Dirichlet series
| L(s) = 1 | − 24·2-s − 27·3-s + 192·4-s + 54·5-s + 648·6-s + 210·7-s + 1.02e3·8-s + 1.51e3·9-s − 1.29e3·10-s + 6.57e3·11-s − 5.18e3·12-s + 1.00e4·13-s − 5.04e3·14-s − 1.45e3·15-s − 3.68e4·16-s − 2.97e4·17-s − 3.62e4·18-s − 1.37e5·19-s + 1.03e4·20-s − 5.67e3·21-s − 1.57e5·22-s − 3.96e4·23-s − 2.76e4·24-s + 1.14e5·25-s − 2.42e5·26-s − 5.10e3·27-s + 4.03e4·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 0.577·3-s + 3/2·4-s + 0.193·5-s + 1.22·6-s + 0.231·7-s + 0.707·8-s + 0.691·9-s − 0.409·10-s + 1.49·11-s − 0.866·12-s + 1.27·13-s − 0.490·14-s − 0.111·15-s − 9/4·16-s − 1.47·17-s − 1.46·18-s − 4.59·19-s + 0.289·20-s − 0.133·21-s − 3.16·22-s − 0.679·23-s − 0.408·24-s + 1.46·25-s − 2.70·26-s − 0.0498·27-s + 0.347·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 34012224 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34012224 ^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(34012224\) = \(2^{6} \cdot 3^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(31606.4\) |
| Root analytic conductor: | \(2.37127\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 34012224,\ (\ :[7/2]^{6}),\ 1)\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(1.060774161\) |
| \(L(\frac12)\) | \(\approx\) | \(1.060774161\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{3} T + p^{6} T^{2} )^{3} \) |
| 3 | \( 1 + p^{3} T - 29 p^{3} T^{2} - 26 p^{7} T^{3} - 29 p^{10} T^{4} + p^{17} T^{5} + p^{21} T^{6} \) | |
| 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
−9.048483516568969319864924867133, −8.875078762933105600224643997188, −8.822664916462969711397730167425, −8.693768077655653023679065929029, −8.456717549390991497212557009785, −8.391809295966026570785869016934, −7.913377819818787234880139802673, −7.31976602036047777613415528422, −6.88860450796962629547455812654, −6.78782905112162086349772481723, −6.71487157940699355519162168474, −6.24701502245867756819897120619, −6.03844712673890818028252219956, −5.54975528757299805263527012361, −4.82955386525724062978557970456, −4.58631577485509991027152994272, −4.16890908398043917345592521042, −4.06657920892999357264187736624, −3.70869283153973158548664744866, −2.40868466635140431755997185209, −2.22929812210252177308239351925, −1.83904679860486403529445483051, −1.06609385576713138706300119940, −0.68644292469734592958609609025, −0.50892215003422368349085739070, 0.50892215003422368349085739070, 0.68644292469734592958609609025, 1.06609385576713138706300119940, 1.83904679860486403529445483051, 2.22929812210252177308239351925, 2.40868466635140431755997185209, 3.70869283153973158548664744866, 4.06657920892999357264187736624, 4.16890908398043917345592521042, 4.58631577485509991027152994272, 4.82955386525724062978557970456, 5.54975528757299805263527012361, 6.03844712673890818028252219956, 6.24701502245867756819897120619, 6.71487157940699355519162168474, 6.78782905112162086349772481723, 6.88860450796962629547455812654, 7.31976602036047777613415528422, 7.913377819818787234880139802673, 8.391809295966026570785869016934, 8.456717549390991497212557009785, 8.693768077655653023679065929029, 8.822664916462969711397730167425, 8.875078762933105600224643997188, 9.048483516568969319864924867133
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.