Dirichlet series
| L(s) = 1 | + 7·3-s + 3·5-s + 52·7-s + 26·9-s − 99·11-s + 21·15-s + 9·17-s − 143·19-s + 364·21-s − 15·23-s − 30·25-s + 73·27-s + 348·29-s + 205·31-s − 693·33-s + 156·35-s + 249·37-s + 78·45-s + 75·47-s + 1.70e3·49-s + 63·51-s + 645·53-s − 297·55-s − 1.00e3·57-s − 321·59-s + 1.70e3·61-s + 1.35e3·63-s + ⋯ |
| L(s) = 1 | + 1.34·3-s + 0.268·5-s + 2.80·7-s + 0.962·9-s − 2.71·11-s + 0.361·15-s + 0.128·17-s − 1.72·19-s + 3.78·21-s − 0.135·23-s − 0.239·25-s + 0.520·27-s + 2.22·29-s + 1.18·31-s − 3.65·33-s + 0.753·35-s + 1.10·37-s + 0.258·45-s + 0.232·47-s + 4.96·49-s + 0.172·51-s + 1.67·53-s − 0.728·55-s − 2.32·57-s − 0.708·59-s + 3.58·61-s + 2.70·63-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{36} \cdot 7^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.41086\times 10^{8}\) |
| Root analytic conductor: | \(5.14128\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{36} \cdot 7^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(6.158456212\) |
| \(L(\frac12)\) | \(\approx\) | \(6.158456212\) |
| \(L(\frac{5}{2})\) | 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 | \( 1 - 52 T + 143 p T^{2} - 2008 p T^{3} + 143 p^{4} T^{4} - 52 p^{6} T^{5} + p^{9} T^{6} \) | |
| good | 3 | \( 1 - 7 T + 23 T^{2} - 52 T^{3} - 481 T^{4} + 3035 T^{5} - 10382 T^{6} + 3035 p^{3} T^{7} - 481 p^{6} T^{8} - 52 p^{9} T^{9} + 23 p^{12} T^{10} - 7 p^{15} T^{11} + p^{18} T^{12} \) |
| 5 | \( 1 - 3 T + 39 T^{2} - 108 T^{3} - 1743 T^{4} - 5721 T^{5} - 3079618 T^{6} - 5721 p^{3} T^{7} - 1743 p^{6} T^{8} - 108 p^{9} T^{9} + 39 p^{12} T^{10} - 3 p^{15} T^{11} + p^{18} T^{12} \) | |
| 11 | \( 1 + 9 p T + 7983 T^{2} + 42444 p T^{3} + 24786639 T^{4} + 1085499369 T^{5} + 42926385890 T^{6} + 1085499369 p^{3} T^{7} + 24786639 p^{6} T^{8} + 42444 p^{10} T^{9} + 7983 p^{12} T^{10} + 9 p^{16} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 - 606 p T^{2} + 33176199 T^{4} - 88766411348 T^{6} + 33176199 p^{6} T^{8} - 606 p^{13} T^{10} + p^{18} T^{12} \) | |
| 17 | \( 1 - 9 T + 99 T^{2} - 648 T^{3} - 221787 T^{4} + 214195761 T^{5} - 237206836690 T^{6} + 214195761 p^{3} T^{7} - 221787 p^{6} T^{8} - 648 p^{9} T^{9} + 99 p^{12} T^{10} - 9 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 + 143 T - 5665 T^{2} - 708 p^{2} T^{3} + 217547935 T^{4} + 7596399877 T^{5} - 1014293563550 T^{6} + 7596399877 p^{3} T^{7} + 217547935 p^{6} T^{8} - 708 p^{11} T^{9} - 5665 p^{12} T^{10} + 143 p^{15} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 + 15 T + 25275 T^{2} + 378000 T^{3} + 329438955 T^{4} + 2856530865 T^{5} + 4126653853346 T^{6} + 2856530865 p^{3} T^{7} + 329438955 p^{6} T^{8} + 378000 p^{9} T^{9} + 25275 p^{12} T^{10} + 15 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( ( 1 - 6 p T + 45663 T^{2} - 3864972 T^{3} + 45663 p^{3} T^{4} - 6 p^{7} T^{5} + p^{9} T^{6} )^{2} \) | |
| 31 | \( 1 - 205 T - 18277 T^{2} + 9001152 T^{3} - 349839533 T^{4} - 89508615347 T^{5} + 21347063732914 T^{6} - 89508615347 p^{3} T^{7} - 349839533 p^{6} T^{8} + 9001152 p^{9} T^{9} - 18277 p^{12} T^{10} - 205 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 - 249 T - 56433 T^{2} + 13334072 T^{3} + 2695969773 T^{4} - 187262229591 T^{5} - 151167420301962 T^{6} - 187262229591 p^{3} T^{7} + 2695969773 p^{6} T^{8} + 13334072 p^{9} T^{9} - 56433 p^{12} T^{10} - 249 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 - 4350 p T^{2} + 20420082255 T^{4} - 1736555393379940 T^{6} + 20420082255 p^{6} T^{8} - 4350 p^{13} T^{10} + p^{18} T^{12} \) | |
| 43 | \( 1 - 225246 T^{2} + 31219429287 T^{4} - 3038313220434308 T^{6} + 31219429287 p^{6} T^{8} - 225246 p^{12} T^{10} + p^{18} T^{12} \) | |
| 47 | \( 1 - 75 T - 271293 T^{2} + 10090392 T^{3} + 46775899635 T^{4} - 768612310653 T^{5} - 5545818129678302 T^{6} - 768612310653 p^{3} T^{7} + 46775899635 p^{6} T^{8} + 10090392 p^{9} T^{9} - 271293 p^{12} T^{10} - 75 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 - 645 T + 85935 T^{2} + 48846432 T^{3} - 24599221275 T^{4} + 5896942109805 T^{5} - 1714098743019578 T^{6} + 5896942109805 p^{3} T^{7} - 24599221275 p^{6} T^{8} + 48846432 p^{9} T^{9} + 85935 p^{12} T^{10} - 645 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 + 321 T - 543849 T^{2} - 57872532 T^{3} + 242992413159 T^{4} + 14127840402531 T^{5} - 55338821261641982 T^{6} + 14127840402531 p^{3} T^{7} + 242992413159 p^{6} T^{8} - 57872532 p^{9} T^{9} - 543849 p^{12} T^{10} + 321 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 - 1707 T + 1959075 T^{2} - 1686160944 T^{3} + 1222226403561 T^{4} - 729202413995805 T^{5} + 376156083256198486 T^{6} - 729202413995805 p^{3} T^{7} + 1222226403561 p^{6} T^{8} - 1686160944 p^{9} T^{9} + 1959075 p^{12} T^{10} - 1707 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 + 447 T + 454983 T^{2} + 173605860 T^{3} + 25906463103 T^{4} - 61606109318883 T^{5} - 14080566334108718 T^{6} - 61606109318883 p^{3} T^{7} + 25906463103 p^{6} T^{8} + 173605860 p^{9} T^{9} + 454983 p^{12} T^{10} + 447 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 - 730518 T^{2} + 329494751679 T^{4} - 139370995079650996 T^{6} + 329494751679 p^{6} T^{8} - 730518 p^{12} T^{10} + p^{18} T^{12} \) | |
| 73 | \( 1 - 705 T + 711327 T^{2} - 384684660 T^{3} + 124877058765 T^{4} + 15186280400661 T^{5} - 17616940504633370 T^{6} + 15186280400661 p^{3} T^{7} + 124877058765 p^{6} T^{8} - 384684660 p^{9} T^{9} + 711327 p^{12} T^{10} - 705 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 + 3447 T + 6344259 T^{2} + 8216462232 T^{3} + 8250767428323 T^{4} + 6929574062947137 T^{5} + 5138476947017253250 T^{6} + 6929574062947137 p^{3} T^{7} + 8250767428323 p^{6} T^{8} + 8216462232 p^{9} T^{9} + 6344259 p^{12} T^{10} + 3447 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( ( 1 + 12 T + 858849 T^{2} - 294490104 T^{3} + 858849 p^{3} T^{4} + 12 p^{6} T^{5} + p^{9} T^{6} )^{2} \) | |
| 89 | \( 1 + 2607 T + 4818303 T^{2} + 6655201740 T^{3} + 7793954699517 T^{4} + 7742714792403333 T^{5} + 6946049341042583942 T^{6} + 7742714792403333 p^{3} T^{7} + 7793954699517 p^{6} T^{8} + 6655201740 p^{9} T^{9} + 4818303 p^{12} T^{10} + 2607 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 - 4437342 T^{2} + 9010629855423 T^{4} - 10556356459893838148 T^{6} + 9010629855423 p^{6} T^{8} - 4437342 p^{12} T^{10} + p^{18} T^{12} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.44185739689888956610981296824, −5.41311768870653114707467489307, −5.15655949628858412992231049661, −4.85037697630174412553817542306, −4.72930687488766897095954039845, −4.67325829709760436784218296000, −4.56697241595566620507695463798, −4.19935171766082638189620472746, −4.12828127765004531441476179961, −3.95446597783833312652246082836, −3.63239462738292536858897683149, −3.57866013518299180197087741998, −2.89213601189049046751371388946, −2.83020449246733956254966384498, −2.75336740594098994772612126508, −2.65873556180255522587541186290, −2.35909788753779185826956389007, −2.11603055682549381615895626981, −2.05994224591930848417137382340, −1.78103710856715080509525701852, −1.33239282148337813697964459715, −1.17109506126344612125174283248, −0.892750704114549771733038379020, −0.62799393182604108627101695847, −0.16328116620057378058318144091, 0.16328116620057378058318144091, 0.62799393182604108627101695847, 0.892750704114549771733038379020, 1.17109506126344612125174283248, 1.33239282148337813697964459715, 1.78103710856715080509525701852, 2.05994224591930848417137382340, 2.11603055682549381615895626981, 2.35909788753779185826956389007, 2.65873556180255522587541186290, 2.75336740594098994772612126508, 2.83020449246733956254966384498, 2.89213601189049046751371388946, 3.57866013518299180197087741998, 3.63239462738292536858897683149, 3.95446597783833312652246082836, 4.12828127765004531441476179961, 4.19935171766082638189620472746, 4.56697241595566620507695463798, 4.67325829709760436784218296000, 4.72930687488766897095954039845, 4.85037697630174412553817542306, 5.15655949628858412992231049661, 5.41311768870653114707467489307, 5.44185739689888956610981296824
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.