Dirichlet series
| L(s) = 1 | + 4·3-s + 8·9-s + 4·11-s + 12·17-s − 12·23-s + 16·27-s − 12·29-s + 16·33-s + 12·37-s − 24·41-s + 48·47-s + 48·51-s + 40·61-s + 8·67-s − 48·69-s − 28·71-s − 48·73-s + 8·79-s + 42·81-s − 48·87-s + 24·89-s + 4·97-s + 32·99-s + 16·101-s − 80·103-s − 12·107-s + 8·109-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 8/3·9-s + 1.20·11-s + 2.91·17-s − 2.50·23-s + 3.07·27-s − 2.22·29-s + 2.78·33-s + 1.97·37-s − 3.74·41-s + 7.00·47-s + 6.72·51-s + 5.12·61-s + 0.977·67-s − 5.77·69-s − 3.32·71-s − 5.61·73-s + 0.900·79-s + 14/3·81-s − 5.14·87-s + 2.54·89-s + 0.406·97-s + 3.21·99-s + 1.59·101-s − 7.88·103-s − 1.16·107-s + 0.766·109-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{12} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{12} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 5^{12} \cdot 17^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.69024\times 10^{12}\) |
| Root analytic conductor: | \(3.29539\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{48} \cdot 5^{12} \cdot 17^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(8.014878697\) |
| \(L(\frac12)\) | \(\approx\) | \(8.014878697\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T^{4} )^{3} \) | |
| 17 | \( 1 - 12 T + 82 T^{2} - 516 T^{3} + 3079 T^{4} - 14512 T^{5} + 60188 T^{6} - 14512 p T^{7} + 3079 p^{2} T^{8} - 516 p^{3} T^{9} + 82 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 3 | \( 1 - 4 T + 8 T^{2} - 16 T^{3} + 22 T^{4} - 16 T^{5} + 16 T^{6} - 28 T^{7} - 29 T^{8} + 304 T^{9} - 760 T^{10} + 1928 T^{11} - 4304 T^{12} + 1928 p T^{13} - 760 p^{2} T^{14} + 304 p^{3} T^{15} - 29 p^{4} T^{16} - 28 p^{5} T^{17} + 16 p^{6} T^{18} - 16 p^{7} T^{19} + 22 p^{8} T^{20} - 16 p^{9} T^{21} + 8 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 - 4 p T^{3} + 6 p T^{4} + 212 T^{5} + 8 p^{2} T^{6} - 928 T^{7} - 3125 T^{8} + 6744 T^{9} + 31992 T^{10} + 888 p T^{11} - 376 p^{3} T^{12} + 888 p^{2} T^{13} + 31992 p^{2} T^{14} + 6744 p^{3} T^{15} - 3125 p^{4} T^{16} - 928 p^{5} T^{17} + 8 p^{8} T^{18} + 212 p^{7} T^{19} + 6 p^{9} T^{20} - 4 p^{10} T^{21} + p^{12} T^{24} \) | |
| 11 | \( 1 - 4 T + 8 T^{2} - 64 T^{3} + 74 T^{4} + 72 p T^{5} - 1712 T^{6} + 13364 T^{7} - 46833 T^{8} - 42384 T^{9} + 129224 T^{10} - 1266064 T^{11} + 10886032 T^{12} - 1266064 p T^{13} + 129224 p^{2} T^{14} - 42384 p^{3} T^{15} - 46833 p^{4} T^{16} + 13364 p^{5} T^{17} - 1712 p^{6} T^{18} + 72 p^{8} T^{19} + 74 p^{8} T^{20} - 64 p^{9} T^{21} + 8 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( ( 1 + 20 T^{2} + 12 T^{3} + 255 T^{4} - 124 T^{5} + 3948 T^{6} - 124 p T^{7} + 255 p^{2} T^{8} + 12 p^{3} T^{9} + 20 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 - 92 T^{2} + 4626 T^{4} - 167116 T^{6} + 4842975 T^{8} - 117605944 T^{10} + 2422842556 T^{12} - 117605944 p^{2} T^{14} + 4842975 p^{4} T^{16} - 167116 p^{6} T^{18} + 4626 p^{8} T^{20} - 92 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( 1 + 12 T + 72 T^{2} + 400 T^{3} + 2878 T^{4} + 18936 T^{5} + 100016 T^{6} + 521948 T^{7} + 2709363 T^{8} + 13154744 T^{9} + 63001992 T^{10} + 314746192 T^{11} + 1561862208 T^{12} + 314746192 p T^{13} + 63001992 p^{2} T^{14} + 13154744 p^{3} T^{15} + 2709363 p^{4} T^{16} + 521948 p^{5} T^{17} + 100016 p^{6} T^{18} + 18936 p^{7} T^{19} + 2878 p^{8} T^{20} + 400 p^{9} T^{21} + 72 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 12 T + 72 T^{2} + 708 T^{3} + 5562 T^{4} + 21252 T^{5} + 105192 T^{6} + 726444 T^{7} + 804479 T^{8} - 8873352 T^{9} - 9334512 T^{10} - 305351640 T^{11} - 3778100564 T^{12} - 305351640 p T^{13} - 9334512 p^{2} T^{14} - 8873352 p^{3} T^{15} + 804479 p^{4} T^{16} + 726444 p^{5} T^{17} + 105192 p^{6} T^{18} + 21252 p^{7} T^{19} + 5562 p^{8} T^{20} + 708 p^{9} T^{21} + 72 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 156 T^{3} - 2754 T^{4} - 452 T^{5} + 12168 T^{6} - 245112 T^{7} + 3961447 T^{8} - 323440 T^{9} - 4624648 T^{10} + 168645768 T^{11} - 4098407752 T^{12} + 168645768 p T^{13} - 4624648 p^{2} T^{14} - 323440 p^{3} T^{15} + 3961447 p^{4} T^{16} - 245112 p^{5} T^{17} + 12168 p^{6} T^{18} - 452 p^{7} T^{19} - 2754 p^{8} T^{20} + 156 p^{9} T^{21} + p^{12} T^{24} \) | |
| 37 | \( 1 - 12 T + 72 T^{2} - 308 T^{3} + 2250 T^{4} - 13364 T^{5} + 45800 T^{6} + 79636 T^{7} - 905 p^{2} T^{8} + 18417192 T^{9} - 170081904 T^{10} + 1483635480 T^{11} - 9533058292 T^{12} + 1483635480 p T^{13} - 170081904 p^{2} T^{14} + 18417192 p^{3} T^{15} - 905 p^{6} T^{16} + 79636 p^{5} T^{17} + 45800 p^{6} T^{18} - 13364 p^{7} T^{19} + 2250 p^{8} T^{20} - 308 p^{9} T^{21} + 72 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 24 T + 288 T^{2} + 2680 T^{3} + 20998 T^{4} + 135832 T^{5} + 803744 T^{6} + 4720696 T^{7} + 28651119 T^{8} + 187625072 T^{9} + 744256 p^{2} T^{10} + 199495088 p T^{11} + 52761079188 T^{12} + 199495088 p^{2} T^{13} + 744256 p^{4} T^{14} + 187625072 p^{3} T^{15} + 28651119 p^{4} T^{16} + 4720696 p^{5} T^{17} + 803744 p^{6} T^{18} + 135832 p^{7} T^{19} + 20998 p^{8} T^{20} + 2680 p^{9} T^{21} + 288 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 184 T^{2} + 18790 T^{4} - 1391328 T^{6} + 81780959 T^{8} - 4113094280 T^{10} + 185743711652 T^{12} - 4113094280 p^{2} T^{14} + 81780959 p^{4} T^{16} - 1391328 p^{6} T^{18} + 18790 p^{8} T^{20} - 184 p^{10} T^{22} + p^{12} T^{24} \) | |
| 47 | \( ( 1 - 24 T + 392 T^{2} - 4556 T^{3} + 43823 T^{4} - 353156 T^{5} + 2577076 T^{6} - 353156 p T^{7} + 43823 p^{2} T^{8} - 4556 p^{3} T^{9} + 392 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 53 | \( 1 - 204 T^{2} + 26562 T^{4} - 2416476 T^{6} + 178063743 T^{8} - 11083541624 T^{10} + 618583405660 T^{12} - 11083541624 p^{2} T^{14} + 178063743 p^{4} T^{16} - 2416476 p^{6} T^{18} + 26562 p^{8} T^{20} - 204 p^{10} T^{22} + p^{12} T^{24} \) | |
| 59 | \( 1 - 468 T^{2} + 102802 T^{4} - 14182372 T^{6} + 1399420159 T^{8} - 107717762600 T^{10} + 6886886586172 T^{12} - 107717762600 p^{2} T^{14} + 1399420159 p^{4} T^{16} - 14182372 p^{6} T^{18} + 102802 p^{8} T^{20} - 468 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( 1 - 40 T + 800 T^{2} - 12504 T^{3} + 184598 T^{4} - 40056 p T^{5} + 28233248 T^{6} - 305052840 T^{7} + 3158111167 T^{8} - 30151790672 T^{9} + 267303164864 T^{10} - 2283320825520 T^{11} + 18527414094644 T^{12} - 2283320825520 p T^{13} + 267303164864 p^{2} T^{14} - 30151790672 p^{3} T^{15} + 3158111167 p^{4} T^{16} - 305052840 p^{5} T^{17} + 28233248 p^{6} T^{18} - 40056 p^{8} T^{19} + 184598 p^{8} T^{20} - 12504 p^{9} T^{21} + 800 p^{10} T^{22} - 40 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( ( 1 - 4 T + 356 T^{2} - 1240 T^{3} + 55175 T^{4} - 159748 T^{5} + 4798716 T^{6} - 159748 p T^{7} + 55175 p^{2} T^{8} - 1240 p^{3} T^{9} + 356 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( 1 + 28 T + 392 T^{2} + 4432 T^{3} + 35314 T^{4} + 151184 T^{5} + 211376 T^{6} - 3536900 T^{7} - 37488993 T^{8} - 144764696 T^{9} - 1069498168 T^{10} - 21338963608 T^{11} - 259039666848 T^{12} - 21338963608 p T^{13} - 1069498168 p^{2} T^{14} - 144764696 p^{3} T^{15} - 37488993 p^{4} T^{16} - 3536900 p^{5} T^{17} + 211376 p^{6} T^{18} + 151184 p^{7} T^{19} + 35314 p^{8} T^{20} + 4432 p^{9} T^{21} + 392 p^{10} T^{22} + 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 48 T + 1152 T^{2} + 19520 T^{3} + 277734 T^{4} + 3608128 T^{5} + 43755776 T^{6} + 498961648 T^{7} + 5387964143 T^{8} + 54979433440 T^{9} + 528636621952 T^{10} + 4817379931648 T^{11} + 41994385065940 T^{12} + 4817379931648 p T^{13} + 528636621952 p^{2} T^{14} + 54979433440 p^{3} T^{15} + 5387964143 p^{4} T^{16} + 498961648 p^{5} T^{17} + 43755776 p^{6} T^{18} + 3608128 p^{7} T^{19} + 277734 p^{8} T^{20} + 19520 p^{9} T^{21} + 1152 p^{10} T^{22} + 48 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 8 T + 32 T^{2} + 1220 T^{3} + 5350 T^{4} - 114868 T^{5} + 1491944 T^{6} + 12162584 T^{7} - 38174937 T^{8} - 233812952 T^{9} + 16545882392 T^{10} + 38512918912 T^{11} - 355031479032 T^{12} + 38512918912 p T^{13} + 16545882392 p^{2} T^{14} - 233812952 p^{3} T^{15} - 38174937 p^{4} T^{16} + 12162584 p^{5} T^{17} + 1491944 p^{6} T^{18} - 114868 p^{7} T^{19} + 5350 p^{8} T^{20} + 1220 p^{9} T^{21} + 32 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 776 T^{2} + 288430 T^{4} - 67950336 T^{6} + 11308890671 T^{8} - 1401355662904 T^{10} + 132600210726644 T^{12} - 1401355662904 p^{2} T^{14} + 11308890671 p^{4} T^{16} - 67950336 p^{6} T^{18} + 288430 p^{8} T^{20} - 776 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( ( 1 - 12 T + 354 T^{2} - 4000 T^{3} + 61415 T^{4} - 604868 T^{5} + 6702448 T^{6} - 604868 p T^{7} + 61415 p^{2} T^{8} - 4000 p^{3} T^{9} + 354 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 - 4 T + 8 T^{2} + 20 T^{3} - 17958 T^{4} + 32004 T^{5} + 15848 T^{6} - 4133236 T^{7} + 241837519 T^{8} - 226090952 T^{9} - 316274672 T^{10} + 41141248808 T^{11} - 2546305955668 T^{12} + 41141248808 p T^{13} - 316274672 p^{2} T^{14} - 226090952 p^{3} T^{15} + 241837519 p^{4} T^{16} - 4133236 p^{5} T^{17} + 15848 p^{6} T^{18} + 32004 p^{7} T^{19} - 17958 p^{8} T^{20} + 20 p^{9} T^{21} + 8 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.87467178306198434771240929169, −2.83747551969600883765772348357, −2.79762820076549052721748167601, −2.78314265792647809048172872431, −2.77239483779136722162990636005, −2.54572457256362479322989396162, −2.47827945187767794705012089245, −2.36416757483348229052412079950, −2.31301590835713114910680254829, −2.19782611371988744974926111053, −2.11163731887521167573984138031, −1.95228827744915838285259316944, −1.92620217363590582974943322330, −1.68922643831731068273160803039, −1.68486601304972518862714741088, −1.37917078669640747816386018710, −1.37169126005033344449620601065, −1.24525131696623548235053632637, −1.20793054891270142365698803698, −1.12080566119254637334010862171, −1.06210918824986083540066322241, −0.76668584570644437326180090136, −0.45777424065147071658379372709, −0.44335027224307723538739865494, −0.10456323054836923581366889134, 0.10456323054836923581366889134, 0.44335027224307723538739865494, 0.45777424065147071658379372709, 0.76668584570644437326180090136, 1.06210918824986083540066322241, 1.12080566119254637334010862171, 1.20793054891270142365698803698, 1.24525131696623548235053632637, 1.37169126005033344449620601065, 1.37917078669640747816386018710, 1.68486601304972518862714741088, 1.68922643831731068273160803039, 1.92620217363590582974943322330, 1.95228827744915838285259316944, 2.11163731887521167573984138031, 2.19782611371988744974926111053, 2.31301590835713114910680254829, 2.36416757483348229052412079950, 2.47827945187767794705012089245, 2.54572457256362479322989396162, 2.77239483779136722162990636005, 2.78314265792647809048172872431, 2.79762820076549052721748167601, 2.83747551969600883765772348357, 2.87467178306198434771240929169
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.