Dirichlet series
| L(s) = 1 | + 4·7-s − 8·9-s + 4·11-s − 24·17-s − 84·19-s + 48·23-s − 38·25-s + 104·29-s − 156·31-s − 68·37-s + 160·43-s + 108·47-s − 32·49-s + 28·53-s + 120·59-s − 252·61-s − 32·63-s − 56·67-s + 208·71-s − 156·73-s + 16·77-s + 160·79-s + 101·81-s + 204·89-s − 32·99-s − 96·101-s + 24·103-s + ⋯ |
| L(s) = 1 | + 4/7·7-s − 8/9·9-s + 4/11·11-s − 1.41·17-s − 4.42·19-s + 2.08·23-s − 1.51·25-s + 3.58·29-s − 5.03·31-s − 1.83·37-s + 3.72·43-s + 2.29·47-s − 0.653·49-s + 0.528·53-s + 2.03·59-s − 4.13·61-s − 0.507·63-s − 0.835·67-s + 2.92·71-s − 2.13·73-s + 0.207·77-s + 2.02·79-s + 1.24·81-s + 2.29·89-s − 0.323·99-s − 0.950·101-s + 0.233·103-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{48} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.93065\times 10^{8}\) |
| Root analytic conductor: | \(3.49386\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{48} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.03885797314\) |
| \(L(\frac12)\) | \(\approx\) | \(0.03885797314\) |
| \(L(2)\) | 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 | \( 1 - 4 T + 48 T^{2} - 332 T^{3} + 722 p T^{4} - 332 p^{2} T^{5} + 48 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) | |
| good | 3 | \( 1 + 8 T^{2} - 37 T^{4} + 4 p T^{5} - 440 T^{6} - 404 p T^{7} - 1112 T^{8} - 404 p^{3} T^{9} - 440 p^{4} T^{10} + 4 p^{7} T^{11} - 37 p^{8} T^{12} + 8 p^{12} T^{14} + p^{16} T^{16} \) |
| 5 | \( 1 + 38 T^{2} + 441 T^{4} + 624 T^{5} - 8618 T^{6} + 48144 T^{7} - 267916 T^{8} + 48144 p^{2} T^{9} - 8618 p^{4} T^{10} + 624 p^{6} T^{11} + 441 p^{8} T^{12} + 38 p^{12} T^{14} + p^{16} T^{16} \) | |
| 11 | \( 1 - 4 T - 200 T^{2} + 392 p T^{3} + 1225 p T^{4} - 62684 p T^{5} + 6839000 T^{6} + 56462416 T^{7} - 1130023208 T^{8} + 56462416 p^{2} T^{9} + 6839000 p^{4} T^{10} - 62684 p^{7} T^{11} + 1225 p^{9} T^{12} + 392 p^{11} T^{13} - 200 p^{12} T^{14} - 4 p^{14} T^{15} + p^{16} T^{16} \) | |
| 13 | \( 1 - 640 T^{2} + 235068 T^{4} - 4641152 p T^{6} + 11718042566 T^{8} - 4641152 p^{5} T^{10} + 235068 p^{8} T^{12} - 640 p^{12} T^{14} + p^{16} T^{16} \) | |
| 17 | \( 1 + 24 T + 990 T^{2} + 19152 T^{3} + 27009 p T^{4} + 9113136 T^{5} + 188222142 T^{6} + 3566193384 T^{7} + 64414768676 T^{8} + 3566193384 p^{2} T^{9} + 188222142 p^{4} T^{10} + 9113136 p^{6} T^{11} + 27009 p^{9} T^{12} + 19152 p^{10} T^{13} + 990 p^{12} T^{14} + 24 p^{14} T^{15} + p^{16} T^{16} \) | |
| 19 | \( 1 + 84 T + 4352 T^{2} + 168000 T^{3} + 5284203 T^{4} + 141943956 T^{5} + 3384825040 T^{6} + 72944180232 T^{7} + 1444176057608 T^{8} + 72944180232 p^{2} T^{9} + 3384825040 p^{4} T^{10} + 141943956 p^{6} T^{11} + 5284203 p^{8} T^{12} + 168000 p^{10} T^{13} + 4352 p^{12} T^{14} + 84 p^{14} T^{15} + p^{16} T^{16} \) | |
| 23 | \( 1 - 48 T - 576 T^{2} + 23016 T^{3} + 1930675 T^{4} - 35477076 T^{5} - 1116060912 T^{6} - 1166650260 T^{7} + 1061708855256 T^{8} - 1166650260 p^{2} T^{9} - 1116060912 p^{4} T^{10} - 35477076 p^{6} T^{11} + 1930675 p^{8} T^{12} + 23016 p^{10} T^{13} - 576 p^{12} T^{14} - 48 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( ( 1 - 52 T + 1680 T^{2} - 32444 T^{3} + 385694 T^{4} - 32444 p^{2} T^{5} + 1680 p^{4} T^{6} - 52 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 31 | \( 1 + 156 T + 13424 T^{2} + 828672 T^{3} + 40603131 T^{4} + 1696628940 T^{5} + 63304318432 T^{6} + 2173306236072 T^{7} + 69665997781640 T^{8} + 2173306236072 p^{2} T^{9} + 63304318432 p^{4} T^{10} + 1696628940 p^{6} T^{11} + 40603131 p^{8} T^{12} + 828672 p^{10} T^{13} + 13424 p^{12} T^{14} + 156 p^{14} T^{15} + p^{16} T^{16} \) | |
| 37 | \( 1 + 68 T + 774 T^{2} - 6200 T^{3} + 537713 T^{4} - 4153032 T^{5} - 2518250 T^{6} - 54255996388 T^{7} - 5388431698620 T^{8} - 54255996388 p^{2} T^{9} - 2518250 p^{4} T^{10} - 4153032 p^{6} T^{11} + 537713 p^{8} T^{12} - 6200 p^{10} T^{13} + 774 p^{12} T^{14} + 68 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( 1 - 7680 T^{2} + 32658684 T^{4} - 90843217920 T^{6} + 180014993441414 T^{8} - 90843217920 p^{4} T^{10} + 32658684 p^{8} T^{12} - 7680 p^{12} T^{14} + p^{16} T^{16} \) | |
| 43 | \( ( 1 - 80 T + 6004 T^{2} - 239120 T^{3} + 12298678 T^{4} - 239120 p^{2} T^{5} + 6004 p^{4} T^{6} - 80 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 47 | \( 1 - 108 T + 11624 T^{2} - 835488 T^{3} + 58240275 T^{4} - 3440424852 T^{5} + 189935197288 T^{6} - 9707415208272 T^{7} + 462352344677336 T^{8} - 9707415208272 p^{2} T^{9} + 189935197288 p^{4} T^{10} - 3440424852 p^{6} T^{11} + 58240275 p^{8} T^{12} - 835488 p^{10} T^{13} + 11624 p^{12} T^{14} - 108 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( 1 - 28 T - 5690 T^{2} + 127240 T^{3} + 12246641 T^{4} - 12533896 T^{5} - 50267298346 T^{6} - 350397212804 T^{7} + 217406608794820 T^{8} - 350397212804 p^{2} T^{9} - 50267298346 p^{4} T^{10} - 12533896 p^{6} T^{11} + 12246641 p^{8} T^{12} + 127240 p^{10} T^{13} - 5690 p^{12} T^{14} - 28 p^{14} T^{15} + p^{16} T^{16} \) | |
| 59 | \( 1 - 120 T + 17968 T^{2} - 1580160 T^{3} + 153965443 T^{4} - 11645905140 T^{5} + 884355553312 T^{6} - 56964357475140 T^{7} + 3583093590199576 T^{8} - 56964357475140 p^{2} T^{9} + 884355553312 p^{4} T^{10} - 11645905140 p^{6} T^{11} + 153965443 p^{8} T^{12} - 1580160 p^{10} T^{13} + 17968 p^{12} T^{14} - 120 p^{14} T^{15} + p^{16} T^{16} \) | |
| 61 | \( 1 + 252 T + 39190 T^{2} + 4541544 T^{3} + 432889153 T^{4} + 35854208712 T^{5} + 2659779396934 T^{6} + 181116202298292 T^{7} + 11432754845543908 T^{8} + 181116202298292 p^{2} T^{9} + 2659779396934 p^{4} T^{10} + 35854208712 p^{6} T^{11} + 432889153 p^{8} T^{12} + 4541544 p^{10} T^{13} + 39190 p^{12} T^{14} + 252 p^{14} T^{15} + p^{16} T^{16} \) | |
| 67 | \( 1 + 56 T - 8216 T^{2} + 89656 T^{3} + 57861563 T^{4} - 1915242028 T^{5} - 169076337880 T^{6} + 5261680334092 T^{7} + 334184050069096 T^{8} + 5261680334092 p^{2} T^{9} - 169076337880 p^{4} T^{10} - 1915242028 p^{6} T^{11} + 57861563 p^{8} T^{12} + 89656 p^{10} T^{13} - 8216 p^{12} T^{14} + 56 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( ( 1 - 104 T + 20596 T^{2} - 1429880 T^{3} + 155212822 T^{4} - 1429880 p^{2} T^{5} + 20596 p^{4} T^{6} - 104 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 73 | \( 1 + 156 T + 28350 T^{2} + 3157128 T^{3} + 362705769 T^{4} + 30616526040 T^{5} + 2796887810814 T^{6} + 202276783979268 T^{7} + 16208192683154804 T^{8} + 202276783979268 p^{2} T^{9} + 2796887810814 p^{4} T^{10} + 30616526040 p^{6} T^{11} + 362705769 p^{8} T^{12} + 3157128 p^{10} T^{13} + 28350 p^{12} T^{14} + 156 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( 1 - 160 T - 368 T^{2} + 1779112 T^{3} - 46559293 T^{4} - 16787347924 T^{5} + 1470529925696 T^{6} + 30567104321884 T^{7} - 9673348948274312 T^{8} + 30567104321884 p^{2} T^{9} + 1470529925696 p^{4} T^{10} - 16787347924 p^{6} T^{11} - 46559293 p^{8} T^{12} + 1779112 p^{10} T^{13} - 368 p^{12} T^{14} - 160 p^{14} T^{15} + p^{16} T^{16} \) | |
| 83 | \( 1 - 31656 T^{2} + 548966172 T^{4} - 6241105138200 T^{6} + 50625160453300358 T^{8} - 6241105138200 p^{4} T^{10} + 548966172 p^{8} T^{12} - 31656 p^{12} T^{14} + p^{16} T^{16} \) | |
| 89 | \( 1 - 204 T + 33606 T^{2} - 4025736 T^{3} + 429000465 T^{4} - 38687764008 T^{5} + 2948831814102 T^{6} - 245790314656644 T^{7} + 19698007802495972 T^{8} - 245790314656644 p^{2} T^{9} + 2948831814102 p^{4} T^{10} - 38687764008 p^{6} T^{11} + 429000465 p^{8} T^{12} - 4025736 p^{10} T^{13} + 33606 p^{12} T^{14} - 204 p^{14} T^{15} + p^{16} T^{16} \) | |
| 97 | \( 1 - 67008 T^{2} + 2033229564 T^{4} - 36439450302528 T^{6} + 421681100650800134 T^{8} - 36439450302528 p^{4} T^{10} + 2033229564 p^{8} T^{12} - 67008 p^{12} T^{14} + p^{16} T^{16} \) | |
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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
−4.52633915337471477812513665585, −4.47057855909640120568989913173, −4.33511476991233043684566222280, −4.26670755365831026875703120304, −4.15573519384517583998151531304, −3.90609309603296343626880902758, −3.75882064895770664270318645796, −3.67049647966279328720509120085, −3.65324093135140226882946210343, −3.27090990616196903787406719973, −3.07704835526022295579356332273, −2.95234214515986214626446715305, −2.68902976280177847803401320757, −2.67667748710734222590389813332, −2.23314235720239699919552199046, −2.19513634077018280200721749825, −2.16247371870585097820417632904, −2.09384548785089041148374532512, −1.66434199086297673970532554291, −1.63310959037689686556455133495, −1.31052761349213325533672598021, −0.821322717474852659864697608790, −0.76843347101631262727614208796, −0.39904882889035918536138934600, −0.02601718708853884978430466774, 0.02601718708853884978430466774, 0.39904882889035918536138934600, 0.76843347101631262727614208796, 0.821322717474852659864697608790, 1.31052761349213325533672598021, 1.63310959037689686556455133495, 1.66434199086297673970532554291, 2.09384548785089041148374532512, 2.16247371870585097820417632904, 2.19513634077018280200721749825, 2.23314235720239699919552199046, 2.67667748710734222590389813332, 2.68902976280177847803401320757, 2.95234214515986214626446715305, 3.07704835526022295579356332273, 3.27090990616196903787406719973, 3.65324093135140226882946210343, 3.67049647966279328720509120085, 3.75882064895770664270318645796, 3.90609309603296343626880902758, 4.15573519384517583998151531304, 4.26670755365831026875703120304, 4.33511476991233043684566222280, 4.47057855909640120568989913173, 4.52633915337471477812513665585
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.