Dirichlet series
| L(s) = 1 | + 2·3-s + 6·7-s + 5·9-s − 6·11-s + 6·19-s + 12·21-s + 2·23-s + 10·25-s + 2·27-s − 8·29-s − 12·33-s + 24·37-s − 12·41-s + 6·43-s + 5·49-s + 20·53-s + 12·57-s − 18·59-s − 4·61-s + 30·63-s + 42·67-s + 4·69-s + 54·71-s + 20·75-s − 36·77-s + 16·79-s − 16·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2.26·7-s + 5/3·9-s − 1.80·11-s + 1.37·19-s + 2.61·21-s + 0.417·23-s + 2·25-s + 0.384·27-s − 1.48·29-s − 2.08·33-s + 3.94·37-s − 1.87·41-s + 0.914·43-s + 5/7·49-s + 2.74·53-s + 1.58·57-s − 2.34·59-s − 0.512·61-s + 3.77·63-s + 5.13·67-s + 0.481·69-s + 6.40·71-s + 2.30·75-s − 4.10·77-s + 1.80·79-s − 1.71·87-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{24} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.84514\times 10^{8}\) |
| Root analytic conductor: | \(3.28569\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{24} \cdot 13^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.439843686\) |
| \(L(\frac12)\) | \(\approx\) | \(2.439843686\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) | |
| good | 3 | \( ( 1 - 4 T + 5 T^{2} + 4 T^{3} - 20 T^{4} + 4 p T^{5} + 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )( 1 + 2 T + 2 T^{2} + 4 T^{3} + 7 T^{4} + 4 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 5 | \( 1 - 2 p T^{2} + 21 p T^{4} - 698 T^{6} + 4004 T^{8} - 698 p^{2} T^{10} + 21 p^{5} T^{12} - 2 p^{7} T^{14} + p^{8} T^{16} \) | |
| 7 | \( 1 - 6 T + 31 T^{2} - 114 T^{3} + 333 T^{4} - 648 T^{5} + 134 p T^{6} + 300 T^{7} - 3022 T^{8} + 300 p T^{9} + 134 p^{3} T^{10} - 648 p^{3} T^{11} + 333 p^{4} T^{12} - 114 p^{5} T^{13} + 31 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 6 T + 47 T^{2} + 210 T^{3} + 981 T^{4} + 3024 T^{5} + 11410 T^{6} + 29100 T^{7} + 109418 T^{8} + 29100 p T^{9} + 11410 p^{2} T^{10} + 3024 p^{3} T^{11} + 981 p^{4} T^{12} + 210 p^{5} T^{13} + 47 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 2 p T^{2} + 481 T^{4} - 194 p T^{6} + 38020 T^{8} - 194 p^{3} T^{10} + 481 p^{4} T^{12} - 2 p^{7} T^{14} + p^{8} T^{16} \) | |
| 19 | \( 1 - 6 T + 35 T^{2} - 138 T^{3} + 433 T^{4} + 24 p T^{5} - 10978 T^{6} + 3996 p T^{7} - 386174 T^{8} + 3996 p^{2} T^{9} - 10978 p^{2} T^{10} + 24 p^{4} T^{11} + 433 p^{4} T^{12} - 138 p^{5} T^{13} + 35 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 2 T - 81 T^{2} + 90 T^{3} + 4109 T^{4} - 2616 T^{5} - 6170 p T^{6} + 22756 T^{7} + 3763506 T^{8} + 22756 p T^{9} - 6170 p^{3} T^{10} - 2616 p^{3} T^{11} + 4109 p^{4} T^{12} + 90 p^{5} T^{13} - 81 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 8 T - 30 T^{2} - 288 T^{3} + 845 T^{4} + 2664 T^{5} - 52246 T^{6} - 15376 T^{7} + 1818396 T^{8} - 15376 p T^{9} - 52246 p^{2} T^{10} + 2664 p^{3} T^{11} + 845 p^{4} T^{12} - 288 p^{5} T^{13} - 30 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 56 T^{2} - 132 T^{4} + 39416 T^{6} - 700282 T^{8} + 39416 p^{2} T^{10} - 132 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \) | |
| 37 | \( 1 - 24 T + 354 T^{2} - 3888 T^{3} + 35917 T^{4} - 291000 T^{5} + 2123946 T^{6} - 14299584 T^{7} + 89708988 T^{8} - 14299584 p T^{9} + 2123946 p^{2} T^{10} - 291000 p^{3} T^{11} + 35917 p^{4} T^{12} - 3888 p^{5} T^{13} + 354 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( ( 1 + 6 T + 81 T^{2} + 414 T^{3} + 3572 T^{4} + 414 p T^{5} + 81 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 43 | \( 1 - 6 T - 69 T^{2} + 726 T^{3} + 793 T^{4} - 26376 T^{5} + 25830 T^{6} + 301236 T^{7} + 148962 T^{8} + 301236 p T^{9} + 25830 p^{2} T^{10} - 26376 p^{3} T^{11} + 793 p^{4} T^{12} + 726 p^{5} T^{13} - 69 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 184 T^{2} + 17148 T^{4} - 1150088 T^{6} + 60931334 T^{8} - 1150088 p^{2} T^{10} + 17148 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \) | |
| 53 | \( ( 1 - 10 T + 133 T^{2} - 746 T^{3} + 6756 T^{4} - 746 p T^{5} + 133 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 59 | \( 1 + 18 T + 279 T^{2} + 3078 T^{3} + 31909 T^{4} + 300384 T^{5} + 2553282 T^{6} + 21591828 T^{7} + 163401402 T^{8} + 21591828 p T^{9} + 2553282 p^{2} T^{10} + 300384 p^{3} T^{11} + 31909 p^{4} T^{12} + 3078 p^{5} T^{13} + 279 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 4 T - 134 T^{2} + 392 T^{3} + 12173 T^{4} - 57344 T^{5} - 6310 p T^{6} + 2268068 T^{7} + 7263196 T^{8} + 2268068 p T^{9} - 6310 p^{3} T^{10} - 57344 p^{3} T^{11} + 12173 p^{4} T^{12} + 392 p^{5} T^{13} - 134 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 42 T + 1003 T^{2} - 17430 T^{3} + 245361 T^{4} - 2971224 T^{5} + 31818974 T^{6} - 305253732 T^{7} + 2633665298 T^{8} - 305253732 p T^{9} + 31818974 p^{2} T^{10} - 2971224 p^{3} T^{11} + 245361 p^{4} T^{12} - 17430 p^{5} T^{13} + 1003 p^{6} T^{14} - 42 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 54 T + 1563 T^{2} - 31914 T^{3} + 509593 T^{4} - 6729840 T^{5} + 76266054 T^{6} - 759952980 T^{7} + 6759291498 T^{8} - 759952980 p T^{9} + 76266054 p^{2} T^{10} - 6729840 p^{3} T^{11} + 509593 p^{4} T^{12} - 31914 p^{5} T^{13} + 1563 p^{6} T^{14} - 54 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 386 T^{2} + 74305 T^{4} - 9208802 T^{6} + 796087780 T^{8} - 9208802 p^{2} T^{10} + 74305 p^{4} T^{12} - 386 p^{6} T^{14} + p^{8} T^{16} \) | |
| 79 | \( ( 1 - 8 T + 156 T^{2} - 1000 T^{3} + 15494 T^{4} - 1000 p T^{5} + 156 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 83 | \( 1 - 416 T^{2} + 78364 T^{4} - 9309536 T^{6} + 846559462 T^{8} - 9309536 p^{2} T^{10} + 78364 p^{4} T^{12} - 416 p^{6} T^{14} + p^{8} T^{16} \) | |
| 89 | \( 1 - 18 T + 179 T^{2} - 1278 T^{3} + 3129 T^{4} + 25596 T^{5} - 468578 T^{6} + 7961112 T^{7} - 85879678 T^{8} + 7961112 p T^{9} - 468578 p^{2} T^{10} + 25596 p^{3} T^{11} + 3129 p^{4} T^{12} - 1278 p^{5} T^{13} + 179 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 30 T + 595 T^{2} + 8850 T^{3} + 102825 T^{4} + 925452 T^{5} + 6324302 T^{6} + 33963720 T^{7} + 208695218 T^{8} + 33963720 p T^{9} + 6324302 p^{2} T^{10} + 925452 p^{3} T^{11} + 102825 p^{4} T^{12} + 8850 p^{5} T^{13} + 595 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.88373192657853701929119726532, −3.87621765068897996468170952308, −3.71742111448679951384238055534, −3.69215210821385787390716775324, −3.68191968289587901233867897517, −3.66518409214208761260527018928, −3.27887828600260924939225756962, −3.19142269827372602574575438762, −3.10125256247150028203922684140, −2.76234791297817438663913753676, −2.54828538626763624814721830188, −2.52411787550612022687924974108, −2.41067965872056017977528390456, −2.35408075602453442066050767093, −2.29814873460689433431943243345, −2.26775174527989200501522215509, −1.85652968182424199270854521903, −1.66727260574101310899267056837, −1.62218364521718091484932005963, −1.30274825585399856070286162313, −1.04136955631968403143114557182, −1.03564627417175221653839681571, −0.857378825353542245798330129982, −0.812615662409074505205398088346, −0.089431085805345631277611551993, 0.089431085805345631277611551993, 0.812615662409074505205398088346, 0.857378825353542245798330129982, 1.03564627417175221653839681571, 1.04136955631968403143114557182, 1.30274825585399856070286162313, 1.62218364521718091484932005963, 1.66727260574101310899267056837, 1.85652968182424199270854521903, 2.26775174527989200501522215509, 2.29814873460689433431943243345, 2.35408075602453442066050767093, 2.41067965872056017977528390456, 2.52411787550612022687924974108, 2.54828538626763624814721830188, 2.76234791297817438663913753676, 3.10125256247150028203922684140, 3.19142269827372602574575438762, 3.27887828600260924939225756962, 3.66518409214208761260527018928, 3.68191968289587901233867897517, 3.69215210821385787390716775324, 3.71742111448679951384238055534, 3.87621765068897996468170952308, 3.88373192657853701929119726532
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.