Dirichlet series
| L(s) = 1 | − 324·3-s − 1.72e3·5-s + 4.84e3·7-s + 6.56e4·9-s − 1.58e4·11-s − 8.24e4·13-s + 5.59e5·15-s + 1.65e5·17-s − 5.39e5·19-s − 1.56e6·21-s + 7.29e5·23-s − 2.30e6·25-s − 1.06e7·27-s − 1.85e6·29-s + 3.19e6·31-s + 5.12e6·33-s − 8.36e6·35-s − 4.18e6·37-s + 2.67e7·39-s + 2.27e5·41-s − 1.60e6·43-s − 1.13e8·45-s + 1.80e7·47-s − 4.01e7·49-s − 5.37e7·51-s − 2.94e7·53-s + 2.73e7·55-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.23·5-s + 0.761·7-s + 10/3·9-s − 0.325·11-s − 0.800·13-s + 2.85·15-s + 0.481·17-s − 0.950·19-s − 1.75·21-s + 0.543·23-s − 1.17·25-s − 3.84·27-s − 0.485·29-s + 0.621·31-s + 0.752·33-s − 0.942·35-s − 0.367·37-s + 1.84·39-s + 0.0125·41-s − 0.0713·43-s − 4.12·45-s + 0.539·47-s − 0.994·49-s − 1.11·51-s − 0.512·53-s + 0.402·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{28} \cdot 3^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.52994\times 10^{9}\) |
| Root analytic conductor: | \(14.0632\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{11}{2})\) | 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 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 1728 T + 5286836 T^{2} + 1344360512 p T^{3} + 585667235526 p^{2} T^{4} + 1344360512 p^{10} T^{5} + 5286836 p^{18} T^{6} + 1728 p^{27} T^{7} + p^{36} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 4840 T + 9079372 p T^{2} - 9122761384 p^{2} T^{3} + 9758964239914 p^{3} T^{4} - 9122761384 p^{11} T^{5} + 9079372 p^{19} T^{6} - 4840 p^{27} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 15824 T + 3272030348 T^{2} + 110832700706512 T^{3} + 11168899925921994134 T^{4} + 110832700706512 p^{9} T^{5} + 3272030348 p^{18} T^{6} + 15824 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 82440 T + 32507843788 T^{2} + 2806427918898776 T^{3} + \)\(46\!\cdots\!74\)\( T^{4} + 2806427918898776 p^{9} T^{5} + 32507843788 p^{18} T^{6} + 82440 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 165912 T + 402742640540 T^{2} - 57573640941296168 T^{3} + \)\(68\!\cdots\!90\)\( T^{4} - 57573640941296168 p^{9} T^{5} + 402742640540 p^{18} T^{6} - 165912 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 28416 p T + 878148739660 T^{2} + 411395194240546048 T^{3} + \)\(35\!\cdots\!02\)\( T^{4} + 411395194240546048 p^{9} T^{5} + 878148739660 p^{18} T^{6} + 28416 p^{28} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 729680 T + 3870147116348 T^{2} - 3090843717350364304 T^{3} + \)\(76\!\cdots\!54\)\( T^{4} - 3090843717350364304 p^{9} T^{5} + 3870147116348 p^{18} T^{6} - 729680 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 1850864 T + 51896421352052 T^{2} + 76884864756743058576 T^{3} + \)\(10\!\cdots\!42\)\( T^{4} + 76884864756743058576 p^{9} T^{5} + 51896421352052 p^{18} T^{6} + 1850864 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 103160 p T + 80766069413044 T^{2} - \)\(19\!\cdots\!92\)\( T^{3} + \)\(28\!\cdots\!74\)\( T^{4} - \)\(19\!\cdots\!92\)\( p^{9} T^{5} + 80766069413044 p^{18} T^{6} - 103160 p^{28} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 4187992 T + 316603511258092 T^{2} + \)\(22\!\cdots\!72\)\( T^{3} + \)\(48\!\cdots\!50\)\( T^{4} + \)\(22\!\cdots\!72\)\( p^{9} T^{5} + 316603511258092 p^{18} T^{6} + 4187992 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 227704 T + 815482715263292 T^{2} - \)\(24\!\cdots\!80\)\( T^{3} + \)\(33\!\cdots\!46\)\( T^{4} - \)\(24\!\cdots\!80\)\( p^{9} T^{5} + 815482715263292 p^{18} T^{6} - 227704 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 1600352 T + 1039425217732396 T^{2} - \)\(25\!\cdots\!40\)\( T^{3} + \)\(54\!\cdots\!14\)\( T^{4} - \)\(25\!\cdots\!40\)\( p^{9} T^{5} + 1039425217732396 p^{18} T^{6} + 1600352 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 18053904 T + 898850271089564 T^{2} - \)\(31\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!54\)\( T^{4} - \)\(31\!\cdots\!80\)\( p^{9} T^{5} + 898850271089564 p^{18} T^{6} - 18053904 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 29418288 T + 8246475311053268 T^{2} + \)\(75\!\cdots\!52\)\( T^{3} + \)\(31\!\cdots\!10\)\( T^{4} + \)\(75\!\cdots\!52\)\( p^{9} T^{5} + 8246475311053268 p^{18} T^{6} + 29418288 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 38300048 T + 12480247782213068 T^{2} + \)\(64\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!66\)\( T^{4} + \)\(64\!\cdots\!60\)\( p^{9} T^{5} + 12480247782213068 p^{18} T^{6} + 38300048 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 99764648 T + 35070113945959756 T^{2} - \)\(25\!\cdots\!08\)\( T^{3} + \)\(53\!\cdots\!54\)\( T^{4} - \)\(25\!\cdots\!08\)\( p^{9} T^{5} + 35070113945959756 p^{18} T^{6} - 99764648 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 183717008 T + 94982362218124012 T^{2} - \)\(13\!\cdots\!36\)\( T^{3} + \)\(56\!\cdots\!98\)\( p T^{4} - \)\(13\!\cdots\!36\)\( p^{9} T^{5} + 94982362218124012 p^{18} T^{6} - 183717008 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 181868080 T + 187559006549429756 T^{2} - \)\(24\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!86\)\( T^{4} - \)\(24\!\cdots\!80\)\( p^{9} T^{5} + 187559006549429756 p^{18} T^{6} - 181868080 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 254539160 T + 208692556022844412 T^{2} - \)\(39\!\cdots\!64\)\( T^{3} + \)\(17\!\cdots\!82\)\( T^{4} - \)\(39\!\cdots\!64\)\( p^{9} T^{5} + 208692556022844412 p^{18} T^{6} - 254539160 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 831578184 T + 541079473633225972 T^{2} - \)\(23\!\cdots\!64\)\( T^{3} + \)\(93\!\cdots\!90\)\( T^{4} - \)\(23\!\cdots\!64\)\( p^{9} T^{5} + 541079473633225972 p^{18} T^{6} - 831578184 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 687923952 T + 569668186782399404 T^{2} - \)\(18\!\cdots\!04\)\( T^{3} + \)\(11\!\cdots\!50\)\( T^{4} - \)\(18\!\cdots\!04\)\( p^{9} T^{5} + 569668186782399404 p^{18} T^{6} - 687923952 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 627627272 T + 1209510335623979132 T^{2} - \)\(64\!\cdots\!44\)\( T^{3} + \)\(60\!\cdots\!14\)\( T^{4} - \)\(64\!\cdots\!44\)\( p^{9} T^{5} + 1209510335623979132 p^{18} T^{6} - 627627272 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 889385880 T + 757736193785874268 T^{2} + \)\(68\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!34\)\( T^{4} + \)\(68\!\cdots\!80\)\( p^{9} T^{5} + 757736193785874268 p^{18} T^{6} + 889385880 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−7.21953944298647234879238008663, −6.70977829897245237584896270327, −6.56811404743840202616116417650, −6.40366981143595671452677332712, −6.40069119132911164184641280463, −5.84300716679751072964329906050, −5.45509633397555028154597785826, −5.43389612054543715067250299516, −5.29653747536534842203075401286, −4.90532976385749784906739601087, −4.73795703653614007245062783312, −4.45795018179924614251708434492, −4.29563866172337722579707232297, −3.98139426579004466978176649471, −3.57644495740852728200879200734, −3.44447708796586852349358038108, −3.40393120353781581012110011707, −2.56856758471296464469941357079, −2.29251621766701040935145204151, −2.08432114502390631600488445208, −2.07997878355662713525033593925, −1.31408191973900606143495366254, −1.13015958075911418184787062208, −1.04489250663729980680634738429, −0.73616208536664821684434958386, 0, 0, 0, 0, 0.73616208536664821684434958386, 1.04489250663729980680634738429, 1.13015958075911418184787062208, 1.31408191973900606143495366254, 2.07997878355662713525033593925, 2.08432114502390631600488445208, 2.29251621766701040935145204151, 2.56856758471296464469941357079, 3.40393120353781581012110011707, 3.44447708796586852349358038108, 3.57644495740852728200879200734, 3.98139426579004466978176649471, 4.29563866172337722579707232297, 4.45795018179924614251708434492, 4.73795703653614007245062783312, 4.90532976385749784906739601087, 5.29653747536534842203075401286, 5.43389612054543715067250299516, 5.45509633397555028154597785826, 5.84300716679751072964329906050, 6.40069119132911164184641280463, 6.40366981143595671452677332712, 6.56811404743840202616116417650, 6.70977829897245237584896270327, 7.21953944298647234879238008663