Dirichlet series
| L(s) = 1 | + 8·2-s + 4·3-s + 36·4-s + 32·6-s + 8·7-s + 120·8-s + 2·9-s + 8·11-s + 144·12-s + 4·13-s + 64·14-s + 330·16-s + 8·17-s + 16·18-s + 16·19-s + 32·21-s + 64·22-s − 8·23-s + 480·24-s − 18·25-s + 32·26-s − 16·27-s + 288·28-s − 8·29-s + 8·31-s + 792·32-s + 32·33-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 2.30·3-s + 18·4-s + 13.0·6-s + 3.02·7-s + 42.4·8-s + 2/3·9-s + 2.41·11-s + 41.5·12-s + 1.10·13-s + 17.1·14-s + 82.5·16-s + 1.94·17-s + 3.77·18-s + 3.67·19-s + 6.98·21-s + 13.6·22-s − 1.66·23-s + 97.9·24-s − 3.59·25-s + 6.27·26-s − 3.07·27-s + 54.4·28-s − 1.48·29-s + 1.43·31-s + 140.·32-s + 5.57·33-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 23^{8} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 23^{8} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 23^{8} \cdot 29^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.65753\times 10^{8}\) |
| Root analytic conductor: | \(3.26374\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 23^{8} \cdot 29^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(5533.534047\) |
| \(L(\frac12)\) | \(\approx\) | \(5533.534047\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{8} \) |
| 23 | \( ( 1 + T )^{8} \) | |
| 29 | \( ( 1 + T )^{8} \) | |
| good | 3 | \( ( 1 - 2 T + 5 T^{2} - 2 p T^{3} + 14 T^{4} - 2 p^{2} T^{5} + 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 + 18 T^{2} + 191 T^{4} + 1438 T^{6} + 8086 T^{8} + 1438 p^{2} T^{10} + 191 p^{4} T^{12} + 18 p^{6} T^{14} + p^{8} T^{16} \) | |
| 7 | \( 1 - 8 T + 58 T^{2} - 284 T^{3} + 1226 T^{4} - 4420 T^{5} + 14526 T^{6} - 42704 T^{7} + 118266 T^{8} - 42704 p T^{9} + 14526 p^{2} T^{10} - 4420 p^{3} T^{11} + 1226 p^{4} T^{12} - 284 p^{5} T^{13} + 58 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 8 T + 72 T^{2} - 344 T^{3} + 1933 T^{4} - 7408 T^{5} + 34476 T^{6} - 115392 T^{7} + 451676 T^{8} - 115392 p T^{9} + 34476 p^{2} T^{10} - 7408 p^{3} T^{11} + 1933 p^{4} T^{12} - 344 p^{5} T^{13} + 72 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 4 T + 56 T^{2} - 176 T^{3} + 1485 T^{4} - 3984 T^{5} + 27556 T^{6} - 65900 T^{7} + 402812 T^{8} - 65900 p T^{9} + 27556 p^{2} T^{10} - 3984 p^{3} T^{11} + 1485 p^{4} T^{12} - 176 p^{5} T^{13} + 56 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 8 T + 82 T^{2} - 436 T^{3} + 3016 T^{4} - 868 p T^{5} + 81738 T^{6} - 20392 p T^{7} + 1581186 T^{8} - 20392 p^{2} T^{9} + 81738 p^{2} T^{10} - 868 p^{4} T^{11} + 3016 p^{4} T^{12} - 436 p^{5} T^{13} + 82 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 16 T + 220 T^{2} - 2048 T^{3} + 16984 T^{4} - 113600 T^{5} + 688852 T^{6} - 3545616 T^{7} + 16670798 T^{8} - 3545616 p T^{9} + 688852 p^{2} T^{10} - 113600 p^{3} T^{11} + 16984 p^{4} T^{12} - 2048 p^{5} T^{13} + 220 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 8 T + 120 T^{2} - 616 T^{3} + 6717 T^{4} - 30744 T^{5} + 306516 T^{6} - 1314040 T^{7} + 11090604 T^{8} - 1314040 p T^{9} + 306516 p^{2} T^{10} - 30744 p^{3} T^{11} + 6717 p^{4} T^{12} - 616 p^{5} T^{13} + 120 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 8 T + 210 T^{2} - 1388 T^{3} + 20754 T^{4} - 114428 T^{5} + 1278022 T^{6} - 5996512 T^{7} + 55332010 T^{8} - 5996512 p T^{9} + 1278022 p^{2} T^{10} - 114428 p^{3} T^{11} + 20754 p^{4} T^{12} - 1388 p^{5} T^{13} + 210 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 8 T + 178 T^{2} - 1068 T^{3} + 16162 T^{4} - 89180 T^{5} + 1036710 T^{6} - 4976272 T^{7} + 48064266 T^{8} - 4976272 p T^{9} + 1036710 p^{2} T^{10} - 89180 p^{3} T^{11} + 16162 p^{4} T^{12} - 1068 p^{5} T^{13} + 178 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 32 T + 640 T^{2} - 9112 T^{3} + 105317 T^{4} - 1026000 T^{5} + 8816228 T^{6} - 67450712 T^{7} + 466341964 T^{8} - 67450712 p T^{9} + 8816228 p^{2} T^{10} - 1026000 p^{3} T^{11} + 105317 p^{4} T^{12} - 9112 p^{5} T^{13} + 640 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 16 T + 304 T^{2} - 72 p T^{3} + 39093 T^{4} - 351848 T^{5} + 3115948 T^{6} - 23754560 T^{7} + 173370876 T^{8} - 23754560 p T^{9} + 3115948 p^{2} T^{10} - 351848 p^{3} T^{11} + 39093 p^{4} T^{12} - 72 p^{6} T^{13} + 304 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 274 T^{2} + 32607 T^{4} + 2362878 T^{6} + 133085750 T^{8} + 2362878 p^{2} T^{10} + 32607 p^{4} T^{12} + 274 p^{6} T^{14} + p^{8} T^{16} \) | |
| 59 | \( 1 + 4 T + 148 T^{2} + 772 T^{3} + 13916 T^{4} + 64932 T^{5} + 982636 T^{6} + 3576516 T^{7} + 64044134 T^{8} + 3576516 p T^{9} + 982636 p^{2} T^{10} + 64932 p^{3} T^{11} + 13916 p^{4} T^{12} + 772 p^{5} T^{13} + 148 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 8 T + 290 T^{2} - 1596 T^{3} + 40866 T^{4} - 177516 T^{5} + 3931414 T^{6} - 14674368 T^{7} + 280101578 T^{8} - 14674368 p T^{9} + 3931414 p^{2} T^{10} - 177516 p^{3} T^{11} + 40866 p^{4} T^{12} - 1596 p^{5} T^{13} + 290 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 20 T + 542 T^{2} - 7448 T^{3} + 119944 T^{4} - 1276008 T^{5} + 227730 p T^{6} - 131365132 T^{7} + 1253417186 T^{8} - 131365132 p T^{9} + 227730 p^{3} T^{10} - 1276008 p^{3} T^{11} + 119944 p^{4} T^{12} - 7448 p^{5} T^{13} + 542 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 24 T + 548 T^{2} + 7960 T^{3} + 115624 T^{4} + 1310488 T^{5} + 14782220 T^{6} + 137213080 T^{7} + 1264237134 T^{8} + 137213080 p T^{9} + 14782220 p^{2} T^{10} + 1310488 p^{3} T^{11} + 115624 p^{4} T^{12} + 7960 p^{5} T^{13} + 548 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 4 T + 166 T^{2} + 1616 T^{3} + 20714 T^{4} + 227016 T^{5} + 2381330 T^{6} + 19617220 T^{7} + 211416058 T^{8} + 19617220 p T^{9} + 2381330 p^{2} T^{10} + 227016 p^{3} T^{11} + 20714 p^{4} T^{12} + 1616 p^{5} T^{13} + 166 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 4 T + 442 T^{2} - 1488 T^{3} + 94647 T^{4} - 265120 T^{5} + 12806582 T^{6} - 30074012 T^{7} + 1200657270 T^{8} - 30074012 p T^{9} + 12806582 p^{2} T^{10} - 265120 p^{3} T^{11} + 94647 p^{4} T^{12} - 1488 p^{5} T^{13} + 442 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 4 T + 324 T^{2} + 1380 T^{3} + 54770 T^{4} + 251692 T^{5} + 6528860 T^{6} + 30260012 T^{7} + 608202698 T^{8} + 30260012 p T^{9} + 6528860 p^{2} T^{10} + 251692 p^{3} T^{11} + 54770 p^{4} T^{12} + 1380 p^{5} T^{13} + 324 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 20 T + 654 T^{2} - 9168 T^{3} + 174656 T^{4} - 1911504 T^{5} + 27336182 T^{6} - 246860548 T^{7} + 2895556322 T^{8} - 246860548 p T^{9} + 27336182 p^{2} T^{10} - 1911504 p^{3} T^{11} + 174656 p^{4} T^{12} - 9168 p^{5} T^{13} + 654 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 4 T + 396 T^{2} - 1788 T^{3} + 88882 T^{4} - 401692 T^{5} + 13784884 T^{6} - 56359460 T^{7} + 1549582730 T^{8} - 56359460 p T^{9} + 13784884 p^{2} T^{10} - 401692 p^{3} T^{11} + 88882 p^{4} T^{12} - 1788 p^{5} T^{13} + 396 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.11827418547782773590553893004, −3.94862115773677661317502471034, −3.82751259500434054604669552351, −3.71617592994336937097326811983, −3.58836654361051238357685002114, −3.54085517550220801354269457242, −3.46327530028933564685028309992, −3.33820455709001209436318448536, −3.31468442321598736365303591978, −3.00390400882677123648500292031, −2.77843605633596583627441648412, −2.74847481465466313659064946121, −2.52421621155069811329269832576, −2.43213726145670448089340932324, −2.42191710818900015975923991608, −2.32813381542216737515317083075, −2.24445137238024824657481029161, −1.82464825608576456761989426236, −1.67432346414410155594991911142, −1.50849061742659401647798522654, −1.46318891412405983995054358863, −1.42451522784511063534015473705, −1.12947347764198509471219618012, −0.862517263467598797787148973189, −0.836543118505236410779992407405, 0.836543118505236410779992407405, 0.862517263467598797787148973189, 1.12947347764198509471219618012, 1.42451522784511063534015473705, 1.46318891412405983995054358863, 1.50849061742659401647798522654, 1.67432346414410155594991911142, 1.82464825608576456761989426236, 2.24445137238024824657481029161, 2.32813381542216737515317083075, 2.42191710818900015975923991608, 2.43213726145670448089340932324, 2.52421621155069811329269832576, 2.74847481465466313659064946121, 2.77843605633596583627441648412, 3.00390400882677123648500292031, 3.31468442321598736365303591978, 3.33820455709001209436318448536, 3.46327530028933564685028309992, 3.54085517550220801354269457242, 3.58836654361051238357685002114, 3.71617592994336937097326811983, 3.82751259500434054604669552351, 3.94862115773677661317502471034, 4.11827418547782773590553893004
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.