Dirichlet series
| L(s) = 1 | + 2.69e4·5-s + 4.12e5·7-s + 4.23e7·11-s + 1.59e8·13-s − 4.22e8·17-s − 1.39e9·19-s + 8.25e9·23-s − 5.63e10·25-s − 2.85e10·29-s + 6.34e10·31-s + 1.11e10·35-s − 2.55e11·37-s + 6.05e11·41-s + 4.78e10·43-s − 4.87e11·47-s − 8.98e12·49-s + 1.41e12·53-s + 1.13e12·55-s + 9.72e12·59-s − 1.91e12·61-s + 4.28e12·65-s + 2.21e13·67-s + 4.16e13·71-s + 1.84e13·73-s + 1.74e13·77-s − 1.22e14·79-s + 4.07e14·83-s + ⋯ |
| L(s) = 1 | + 0.154·5-s + 0.189·7-s + 0.654·11-s + 0.703·13-s − 0.249·17-s − 0.359·19-s + 0.505·23-s − 1.84·25-s − 0.307·29-s + 0.414·31-s + 0.0291·35-s − 0.443·37-s + 0.485·41-s + 0.0268·43-s − 0.140·47-s − 1.89·49-s + 0.165·53-s + 0.100·55-s + 0.508·59-s − 0.0781·61-s + 0.108·65-s + 0.447·67-s + 0.543·71-s + 0.195·73-s + 0.123·77-s − 0.718·79-s + 1.64·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+15/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(26873856\) = \(2^{12} \cdot 3^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.11415\times 10^{8}\) |
| Root analytic conductor: | \(10.1360\) |
| Motivic weight: | \(15\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 26873856,\ (\ :15/2, 15/2, 15/2, 15/2),\ 1)\) |
Particular Values
| \(L(8)\) | \(\approx\) | \(5.702907857\) |
| \(L(\frac12)\) | \(\approx\) | \(5.702907857\) |
| \(L(\frac{17}{2})\) | 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 | \( 1 \) | ||
| good | 5 | $C_2 \wr S_4$ | \( 1 - 26944 T + 11406727588 p T^{2} - 393198269220544 p^{2} T^{3} + 11217561351935992238 p^{3} T^{4} - 393198269220544 p^{17} T^{5} + 11406727588 p^{31} T^{6} - 26944 p^{45} T^{7} + p^{60} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 412176 T + 1307143333348 p T^{2} - 3574247980510800 p^{2} T^{3} + \)\(13\!\cdots\!82\)\( p^{3} T^{4} - 3574247980510800 p^{17} T^{5} + 1307143333348 p^{31} T^{6} - 412176 p^{45} T^{7} + p^{60} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 42301696 T + 733128753318436 p T^{2} - \)\(40\!\cdots\!52\)\( p^{2} T^{3} + \)\(24\!\cdots\!62\)\( p^{3} T^{4} - \)\(40\!\cdots\!52\)\( p^{17} T^{5} + 733128753318436 p^{31} T^{6} - 42301696 p^{45} T^{7} + p^{60} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 159108904 T + 6735049690414876 p T^{2} - \)\(14\!\cdots\!32\)\( p^{2} T^{3} + \)\(12\!\cdots\!14\)\( p^{3} T^{4} - \)\(14\!\cdots\!32\)\( p^{17} T^{5} + 6735049690414876 p^{31} T^{6} - 159108904 p^{45} T^{7} + p^{60} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 422803840 T + 2448515386798788548 T^{2} + \)\(15\!\cdots\!48\)\( p T^{3} + \)\(20\!\cdots\!42\)\( T^{4} + \)\(15\!\cdots\!48\)\( p^{16} T^{5} + 2448515386798788548 p^{30} T^{6} + 422803840 p^{45} T^{7} + p^{60} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 1399441312 T + 28675996312933468396 T^{2} + \)\(45\!\cdots\!32\)\( T^{3} + \)\(34\!\cdots\!90\)\( p T^{4} + \)\(45\!\cdots\!32\)\( p^{15} T^{5} + 28675996312933468396 p^{30} T^{6} + 1399441312 p^{45} T^{7} + p^{60} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 8256921088 T + \)\(52\!\cdots\!96\)\( T^{2} - \)\(10\!\cdots\!88\)\( T^{3} + \)\(13\!\cdots\!74\)\( T^{4} - \)\(10\!\cdots\!88\)\( p^{15} T^{5} + \)\(52\!\cdots\!96\)\( p^{30} T^{6} - 8256921088 p^{45} T^{7} + p^{60} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 28527814464 T + \)\(23\!\cdots\!00\)\( T^{2} + \)\(11\!\cdots\!24\)\( T^{3} + \)\(25\!\cdots\!38\)\( T^{4} + \)\(11\!\cdots\!24\)\( p^{15} T^{5} + \)\(23\!\cdots\!00\)\( p^{30} T^{6} + 28527814464 p^{45} T^{7} + p^{60} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 63433875344 T + \)\(33\!\cdots\!04\)\( T^{2} - \)\(41\!\cdots\!32\)\( T^{3} + \)\(71\!\cdots\!06\)\( T^{4} - \)\(41\!\cdots\!32\)\( p^{15} T^{5} + \)\(33\!\cdots\!04\)\( p^{30} T^{6} - 63433875344 p^{45} T^{7} + p^{60} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 255928251912 T + \)\(67\!\cdots\!08\)\( T^{2} + \)\(29\!\cdots\!80\)\( T^{3} + \)\(23\!\cdots\!90\)\( T^{4} + \)\(29\!\cdots\!80\)\( p^{15} T^{5} + \)\(67\!\cdots\!08\)\( p^{30} T^{6} + 255928251912 p^{45} T^{7} + p^{60} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 605224516992 T + \)\(39\!\cdots\!80\)\( T^{2} - \)\(19\!\cdots\!68\)\( T^{3} + \)\(86\!\cdots\!58\)\( T^{4} - \)\(19\!\cdots\!68\)\( p^{15} T^{5} + \)\(39\!\cdots\!80\)\( p^{30} T^{6} - 605224516992 p^{45} T^{7} + p^{60} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 47867311456 T + \)\(42\!\cdots\!28\)\( T^{2} + \)\(28\!\cdots\!24\)\( T^{3} + \)\(11\!\cdots\!94\)\( T^{4} + \)\(28\!\cdots\!24\)\( p^{15} T^{5} + \)\(42\!\cdots\!28\)\( p^{30} T^{6} - 47867311456 p^{45} T^{7} + p^{60} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 487138638336 T + \)\(20\!\cdots\!08\)\( T^{2} + \)\(37\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!26\)\( T^{4} + \)\(37\!\cdots\!60\)\( p^{15} T^{5} + \)\(20\!\cdots\!08\)\( p^{30} T^{6} + 487138638336 p^{45} T^{7} + p^{60} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 1419343288000 T + \)\(16\!\cdots\!72\)\( T^{2} - \)\(74\!\cdots\!08\)\( T^{3} + \)\(12\!\cdots\!82\)\( T^{4} - \)\(74\!\cdots\!08\)\( p^{15} T^{5} + \)\(16\!\cdots\!72\)\( p^{30} T^{6} - 1419343288000 p^{45} T^{7} + p^{60} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 9727633870336 T + \)\(46\!\cdots\!40\)\( T^{2} - \)\(12\!\cdots\!28\)\( T^{3} + \)\(20\!\cdots\!86\)\( T^{4} - \)\(12\!\cdots\!28\)\( p^{15} T^{5} + \)\(46\!\cdots\!40\)\( p^{30} T^{6} - 9727633870336 p^{45} T^{7} + p^{60} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 1917295601768 T + \)\(11\!\cdots\!52\)\( T^{2} + \)\(10\!\cdots\!64\)\( T^{3} + \)\(82\!\cdots\!30\)\( T^{4} + \)\(10\!\cdots\!64\)\( p^{15} T^{5} + \)\(11\!\cdots\!52\)\( p^{30} T^{6} + 1917295601768 p^{45} T^{7} + p^{60} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 22198458997568 T + \)\(44\!\cdots\!68\)\( T^{2} + \)\(48\!\cdots\!56\)\( T^{3} + \)\(92\!\cdots\!18\)\( T^{4} + \)\(48\!\cdots\!56\)\( p^{15} T^{5} + \)\(44\!\cdots\!68\)\( p^{30} T^{6} - 22198458997568 p^{45} T^{7} + p^{60} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 41653926201344 T + \)\(16\!\cdots\!64\)\( T^{2} - \)\(75\!\cdots\!76\)\( T^{3} + \)\(12\!\cdots\!10\)\( T^{4} - \)\(75\!\cdots\!76\)\( p^{15} T^{5} + \)\(16\!\cdots\!64\)\( p^{30} T^{6} - 41653926201344 p^{45} T^{7} + p^{60} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 18428980892440 T + \)\(31\!\cdots\!60\)\( T^{2} - \)\(57\!\cdots\!80\)\( T^{3} + \)\(40\!\cdots\!98\)\( T^{4} - \)\(57\!\cdots\!80\)\( p^{15} T^{5} + \)\(31\!\cdots\!60\)\( p^{30} T^{6} - 18428980892440 p^{45} T^{7} + p^{60} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 122607797063024 T + \)\(41\!\cdots\!76\)\( T^{2} - \)\(70\!\cdots\!08\)\( T^{3} + \)\(60\!\cdots\!30\)\( T^{4} - \)\(70\!\cdots\!08\)\( p^{15} T^{5} + \)\(41\!\cdots\!76\)\( p^{30} T^{6} + 122607797063024 p^{45} T^{7} + p^{60} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 407113695568640 T + \)\(25\!\cdots\!12\)\( T^{2} - \)\(72\!\cdots\!92\)\( T^{3} + \)\(23\!\cdots\!02\)\( T^{4} - \)\(72\!\cdots\!92\)\( p^{15} T^{5} + \)\(25\!\cdots\!12\)\( p^{30} T^{6} - 407113695568640 p^{45} T^{7} + p^{60} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 516461222030592 T + \)\(18\!\cdots\!76\)\( T^{2} - \)\(11\!\cdots\!24\)\( T^{3} + \)\(74\!\cdots\!46\)\( T^{4} - \)\(11\!\cdots\!24\)\( p^{15} T^{5} + \)\(18\!\cdots\!76\)\( p^{30} T^{6} - 516461222030592 p^{45} T^{7} + p^{60} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 127031721196744 T + \)\(18\!\cdots\!04\)\( T^{2} + \)\(23\!\cdots\!92\)\( T^{3} + \)\(16\!\cdots\!98\)\( T^{4} + \)\(23\!\cdots\!92\)\( p^{15} T^{5} + \)\(18\!\cdots\!04\)\( p^{30} T^{6} + 127031721196744 p^{45} T^{7} + p^{60} T^{8} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.87481769365116935436431472383, −7.51495461847324843858834559600, −7.13748543940502818729281250415, −6.95508171894614448358633170376, −6.74969413713599435392219229951, −6.11578407610013074581826448350, −5.96565955195264719578951020386, −5.90454163451424189106177682768, −5.62695544553988223601921660146, −4.86287177277616055523410275614, −4.79947101432277937131018988196, −4.41980381138221418571958020394, −4.37842972078737925181157795635, −3.63005280860463740649400199990, −3.46684776080782949632294881148, −3.27060768240463266341122187734, −3.12436986315003711122796924897, −2.15544741891414744109199357449, −2.08940819503213261546909401297, −1.98287263031550018529870689784, −1.74640063851180921466567253817, −0.947255955311532678451514479096, −0.928637890151860421445701856588, −0.60049520363929348204175550464, −0.25682654783024653066908857550, 0.25682654783024653066908857550, 0.60049520363929348204175550464, 0.928637890151860421445701856588, 0.947255955311532678451514479096, 1.74640063851180921466567253817, 1.98287263031550018529870689784, 2.08940819503213261546909401297, 2.15544741891414744109199357449, 3.12436986315003711122796924897, 3.27060768240463266341122187734, 3.46684776080782949632294881148, 3.63005280860463740649400199990, 4.37842972078737925181157795635, 4.41980381138221418571958020394, 4.79947101432277937131018988196, 4.86287177277616055523410275614, 5.62695544553988223601921660146, 5.90454163451424189106177682768, 5.96565955195264719578951020386, 6.11578407610013074581826448350, 6.74969413713599435392219229951, 6.95508171894614448358633170376, 7.13748543940502818729281250415, 7.51495461847324843858834559600, 7.87481769365116935436431472383