^{28} T^{10} - 2504853 p^{35} T^{11} + p^{42} T^{12} \) | |
| 61 | \( 1 - 7309038 T + 28312337732736 T^{2} - 75450647408931530320 T^{3} + \)\(15\!\cdots\!00\)\( T^{4} - \)\(27\!\cdots\!26\)\( T^{5} + \)\(47\!\cdots\!38\)\( T^{6} - \)\(27\!\cdots\!26\)\( p^{7} T^{7} + \)\(15\!\cdots\!00\)\( p^{14} T^{8} - 75450647408931530320 p^{21} T^{9} + 28312337732736 p^{28} T^{10} - 7309038 p^{35} T^{11} + p^{42} T^{12} \) | |
| 67 | \( 1 + 3433035 T - 7140644277387 T^{2} - 18205303852289773562 T^{3} + \)\(10\!\cdots\!83\)\( T^{4} + \)\(12\!\cdots\!87\)\( T^{5} - \)\(42\!\cdots\!22\)\( T^{6} + \)\(12\!\cdots\!87\)\( p^{7} T^{7} + \)\(10\!\cdots\!83\)\( p^{14} T^{8} - 18205303852289773562 p^{21} T^{9} - 7140644277387 p^{28} T^{10} + 3433035 p^{35} T^{11} + p^{42} T^{12} \) | |
| 71 | \( ( 1 - 1134684 T + 22664106615237 T^{2} - 15919744395740599944 T^{3} + 22664106615237 p^{7} T^{4} - 1134684 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 73 | \( ( 1 + 7950873 T + 48400849544307 T^{2} + \)\(18\!\cdots\!70\)\( T^{3} + 48400849544307 p^{7} T^{4} + 7950873 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 79 | \( 1 + 7076928 T - 1581056048724 T^{2} - \)\(11\!\cdots\!40\)\( T^{3} - \)\(36\!\cdots\!04\)\( T^{4} + \)\(10\!\cdots\!92\)\( T^{5} + \)\(30\!\cdots\!02\)\( T^{6} + \)\(10\!\cdots\!92\)\( p^{7} T^{7} - \)\(36\!\cdots\!04\)\( p^{14} T^{8} - \)\(11\!\cdots\!40\)\( p^{21} T^{9} - 1581056048724 p^{28} T^{10} + 7076928 p^{35} T^{11} + p^{42} T^{12} \) | |
| 83 | \( 1 + 10914444 T + 46744190612508 T^{2} - 16352888943942459864 T^{3} - \)\(11\!\cdots\!84\)\( T^{4} - \)\(56\!\cdots\!08\)\( T^{5} - \)\(27\!\cdots\!42\)\( T^{6} - \)\(56\!\cdots\!08\)\( p^{7} T^{7} - \)\(11\!\cdots\!84\)\( p^{14} T^{8} - 16352888943942459864 p^{21} T^{9} + 46744190612508 p^{28} T^{10} + 10914444 p^{35} T^{11} + p^{42} T^{12} \) | |
| 89 | \( ( 1 + 11900178 T + 136066730426583 T^{2} + \)\(10\!\cdots\!24\)\( T^{3} + 136066730426583 p^{7} T^{4} + 11900178 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 97 | \( 1 - 519357 T - 27215275056513 T^{2} - \)\(17\!\cdots\!32\)\( T^{3} - \)\(10\!\cdots\!19\)\( T^{4} + \)\(22\!\cdots\!89\)\( T^{5} + \)\(17\!\cdots\!42\)\( T^{6} + \)\(22\!\cdots\!89\)\( p^{7} T^{7} - \)\(10\!\cdots\!19\)\( p^{14} T^{8} - \)\(17\!\cdots\!32\)\( p^{21} T^{9} - 27215275056513 p^{28} T^{10} - 519357 p^{35} T^{11} + p^{42} T^{12} \) | |
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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
−5.02100463712436071850786221918, −4.56433564199836054868380783486, −4.37573196991443872103868683031, −4.24015199981242557012799737764, −3.96408696951463852955814177448, −3.79315463217114025248997933724, −3.71868444712434663351002335869, −3.55549937835500811882646019412, −3.37282873338070240208771531433, −3.13723578980690006150700732138, −3.07173303349426375697195002007, −2.98135767105303111243189199737, −2.74366390329293809868146725089, −2.43691995015679059763421811215, −2.03066656906483996634930690917, −1.89197500396261302664439635865, −1.67960340683779980255703159111, −1.49690600145790884677320627357, −1.24844880543048479850073580922, −1.20088092049182606394106911291, −0.989356756990040631240047755378, −0.928940366835579338430176094553, −0.76538538512403006755626136004, −0.092632405389633420982764156844, −0.01456041034430219966166876626, 0.01456041034430219966166876626, 0.092632405389633420982764156844, 0.76538538512403006755626136004, 0.928940366835579338430176094553, 0.989356756990040631240047755378, 1.20088092049182606394106911291, 1.24844880543048479850073580922, 1.49690600145790884677320627357, 1.67960340683779980255703159111, 1.89197500396261302664439635865, 2.03066656906483996634930690917, 2.43691995015679059763421811215, 2.74366390329293809868146725089, 2.98135767105303111243189199737, 3.07173303349426375697195002007, 3.13723578980690006150700732138, 3.37282873338070240208771531433, 3.55549937835500811882646019412, 3.71868444712434663351002335869, 3.79315463217114025248997933724, 3.96408696951463852955814177448, 4.24015199981242557012799737764, 4.37573196991443872103868683031, 4.56433564199836054868380783486, 5.02100463712436071850786221918
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.