Dirichlet series
| L(s) = 1 | + 2-s − 5·3-s − 4·4-s − 8·5-s − 5·6-s + 9·7-s − 4·8-s + 2·9-s − 8·10-s + 20·12-s − 8·13-s + 9·14-s + 40·15-s + 7·16-s + 4·17-s + 2·18-s + 16·19-s + 32·20-s − 45·21-s − 9·23-s + 20·24-s + 36·25-s − 8·26-s + 33·27-s − 36·28-s − 13·29-s + 40·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 2.88·3-s − 2·4-s − 3.57·5-s − 2.04·6-s + 3.40·7-s − 1.41·8-s + 2/3·9-s − 2.52·10-s + 5.77·12-s − 2.21·13-s + 2.40·14-s + 10.3·15-s + 7/4·16-s + 0.970·17-s + 0.471·18-s + 3.67·19-s + 7.15·20-s − 9.81·21-s − 1.87·23-s + 4.08·24-s + 36/5·25-s − 1.56·26-s + 6.35·27-s − 6.80·28-s − 2.41·29-s + 7.30·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(5^{8} \cdot 11^{16} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.41995\times 10^{14}\) |
| Root analytic conductor: | \(7.92479\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 5^{8} \cdot 11^{16} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.366726707\) |
| \(L(\frac12)\) | \(\approx\) | \(1.366726707\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + T )^{8} \) |
| 11 | \( 1 \) | |
| 13 | \( ( 1 + T )^{8} \) | |
| good | 2 | \( 1 - T + 5 T^{2} - 5 T^{3} + 7 p T^{4} - 7 T^{5} + 25 T^{6} - T^{7} + 41 T^{8} - p T^{9} + 25 p^{2} T^{10} - 7 p^{3} T^{11} + 7 p^{5} T^{12} - 5 p^{5} T^{13} + 5 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) |
| 3 | \( 1 + 5 T + 23 T^{2} + 8 p^{2} T^{3} + 208 T^{4} + 167 p T^{5} + 124 p^{2} T^{6} + 2207 T^{7} + 4031 T^{8} + 2207 p T^{9} + 124 p^{4} T^{10} + 167 p^{4} T^{11} + 208 p^{4} T^{12} + 8 p^{7} T^{13} + 23 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 - 9 T + 57 T^{2} - 260 T^{3} + 1021 T^{4} - 3334 T^{5} + 9955 T^{6} - 27203 T^{7} + 73596 T^{8} - 27203 p T^{9} + 9955 p^{2} T^{10} - 3334 p^{3} T^{11} + 1021 p^{4} T^{12} - 260 p^{5} T^{13} + 57 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 4 T + 80 T^{2} - 351 T^{3} + 3232 T^{4} - 13477 T^{5} + 88696 T^{6} - 319075 T^{7} + 1766337 T^{8} - 319075 p T^{9} + 88696 p^{2} T^{10} - 13477 p^{3} T^{11} + 3232 p^{4} T^{12} - 351 p^{5} T^{13} + 80 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 16 T + 162 T^{2} - 1165 T^{3} + 7555 T^{4} - 43451 T^{5} + 12214 p T^{6} - 1091786 T^{7} + 4919072 T^{8} - 1091786 p T^{9} + 12214 p^{3} T^{10} - 43451 p^{3} T^{11} + 7555 p^{4} T^{12} - 1165 p^{5} T^{13} + 162 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 9 T + 108 T^{2} + 827 T^{3} + 6552 T^{4} + 76 p^{2} T^{5} + 251580 T^{6} + 1307794 T^{7} + 6820589 T^{8} + 1307794 p T^{9} + 251580 p^{2} T^{10} + 76 p^{5} T^{11} + 6552 p^{4} T^{12} + 827 p^{5} T^{13} + 108 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 13 T + 205 T^{2} + 1904 T^{3} + 17640 T^{4} + 125917 T^{5} + 886436 T^{6} + 5165401 T^{7} + 30327763 T^{8} + 5165401 p T^{9} + 886436 p^{2} T^{10} + 125917 p^{3} T^{11} + 17640 p^{4} T^{12} + 1904 p^{5} T^{13} + 205 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 24 T^{2} - 157 T^{3} + 1623 T^{4} - 283 T^{5} + 66616 T^{6} - 194434 T^{7} + 1473804 T^{8} - 194434 p T^{9} + 66616 p^{2} T^{10} - 283 p^{3} T^{11} + 1623 p^{4} T^{12} - 157 p^{5} T^{13} + 24 p^{6} T^{14} + p^{8} T^{16} \) | |
| 37 | \( 1 + 5 T + 202 T^{2} + 895 T^{3} + 19493 T^{4} + 77749 T^{5} + 1209862 T^{6} + 4276447 T^{7} + 52873448 T^{8} + 4276447 p T^{9} + 1209862 p^{2} T^{10} + 77749 p^{3} T^{11} + 19493 p^{4} T^{12} + 895 p^{5} T^{13} + 202 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 13 T + 6 p T^{2} - 2375 T^{3} + 28325 T^{4} - 223233 T^{5} + 2033322 T^{6} - 13384187 T^{7} + 99712568 T^{8} - 13384187 p T^{9} + 2033322 p^{2} T^{10} - 223233 p^{3} T^{11} + 28325 p^{4} T^{12} - 2375 p^{5} T^{13} + 6 p^{7} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 5 T + 236 T^{2} - 1181 T^{3} + 25814 T^{4} - 133246 T^{5} + 1796672 T^{6} - 9059548 T^{7} + 89744511 T^{8} - 9059548 p T^{9} + 1796672 p^{2} T^{10} - 133246 p^{3} T^{11} + 25814 p^{4} T^{12} - 1181 p^{5} T^{13} + 236 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 10 T + 222 T^{2} - 2421 T^{3} + 27373 T^{4} - 261423 T^{5} + 2318658 T^{6} - 17423216 T^{7} + 133475496 T^{8} - 17423216 p T^{9} + 2318658 p^{2} T^{10} - 261423 p^{3} T^{11} + 27373 p^{4} T^{12} - 2421 p^{5} T^{13} + 222 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 14 T + 268 T^{2} - 2075 T^{3} + 22184 T^{4} - 93431 T^{5} + 834056 T^{6} - 19941 p T^{7} + 27908897 T^{8} - 19941 p^{2} T^{9} + 834056 p^{2} T^{10} - 93431 p^{3} T^{11} + 22184 p^{4} T^{12} - 2075 p^{5} T^{13} + 268 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 20 T + 349 T^{2} + 4336 T^{3} + 48987 T^{4} + 446456 T^{5} + 4036095 T^{6} + 31641692 T^{7} + 253999300 T^{8} + 31641692 p T^{9} + 4036095 p^{2} T^{10} + 446456 p^{3} T^{11} + 48987 p^{4} T^{12} + 4336 p^{5} T^{13} + 349 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 6 T + 314 T^{2} - 2097 T^{3} + 47212 T^{4} - 346189 T^{5} + 4575416 T^{6} - 33642035 T^{7} + 321550313 T^{8} - 33642035 p T^{9} + 4575416 p^{2} T^{10} - 346189 p^{3} T^{11} + 47212 p^{4} T^{12} - 2097 p^{5} T^{13} + 314 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 4 T + 282 T^{2} - 1123 T^{3} + 43183 T^{4} - 150305 T^{5} + 4518838 T^{6} - 13938626 T^{7} + 347011868 T^{8} - 13938626 p T^{9} + 4518838 p^{2} T^{10} - 150305 p^{3} T^{11} + 43183 p^{4} T^{12} - 1123 p^{5} T^{13} + 282 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 9 T + 315 T^{2} + 3236 T^{3} + 57881 T^{4} + 529622 T^{5} + 7001241 T^{6} + 56038399 T^{7} + 585217952 T^{8} + 56038399 p T^{9} + 7001241 p^{2} T^{10} + 529622 p^{3} T^{11} + 57881 p^{4} T^{12} + 3236 p^{5} T^{13} + 315 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 10 T + 138 T^{2} - 53 T^{3} - 221 T^{4} + 31971 T^{5} + 165274 T^{6} - 2954684 T^{7} + 32251536 T^{8} - 2954684 p T^{9} + 165274 p^{2} T^{10} + 31971 p^{3} T^{11} - 221 p^{4} T^{12} - 53 p^{5} T^{13} + 138 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 13 T + 460 T^{2} - 3985 T^{3} + 84912 T^{4} - 519406 T^{5} + 9415332 T^{6} - 44710832 T^{7} + 804181589 T^{8} - 44710832 p T^{9} + 9415332 p^{2} T^{10} - 519406 p^{3} T^{11} + 84912 p^{4} T^{12} - 3985 p^{5} T^{13} + 460 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 26 T + 544 T^{2} - 7160 T^{3} + 86171 T^{4} - 831396 T^{5} + 8517920 T^{6} - 77729350 T^{7} + 761653156 T^{8} - 77729350 p T^{9} + 8517920 p^{2} T^{10} - 831396 p^{3} T^{11} + 86171 p^{4} T^{12} - 7160 p^{5} T^{13} + 544 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 6 T + 520 T^{2} + 2073 T^{3} + 122389 T^{4} + 328953 T^{5} + 18045388 T^{6} + 35719040 T^{7} + 1881059484 T^{8} + 35719040 p T^{9} + 18045388 p^{2} T^{10} + 328953 p^{3} T^{11} + 122389 p^{4} T^{12} + 2073 p^{5} T^{13} + 520 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 26 T + 912 T^{2} - 16420 T^{3} + 331251 T^{4} - 4533616 T^{5} + 65992796 T^{6} - 712968338 T^{7} + 8056776688 T^{8} - 712968338 p T^{9} + 65992796 p^{2} T^{10} - 4533616 p^{3} T^{11} + 331251 p^{4} T^{12} - 16420 p^{5} T^{13} + 912 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.25578741658008832254732965878, −3.08331055617778576778748995689, −3.04538038407143513432440223595, −2.98315777433672340168562829104, −2.95528310634326640430009996864, −2.71236850025928161053286785186, −2.45559000671993333861959268811, −2.42580773336251946351047053393, −2.35681441597497414679087886561, −2.17408013709407639450569792121, −2.03762503933140050132724958741, −1.93026312931214168692546974967, −1.65469658109729700227099910415, −1.65128986946891223574937185275, −1.63561595426748864524892478576, −1.40330967008780789283774305152, −1.32338337926324195576469977149, −0.910673494017715710462125443600, −0.65410086952590584005173269322, −0.63337345240438647755420241704, −0.63230822193216780483777177130, −0.60990069112063588162153645653, −0.48831242915457294023102915130, −0.36429946897431506640744254740, −0.21420646694569705208904566415, 0.21420646694569705208904566415, 0.36429946897431506640744254740, 0.48831242915457294023102915130, 0.60990069112063588162153645653, 0.63230822193216780483777177130, 0.63337345240438647755420241704, 0.65410086952590584005173269322, 0.910673494017715710462125443600, 1.32338337926324195576469977149, 1.40330967008780789283774305152, 1.63561595426748864524892478576, 1.65128986946891223574937185275, 1.65469658109729700227099910415, 1.93026312931214168692546974967, 2.03762503933140050132724958741, 2.17408013709407639450569792121, 2.35681441597497414679087886561, 2.42580773336251946351047053393, 2.45559000671993333861959268811, 2.71236850025928161053286785186, 2.95528310634326640430009996864, 2.98315777433672340168562829104, 3.04538038407143513432440223595, 3.08331055617778576778748995689, 3.25578741658008832254732965878
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.