Dirichlet series
| L(s) = 1 | − 4·3-s + 8·7-s + 8·9-s + 4·11-s + 8·13-s + 4·19-s − 32·21-s + 8·23-s − 4·25-s − 12·27-s − 32·31-s − 16·33-s + 8·37-s − 32·39-s + 8·41-s − 12·43-s + 32·49-s − 8·53-s − 16·57-s − 20·59-s − 24·61-s + 64·63-s − 36·67-s − 32·69-s + 24·71-s − 32·73-s + 16·75-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 3.02·7-s + 8/3·9-s + 1.20·11-s + 2.21·13-s + 0.917·19-s − 6.98·21-s + 1.66·23-s − 4/5·25-s − 2.30·27-s − 5.74·31-s − 2.78·33-s + 1.31·37-s − 5.12·39-s + 1.24·41-s − 1.82·43-s + 32/7·49-s − 1.09·53-s − 2.11·57-s − 2.60·59-s − 3.07·61-s + 8.06·63-s − 4.39·67-s − 3.85·69-s + 2.84·71-s − 3.74·73-s + 1.84·75-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{64}\) |
| Sign: | $1$ |
| Analytic conductor: | \(304.885\) |
| Root analytic conductor: | \(1.42974\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{64} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.595547523\) |
| \(L(\frac12)\) | \(\approx\) | \(1.595547523\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + 16 T^{4} + 4 p T^{5} + 8 p T^{6} + 68 T^{7} + 142 T^{8} + 68 p T^{9} + 8 p^{3} T^{10} + 4 p^{4} T^{11} + 16 p^{4} T^{12} + 4 p^{6} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( ( 1 + 2 T^{2} - 16 T^{3} + 2 T^{4} - 16 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) | |
| 7 | \( 1 - 8 T + 32 T^{2} - 104 T^{3} + 36 p T^{4} - 8 p^{2} T^{5} + 480 T^{6} - 24 p T^{7} - 26 p^{2} T^{8} - 24 p^{2} T^{9} + 480 p^{2} T^{10} - 8 p^{5} T^{11} + 36 p^{5} T^{12} - 104 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 4 T + 8 T^{2} + 68 T^{3} - 304 T^{4} + 804 T^{5} + 1304 T^{6} - 7396 T^{7} + 43982 T^{8} - 7396 p T^{9} + 1304 p^{2} T^{10} + 804 p^{3} T^{11} - 304 p^{4} T^{12} + 68 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 8 T + 36 T^{2} - 8 p T^{3} + 200 T^{4} + 72 T^{5} - 4084 T^{6} + 2632 p T^{7} - 153618 T^{8} + 2632 p^{2} T^{9} - 4084 p^{2} T^{10} + 72 p^{3} T^{11} + 200 p^{4} T^{12} - 8 p^{6} T^{13} + 36 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 72 T^{2} + 2620 T^{4} - 64376 T^{6} + 1215110 T^{8} - 64376 p^{2} T^{10} + 2620 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} \) | |
| 19 | \( 1 - 4 T - 8 T^{2} - 28 T^{3} + 336 T^{4} - 1308 T^{5} + 14056 T^{6} - 28228 T^{7} - 115058 T^{8} - 28228 p T^{9} + 14056 p^{2} T^{10} - 1308 p^{3} T^{11} + 336 p^{4} T^{12} - 28 p^{5} T^{13} - 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 8 T + 32 T^{2} - 200 T^{3} + 2108 T^{4} - 11560 T^{5} + 45024 T^{6} - 277224 T^{7} + 1704454 T^{8} - 277224 p T^{9} + 45024 p^{2} T^{10} - 11560 p^{3} T^{11} + 2108 p^{4} T^{12} - 200 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 12 T^{2} + 64 T^{3} + 72 T^{4} - 4416 T^{5} + 14140 T^{6} + 57344 T^{7} - 693650 T^{8} + 57344 p T^{9} + 14140 p^{2} T^{10} - 4416 p^{3} T^{11} + 72 p^{4} T^{12} + 64 p^{5} T^{13} - 12 p^{6} T^{14} + p^{8} T^{16} \) | |
| 31 | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) | |
| 37 | \( 1 - 8 T - 44 T^{2} + 760 T^{3} - 1272 T^{4} - 22872 T^{5} + 112156 T^{6} + 219496 T^{7} - 4000082 T^{8} + 219496 p T^{9} + 112156 p^{2} T^{10} - 22872 p^{3} T^{11} - 1272 p^{4} T^{12} + 760 p^{5} T^{13} - 44 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 8 T + 32 T^{2} - 88 T^{3} + 1788 T^{4} - 23048 T^{5} + 131040 T^{6} - 18264 p T^{7} + 2278 p^{2} T^{8} - 18264 p^{2} T^{9} + 131040 p^{2} T^{10} - 23048 p^{3} T^{11} + 1788 p^{4} T^{12} - 88 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 12 T + 56 T^{2} - 604 T^{3} - 8560 T^{4} - 59644 T^{5} - 32984 T^{6} + 2234540 T^{7} + 25213774 T^{8} + 2234540 p T^{9} - 32984 p^{2} T^{10} - 59644 p^{3} T^{11} - 8560 p^{4} T^{12} - 604 p^{5} T^{13} + 56 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 312 T^{2} + 44348 T^{4} - 3794696 T^{6} + 215798406 T^{8} - 3794696 p^{2} T^{10} + 44348 p^{4} T^{12} - 312 p^{6} T^{14} + p^{8} T^{16} \) | |
| 53 | \( 1 + 8 T + 100 T^{2} + 8 p T^{3} + 3016 T^{4} + 19768 T^{5} + 118540 T^{6} + 2296472 T^{7} + 11779502 T^{8} + 2296472 p T^{9} + 118540 p^{2} T^{10} + 19768 p^{3} T^{11} + 3016 p^{4} T^{12} + 8 p^{6} T^{13} + 100 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 20 T + 136 T^{2} - 1124 T^{3} - 27632 T^{4} - 217412 T^{5} - 194408 T^{6} + 12195572 T^{7} + 142178894 T^{8} + 12195572 p T^{9} - 194408 p^{2} T^{10} - 217412 p^{3} T^{11} - 27632 p^{4} T^{12} - 1124 p^{5} T^{13} + 136 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 24 T + 132 T^{2} - 648 T^{3} + 72 T^{4} + 138600 T^{5} + 1067820 T^{6} + 2497416 T^{7} + 88430 T^{8} + 2497416 p T^{9} + 1067820 p^{2} T^{10} + 138600 p^{3} T^{11} + 72 p^{4} T^{12} - 648 p^{5} T^{13} + 132 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 36 T + 504 T^{2} + 2652 T^{3} - 10800 T^{4} - 217380 T^{5} - 1006488 T^{6} - 637596 T^{7} + 11635982 T^{8} - 637596 p T^{9} - 1006488 p^{2} T^{10} - 217380 p^{3} T^{11} - 10800 p^{4} T^{12} + 2652 p^{5} T^{13} + 504 p^{6} T^{14} + 36 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 24 T + 288 T^{2} - 2904 T^{3} + 21308 T^{4} - 62712 T^{5} - 415008 T^{6} + 11818440 T^{7} - 144298362 T^{8} + 11818440 p T^{9} - 415008 p^{2} T^{10} - 62712 p^{3} T^{11} + 21308 p^{4} T^{12} - 2904 p^{5} T^{13} + 288 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 32 T + 512 T^{2} + 96 p T^{3} + 95708 T^{4} + 1109024 T^{5} + 11042304 T^{6} + 107800160 T^{7} + 989395590 T^{8} + 107800160 p T^{9} + 11042304 p^{2} T^{10} + 1109024 p^{3} T^{11} + 95708 p^{4} T^{12} + 96 p^{6} T^{13} + 512 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 120 T^{2} + 10940 T^{4} + 170552 T^{6} - 23710074 T^{8} + 170552 p^{2} T^{10} + 10940 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} \) | |
| 83 | \( 1 - 20 T + 184 T^{2} - 2684 T^{3} + 37008 T^{4} - 337916 T^{5} + 3624872 T^{6} - 38211284 T^{7} + 335969678 T^{8} - 38211284 p T^{9} + 3624872 p^{2} T^{10} - 337916 p^{3} T^{11} + 37008 p^{4} T^{12} - 2684 p^{5} T^{13} + 184 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 16 T + 128 T^{2} + 2096 T^{3} + 18652 T^{4} - 22640 T^{5} - 553088 T^{6} - 12396240 T^{7} - 224072442 T^{8} - 12396240 p T^{9} - 553088 p^{2} T^{10} - 22640 p^{3} T^{11} + 18652 p^{4} T^{12} + 2096 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( ( 1 - 16 T + 428 T^{2} - 4368 T^{3} + 63222 T^{4} - 4368 p T^{5} + 428 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.47364135744340532531006203128, −5.30716561134383202819928653108, −5.02603819273900233487019414136, −4.90498100688558481172343842555, −4.88207364666038464573694613481, −4.79882858504322103384065343789, −4.58965041661086261495962137298, −4.50992329846976011838640936671, −4.14103730336549341602736423694, −4.13754776064998006319109655548, −4.10581246246510330798248825213, −3.57436855186128655525494349831, −3.52541632529578599828463967728, −3.44392181827314499916270087398, −3.15994750966775071054812577723, −3.03686114827668590796831728618, −2.99120739169023281054858749985, −2.26865181478770828047109475372, −1.98036305591451127709235678780, −1.77717380933825874944991177655, −1.73629161549614519628228408764, −1.49318536335302826785765139502, −1.43127219942881954151784623146, −1.04332155301220220846036178964, −0.47603890690648962035794115000, 0.47603890690648962035794115000, 1.04332155301220220846036178964, 1.43127219942881954151784623146, 1.49318536335302826785765139502, 1.73629161549614519628228408764, 1.77717380933825874944991177655, 1.98036305591451127709235678780, 2.26865181478770828047109475372, 2.99120739169023281054858749985, 3.03686114827668590796831728618, 3.15994750966775071054812577723, 3.44392181827314499916270087398, 3.52541632529578599828463967728, 3.57436855186128655525494349831, 4.10581246246510330798248825213, 4.13754776064998006319109655548, 4.14103730336549341602736423694, 4.50992329846976011838640936671, 4.58965041661086261495962137298, 4.79882858504322103384065343789, 4.88207364666038464573694613481, 4.90498100688558481172343842555, 5.02603819273900233487019414136, 5.30716561134383202819928653108, 5.47364135744340532531006203128
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.