Dirichlet series
| L(s) = 1 | − 2·9-s − 12·11-s + 16·23-s − 10·25-s + 12·29-s + 28·37-s − 12·43-s + 36·53-s − 20·67-s + 20·71-s − 16·79-s + 9·81-s + 24·99-s + 68·107-s + 24·109-s + 64·113-s + 78·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + ⋯ |
| L(s) = 1 | − 2/3·9-s − 3.61·11-s + 3.33·23-s − 2·25-s + 2.22·29-s + 4.60·37-s − 1.82·43-s + 4.94·53-s − 2.44·67-s + 2.37·71-s − 1.80·79-s + 81-s + 2.41·99-s + 6.57·107-s + 2.29·109-s + 6.02·113-s + 7.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{24} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{24} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{36} \cdot 7^{24} \cdot 11^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.77632\times 10^{18}\) |
| Root analytic conductor: | \(5.86783\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{36} \cdot 7^{24} \cdot 11^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(52.49790790\) |
| \(L(\frac12)\) | \(\approx\) | \(52.49790790\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) | |
| 11 | \( ( 1 + T )^{12} \) | |
| good | 3 | \( 1 + 2 T^{2} - 5 T^{4} + 28 T^{6} + 139 T^{8} + 106 T^{10} + 34 T^{12} + 106 p^{2} T^{14} + 139 p^{4} T^{16} + 28 p^{6} T^{18} - 5 p^{8} T^{20} + 2 p^{10} T^{22} + p^{12} T^{24} \) |
| 5 | \( 1 + 2 p T^{2} + 107 T^{4} + 556 T^{6} + 123 p^{2} T^{8} + 9554 T^{10} + 52274 T^{12} + 9554 p^{2} T^{14} + 123 p^{6} T^{16} + 556 p^{6} T^{18} + 107 p^{8} T^{20} + 2 p^{11} T^{22} + p^{12} T^{24} \) | |
| 13 | \( 1 + 10 T^{2} + 462 T^{4} + 2258 T^{6} + 81327 T^{8} + 16308 T^{10} + 11271716 T^{12} + 16308 p^{2} T^{14} + 81327 p^{4} T^{16} + 2258 p^{6} T^{18} + 462 p^{8} T^{20} + 10 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( 1 + 4 T^{2} + 642 T^{4} + 7284 T^{6} + 216239 T^{8} + 4648968 T^{10} + 62746652 T^{12} + 4648968 p^{2} T^{14} + 216239 p^{4} T^{16} + 7284 p^{6} T^{18} + 642 p^{8} T^{20} + 4 p^{10} T^{22} + p^{12} T^{24} \) | |
| 19 | \( 1 + 82 T^{2} + 4518 T^{4} + 178202 T^{6} + 5527823 T^{8} + 139984868 T^{10} + 2907909556 T^{12} + 139984868 p^{2} T^{14} + 5527823 p^{4} T^{16} + 178202 p^{6} T^{18} + 4518 p^{8} T^{20} + 82 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( ( 1 - 8 T + 63 T^{2} - 376 T^{3} + 1619 T^{4} - 6672 T^{5} + 32810 T^{6} - 6672 p T^{7} + 1619 p^{2} T^{8} - 376 p^{3} T^{9} + 63 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( ( 1 - 6 T + 78 T^{2} - 278 T^{3} + 3223 T^{4} - 14284 T^{5} + 127172 T^{6} - 14284 p T^{7} + 3223 p^{2} T^{8} - 278 p^{3} T^{9} + 78 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( 1 + 184 T^{2} + 15987 T^{4} + 925560 T^{6} + 42602499 T^{8} + 1679682896 T^{10} + 56637012002 T^{12} + 1679682896 p^{2} T^{14} + 42602499 p^{4} T^{16} + 925560 p^{6} T^{18} + 15987 p^{8} T^{20} + 184 p^{10} T^{22} + p^{12} T^{24} \) | |
| 37 | \( ( 1 - 14 T + 191 T^{2} - 1650 T^{3} + 14667 T^{4} - 99560 T^{5} + 18434 p T^{6} - 99560 p T^{7} + 14667 p^{2} T^{8} - 1650 p^{3} T^{9} + 191 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 + 260 T^{2} + 33186 T^{4} + 2842164 T^{6} + 185332175 T^{8} + 9791798024 T^{10} + 434803859036 T^{12} + 9791798024 p^{2} T^{14} + 185332175 p^{4} T^{16} + 2842164 p^{6} T^{18} + 33186 p^{8} T^{20} + 260 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( ( 1 + 6 T + 162 T^{2} + 1034 T^{3} + 325 p T^{4} + 73308 T^{5} + 763836 T^{6} + 73308 p T^{7} + 325 p^{3} T^{8} + 1034 p^{3} T^{9} + 162 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 + 352 T^{2} + 62022 T^{4} + 7218528 T^{6} + 618328527 T^{8} + 41022309696 T^{10} + 2157971955348 T^{12} + 41022309696 p^{2} T^{14} + 618328527 p^{4} T^{16} + 7218528 p^{6} T^{18} + 62022 p^{8} T^{20} + 352 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( ( 1 - 18 T + 286 T^{2} - 3426 T^{3} + 34839 T^{4} - 312404 T^{5} + 2416708 T^{6} - 312404 p T^{7} + 34839 p^{2} T^{8} - 3426 p^{3} T^{9} + 286 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 + 90 T^{2} + 2299 T^{4} + 201532 T^{6} + 29870571 T^{8} + 1590285330 T^{10} + 65172156898 T^{12} + 1590285330 p^{2} T^{14} + 29870571 p^{4} T^{16} + 201532 p^{6} T^{18} + 2299 p^{8} T^{20} + 90 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( 1 + 226 T^{2} + 35454 T^{4} + 3755818 T^{6} + 332670575 T^{8} + 23967317252 T^{10} + 1573801630916 T^{12} + 23967317252 p^{2} T^{14} + 332670575 p^{4} T^{16} + 3755818 p^{6} T^{18} + 35454 p^{8} T^{20} + 226 p^{10} T^{22} + p^{12} T^{24} \) | |
| 67 | \( ( 1 + 10 T + 211 T^{2} + 952 T^{3} + 15487 T^{4} + 28110 T^{5} + 937498 T^{6} + 28110 p T^{7} + 15487 p^{2} T^{8} + 952 p^{3} T^{9} + 211 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( ( 1 - 10 T + 147 T^{2} - 1624 T^{3} + 22999 T^{4} - 177726 T^{5} + 1691114 T^{6} - 177726 p T^{7} + 22999 p^{2} T^{8} - 1624 p^{3} T^{9} + 147 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 73 | \( 1 + 364 T^{2} + 62066 T^{4} + 6382716 T^{6} + 424031247 T^{8} + 18921128408 T^{10} + 892132116796 T^{12} + 18921128408 p^{2} T^{14} + 424031247 p^{4} T^{16} + 6382716 p^{6} T^{18} + 62066 p^{8} T^{20} + 364 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( ( 1 + 8 T + 270 T^{2} + 2136 T^{3} + 40287 T^{4} + 275664 T^{5} + 3948516 T^{6} + 275664 p T^{7} + 40287 p^{2} T^{8} + 2136 p^{3} T^{9} + 270 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 + 658 T^{2} + 220534 T^{4} + 48528890 T^{6} + 7737058575 T^{8} + 936117076420 T^{10} + 87906163117588 T^{12} + 936117076420 p^{2} T^{14} + 7737058575 p^{4} T^{16} + 48528890 p^{6} T^{18} + 220534 p^{8} T^{20} + 658 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( 1 + 184 T^{2} + 20587 T^{4} + 2498488 T^{6} + 329422595 T^{8} + 29630016528 T^{10} + 2362154448690 T^{12} + 29630016528 p^{2} T^{14} + 329422595 p^{4} T^{16} + 2498488 p^{6} T^{18} + 20587 p^{8} T^{20} + 184 p^{10} T^{22} + p^{12} T^{24} \) | |
| 97 | \( 1 + 872 T^{2} + 366779 T^{4} + 98491128 T^{6} + 18817342307 T^{8} + 2695919020512 T^{10} + 297053019336018 T^{12} + 2695919020512 p^{2} T^{14} + 18817342307 p^{4} T^{16} + 98491128 p^{6} T^{18} + 366779 p^{8} T^{20} + 872 p^{10} T^{22} + p^{12} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.59522292242607321481492520757, −2.34082103064399760859517111843, −2.31214937545771632638475787785, −2.18735422826193854532166376230, −2.11912710856484632355875744761, −2.08566595436442239892064146479, −2.07714349118545464015989948785, −2.05162640000562220282145171398, −2.03614283282534671026393178476, −1.79280280322521149559749691111, −1.67935252285718734755828766332, −1.63880323947709115387362507530, −1.41063579915323219970893371512, −1.33566276669642042674786278824, −1.32194797555044010042803647285, −0.961042828544102797312994870597, −0.901042220492683996953947303911, −0.879085094037521925172572599637, −0.845733748701363258484782901800, −0.797546217393079257243501019213, −0.55046800853768573913198680815, −0.46310075513013093243686814541, −0.44933604302240146544124678568, −0.38901992714681213213443584707, −0.26283415319915686589714730977, 0.26283415319915686589714730977, 0.38901992714681213213443584707, 0.44933604302240146544124678568, 0.46310075513013093243686814541, 0.55046800853768573913198680815, 0.797546217393079257243501019213, 0.845733748701363258484782901800, 0.879085094037521925172572599637, 0.901042220492683996953947303911, 0.961042828544102797312994870597, 1.32194797555044010042803647285, 1.33566276669642042674786278824, 1.41063579915323219970893371512, 1.63880323947709115387362507530, 1.67935252285718734755828766332, 1.79280280322521149559749691111, 2.03614283282534671026393178476, 2.05162640000562220282145171398, 2.07714349118545464015989948785, 2.08566595436442239892064146479, 2.11912710856484632355875744761, 2.18735422826193854532166376230, 2.31214937545771632638475787785, 2.34082103064399760859517111843, 2.59522292242607321481492520757
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.