Dirichlet series
| L(s) = 1 | + 8.74e3·3-s + 2.75e5·5-s − 3.29e6·7-s + 4.78e7·9-s − 6.15e7·11-s + 3.69e8·13-s + 2.40e9·15-s − 1.45e9·17-s − 2.69e9·19-s − 2.88e10·21-s + 1.80e10·23-s − 1.88e10·25-s + 2.09e11·27-s − 1.43e10·29-s + 2.71e11·31-s − 5.38e11·33-s − 9.07e11·35-s + 1.71e12·37-s + 3.22e12·39-s + 2.78e12·41-s + 1.89e12·43-s + 1.31e13·45-s + 1.87e12·47-s + 6.78e12·49-s − 1.27e13·51-s + 4.74e12·53-s − 1.69e13·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.57·5-s − 1.51·7-s + 10/3·9-s − 0.952·11-s + 1.63·13-s + 3.64·15-s − 0.862·17-s − 0.690·19-s − 3.49·21-s + 1.10·23-s − 0.618·25-s + 3.84·27-s − 0.154·29-s + 1.77·31-s − 2.19·33-s − 2.38·35-s + 2.97·37-s + 3.76·39-s + 2.23·41-s + 1.06·43-s + 5.25·45-s + 0.539·47-s + 10/7·49-s − 1.99·51-s + 0.554·53-s − 1.50·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+15/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(49787136\) = \(2^{8} \cdot 3^{4} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.06411\times 10^{8}\) |
| Root analytic conductor: | \(10.9481\) |
| Motivic weight: | \(15\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 49787136,\ (\ :15/2, 15/2, 15/2, 15/2),\ 1)\) |
Particular Values
| \(L(8)\) | \(\approx\) | \(52.14591546\) |
| \(L(\frac12)\) | \(\approx\) | \(52.14591546\) |
| \(L(\frac{17}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| bad | 2 | \( 1 \) | |
| 3 | $C_1$ | \( ( 1 - p^{7} T )^{4} \) | |
| 7 | $C_1$ | \( ( 1 + p^{7} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 275436 T + 18948660708 p T^{2} - 955931906687076 p^{2} T^{3} + 32075049300920943838 p^{3} T^{4} - 955931906687076 p^{17} T^{5} + 18948660708 p^{31} T^{6} - 275436 p^{45} T^{7} + p^{60} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 61540668 T + 1927177008793020 T^{2} + \)\(19\!\cdots\!76\)\( p T^{3} + \)\(28\!\cdots\!26\)\( p^{2} T^{4} + \)\(19\!\cdots\!76\)\( p^{16} T^{5} + 1927177008793020 p^{30} T^{6} + 61540668 p^{45} T^{7} + p^{60} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 28385112 p T + 585576676069324 p^{2} T^{2} - \)\(73\!\cdots\!64\)\( p^{3} T^{3} + \)\(10\!\cdots\!70\)\( p^{4} T^{4} - \)\(73\!\cdots\!64\)\( p^{18} T^{5} + 585576676069324 p^{32} T^{6} - 28385112 p^{46} T^{7} + p^{60} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 1458569484 T + 4959433432206976068 T^{2} + \)\(84\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!06\)\( T^{4} + \)\(84\!\cdots\!80\)\( p^{15} T^{5} + 4959433432206976068 p^{30} T^{6} + 1458569484 p^{45} T^{7} + p^{60} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 141644536 p T + 879178681304490340 p T^{2} + \)\(19\!\cdots\!68\)\( p T^{3} + \)\(14\!\cdots\!46\)\( T^{4} + \)\(19\!\cdots\!68\)\( p^{16} T^{5} + 879178681304490340 p^{31} T^{6} + 141644536 p^{46} T^{7} + p^{60} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 18028282212 T + \)\(65\!\cdots\!36\)\( T^{2} - \)\(78\!\cdots\!72\)\( T^{3} + \)\(19\!\cdots\!94\)\( T^{4} - \)\(78\!\cdots\!72\)\( p^{15} T^{5} + \)\(65\!\cdots\!36\)\( p^{30} T^{6} - 18028282212 p^{45} T^{7} + p^{60} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 14387019264 T + \)\(24\!\cdots\!32\)\( T^{2} + \)\(37\!\cdots\!92\)\( T^{3} + \)\(29\!\cdots\!70\)\( T^{4} + \)\(37\!\cdots\!92\)\( p^{15} T^{5} + \)\(24\!\cdots\!32\)\( p^{30} T^{6} + 14387019264 p^{45} T^{7} + p^{60} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 271402618248 T + \)\(81\!\cdots\!56\)\( T^{2} - \)\(15\!\cdots\!68\)\( T^{3} + \)\(29\!\cdots\!54\)\( T^{4} - \)\(15\!\cdots\!68\)\( p^{15} T^{5} + \)\(81\!\cdots\!56\)\( p^{30} T^{6} - 271402618248 p^{45} T^{7} + p^{60} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 1719450823976 T + \)\(18\!\cdots\!04\)\( T^{2} - \)\(13\!\cdots\!52\)\( T^{3} + \)\(86\!\cdots\!06\)\( T^{4} - \)\(13\!\cdots\!52\)\( p^{15} T^{5} + \)\(18\!\cdots\!04\)\( p^{30} T^{6} - 1719450823976 p^{45} T^{7} + p^{60} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 2784989291868 T + \)\(83\!\cdots\!64\)\( T^{2} - \)\(13\!\cdots\!04\)\( T^{3} + \)\(21\!\cdots\!26\)\( T^{4} - \)\(13\!\cdots\!04\)\( p^{15} T^{5} + \)\(83\!\cdots\!64\)\( p^{30} T^{6} - 2784989291868 p^{45} T^{7} + p^{60} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 1897932676640 T + \)\(80\!\cdots\!04\)\( T^{2} - \)\(14\!\cdots\!24\)\( T^{3} + \)\(33\!\cdots\!50\)\( T^{4} - \)\(14\!\cdots\!24\)\( p^{15} T^{5} + \)\(80\!\cdots\!04\)\( p^{30} T^{6} - 1897932676640 p^{45} T^{7} + p^{60} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 1872077464008 T + \)\(28\!\cdots\!60\)\( T^{2} - \)\(89\!\cdots\!52\)\( T^{3} + \)\(40\!\cdots\!90\)\( T^{4} - \)\(89\!\cdots\!52\)\( p^{15} T^{5} + \)\(28\!\cdots\!60\)\( p^{30} T^{6} - 1872077464008 p^{45} T^{7} + p^{60} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 4740642191736 T + \)\(24\!\cdots\!28\)\( T^{2} - \)\(95\!\cdots\!44\)\( T^{3} + \)\(25\!\cdots\!94\)\( T^{4} - \)\(95\!\cdots\!44\)\( p^{15} T^{5} + \)\(24\!\cdots\!28\)\( p^{30} T^{6} - 4740642191736 p^{45} T^{7} + p^{60} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 13984487218296 T + \)\(19\!\cdots\!52\)\( p T^{2} - \)\(14\!\cdots\!72\)\( T^{3} + \)\(56\!\cdots\!70\)\( T^{4} - \)\(14\!\cdots\!72\)\( p^{15} T^{5} + \)\(19\!\cdots\!52\)\( p^{31} T^{6} - 13984487218296 p^{45} T^{7} + p^{60} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 32810552663568 T + \)\(65\!\cdots\!52\)\( T^{2} + \)\(18\!\cdots\!36\)\( T^{3} - \)\(60\!\cdots\!70\)\( T^{4} + \)\(18\!\cdots\!36\)\( p^{15} T^{5} + \)\(65\!\cdots\!52\)\( p^{30} T^{6} - 32810552663568 p^{45} T^{7} + p^{60} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 4158516498856 T + \)\(75\!\cdots\!44\)\( T^{2} - \)\(22\!\cdots\!04\)\( T^{3} + \)\(26\!\cdots\!74\)\( T^{4} - \)\(22\!\cdots\!04\)\( p^{15} T^{5} + \)\(75\!\cdots\!44\)\( p^{30} T^{6} - 4158516498856 p^{45} T^{7} + p^{60} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 70250046828444 T + \)\(21\!\cdots\!28\)\( T^{2} - \)\(11\!\cdots\!12\)\( T^{3} + \)\(18\!\cdots\!54\)\( T^{4} - \)\(11\!\cdots\!12\)\( p^{15} T^{5} + \)\(21\!\cdots\!28\)\( p^{30} T^{6} - 70250046828444 p^{45} T^{7} + p^{60} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 27064251481456 T + \)\(23\!\cdots\!60\)\( T^{2} - \)\(76\!\cdots\!64\)\( T^{3} + \)\(25\!\cdots\!22\)\( T^{4} - \)\(76\!\cdots\!64\)\( p^{15} T^{5} + \)\(23\!\cdots\!60\)\( p^{30} T^{6} - 27064251481456 p^{45} T^{7} + p^{60} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 238381767283992 T + \)\(10\!\cdots\!52\)\( T^{2} + \)\(18\!\cdots\!64\)\( T^{3} + \)\(47\!\cdots\!94\)\( T^{4} + \)\(18\!\cdots\!64\)\( p^{15} T^{5} + \)\(10\!\cdots\!52\)\( p^{30} T^{6} + 238381767283992 p^{45} T^{7} + p^{60} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 233193205883808 T + \)\(23\!\cdots\!76\)\( T^{2} + \)\(41\!\cdots\!96\)\( T^{3} + \)\(21\!\cdots\!78\)\( T^{4} + \)\(41\!\cdots\!96\)\( p^{15} T^{5} + \)\(23\!\cdots\!76\)\( p^{30} T^{6} + 233193205883808 p^{45} T^{7} + p^{60} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 316154161239228 T + \)\(44\!\cdots\!08\)\( T^{2} - \)\(72\!\cdots\!84\)\( T^{3} + \)\(92\!\cdots\!98\)\( T^{4} - \)\(72\!\cdots\!84\)\( p^{15} T^{5} + \)\(44\!\cdots\!08\)\( p^{30} T^{6} - 316154161239228 p^{45} T^{7} + p^{60} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 529302182472448 T + \)\(16\!\cdots\!32\)\( T^{2} - \)\(95\!\cdots\!28\)\( T^{3} + \)\(13\!\cdots\!54\)\( T^{4} - \)\(95\!\cdots\!28\)\( p^{15} T^{5} + \)\(16\!\cdots\!32\)\( p^{30} T^{6} - 529302182472448 p^{45} T^{7} + p^{60} T^{8} \) | |
| show more | |||
| show less | |||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.79576321376196731235534120222, −7.31723290468452156485409067592, −7.00474591344880371006682967496, −6.76123850394837565104374994692, −6.57068068223557112181532047825, −5.92946232430840995416595096322, −5.91591186227739977284445829569, −5.78260142731300809139965632962, −5.56609550406093171124310015586, −4.57440728644932762090186117366, −4.45035249154363372162007378454, −4.20999357860698643384200480872, −4.20988060423220080496113215531, −3.34298759106725210312896679390, −3.33329884581505037796558091228, −3.05410065571684765264538427838, −2.81662347270795369612948736317, −2.27919709855140855342368825951, −2.19720444290643574878759322151, −2.01588536529441972758175519266, −1.91800011126978696166160441087, −0.943946426059551794736821978517, −0.907051796159301597283130703656, −0.73832732718931652889904726124, −0.49275352232307030656316525749, 0.49275352232307030656316525749, 0.73832732718931652889904726124, 0.907051796159301597283130703656, 0.943946426059551794736821978517, 1.91800011126978696166160441087, 2.01588536529441972758175519266, 2.19720444290643574878759322151, 2.27919709855140855342368825951, 2.81662347270795369612948736317, 3.05410065571684765264538427838, 3.33329884581505037796558091228, 3.34298759106725210312896679390, 4.20988060423220080496113215531, 4.20999357860698643384200480872, 4.45035249154363372162007378454, 4.57440728644932762090186117366, 5.56609550406093171124310015586, 5.78260142731300809139965632962, 5.91591186227739977284445829569, 5.92946232430840995416595096322, 6.57068068223557112181532047825, 6.76123850394837565104374994692, 7.00474591344880371006682967496, 7.31723290468452156485409067592, 7.79576321376196731235534120222