Dirichlet series
| L(s) = 1 | − 7·3-s + 5·5-s − 7·7-s + 28·9-s + 6·13-s − 35·15-s + 6·17-s + 2·19-s + 49·21-s + 25-s − 84·27-s − 4·29-s − 7·31-s − 35·35-s − 3·37-s − 42·39-s − 12·41-s + 2·43-s + 140·45-s + 11·47-s + 5·49-s − 42·51-s + 53-s − 14·57-s + 19·59-s + 12·61-s − 196·63-s + ⋯ |
| L(s) = 1 | − 4.04·3-s + 2.23·5-s − 2.64·7-s + 28/3·9-s + 1.66·13-s − 9.03·15-s + 1.45·17-s + 0.458·19-s + 10.6·21-s + 1/5·25-s − 16.1·27-s − 0.742·29-s − 1.25·31-s − 5.91·35-s − 0.493·37-s − 6.72·39-s − 1.87·41-s + 0.304·43-s + 20.8·45-s + 1.60·47-s + 5/7·49-s − 5.88·51-s + 0.137·53-s − 1.85·57-s + 2.47·59-s + 1.53·61-s − 24.6·63-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{7} \cdot 167^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{7} \cdot 167^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{28} \cdot 3^{7} \cdot 167^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.40193\times 10^{12}\) |
| Root analytic conductor: | \(8.00050\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{28} \cdot 3^{7} \cdot 167^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.118727396\) |
| \(L(\frac12)\) | \(\approx\) | \(1.118727396\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T )^{7} \) | |
| 167 | \( ( 1 + T )^{7} \) | |
| good | 5 | \( 1 - p T + 24 T^{2} - 57 T^{3} + 28 p T^{4} - p^{3} T^{5} + 183 T^{6} + 286 T^{7} + 183 p T^{8} - p^{5} T^{9} + 28 p^{4} T^{10} - 57 p^{4} T^{11} + 24 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) |
| 7 | \( 1 + p T + 44 T^{2} + 195 T^{3} + 118 p T^{4} + 403 p T^{5} + 9063 T^{6} + 24610 T^{7} + 9063 p T^{8} + 403 p^{3} T^{9} + 118 p^{4} T^{10} + 195 p^{4} T^{11} + 44 p^{5} T^{12} + p^{7} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 29 T^{2} - 12 T^{3} + 525 T^{4} - 160 T^{5} + 7753 T^{6} - 1768 T^{7} + 7753 p T^{8} - 160 p^{2} T^{9} + 525 p^{3} T^{10} - 12 p^{4} T^{11} + 29 p^{5} T^{12} + p^{7} T^{14} \) | |
| 13 | \( 1 - 6 T + 61 T^{2} - 224 T^{3} + 1311 T^{4} - 3074 T^{5} + 16051 T^{6} - 30960 T^{7} + 16051 p T^{8} - 3074 p^{2} T^{9} + 1311 p^{3} T^{10} - 224 p^{4} T^{11} + 61 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 6 T + 63 T^{2} - 428 T^{3} + 2417 T^{4} - 13658 T^{5} + 63663 T^{6} - 277752 T^{7} + 63663 p T^{8} - 13658 p^{2} T^{9} + 2417 p^{3} T^{10} - 428 p^{4} T^{11} + 63 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 2 T + 17 T^{2} - 184 T^{3} + 497 T^{4} - 4174 T^{5} + 22745 T^{6} - 45392 T^{7} + 22745 p T^{8} - 4174 p^{2} T^{9} + 497 p^{3} T^{10} - 184 p^{4} T^{11} + 17 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 3 p T^{2} + 140 T^{3} + 2337 T^{4} + 9584 T^{5} + 59309 T^{6} + 291720 T^{7} + 59309 p T^{8} + 9584 p^{2} T^{9} + 2337 p^{3} T^{10} + 140 p^{4} T^{11} + 3 p^{6} T^{12} + p^{7} T^{14} \) | |
| 29 | \( 1 + 4 T + 43 T^{2} + 300 T^{3} + 2397 T^{4} + 13692 T^{5} + 90543 T^{6} + 417320 T^{7} + 90543 p T^{8} + 13692 p^{2} T^{9} + 2397 p^{3} T^{10} + 300 p^{4} T^{11} + 43 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 7 T + 148 T^{2} + 931 T^{3} + 11198 T^{4} + 58509 T^{5} + 519739 T^{6} + 2272786 T^{7} + 519739 p T^{8} + 58509 p^{2} T^{9} + 11198 p^{3} T^{10} + 931 p^{4} T^{11} + 148 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 3 T + 114 T^{2} + 211 T^{3} + 8476 T^{4} + 16119 T^{5} + 421173 T^{6} + 585550 T^{7} + 421173 p T^{8} + 16119 p^{2} T^{9} + 8476 p^{3} T^{10} + 211 p^{4} T^{11} + 114 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 12 T + 183 T^{2} + 1700 T^{3} + 17105 T^{4} + 131452 T^{5} + 1040039 T^{6} + 6545832 T^{7} + 1040039 p T^{8} + 131452 p^{2} T^{9} + 17105 p^{3} T^{10} + 1700 p^{4} T^{11} + 183 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 2 T + 167 T^{2} - 660 T^{3} + 14931 T^{4} - 67206 T^{5} + 899789 T^{6} - 3782856 T^{7} + 899789 p T^{8} - 67206 p^{2} T^{9} + 14931 p^{3} T^{10} - 660 p^{4} T^{11} + 167 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 11 T + 192 T^{2} - 1847 T^{3} + 17634 T^{4} - 150193 T^{5} + 1120915 T^{6} - 8253386 T^{7} + 1120915 p T^{8} - 150193 p^{2} T^{9} + 17634 p^{3} T^{10} - 1847 p^{4} T^{11} + 192 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - T + 270 T^{2} - 231 T^{3} + 34242 T^{4} - 26021 T^{5} + 2690283 T^{6} - 1753150 T^{7} + 2690283 p T^{8} - 26021 p^{2} T^{9} + 34242 p^{3} T^{10} - 231 p^{4} T^{11} + 270 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 19 T + 364 T^{2} - 4667 T^{3} + 57934 T^{4} - 561883 T^{5} + 5308991 T^{6} - 41425142 T^{7} + 5308991 p T^{8} - 561883 p^{2} T^{9} + 57934 p^{3} T^{10} - 4667 p^{4} T^{11} + 364 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 12 T + 295 T^{2} - 3288 T^{3} + 43865 T^{4} - 410260 T^{5} + 4116935 T^{6} - 31032016 T^{7} + 4116935 p T^{8} - 410260 p^{2} T^{9} + 43865 p^{3} T^{10} - 3288 p^{4} T^{11} + 295 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 17 T + 98 T^{2} - 137 T^{3} + 4010 T^{4} - 81897 T^{5} + 561533 T^{6} - 2400242 T^{7} + 561533 p T^{8} - 81897 p^{2} T^{9} + 4010 p^{3} T^{10} - 137 p^{4} T^{11} + 98 p^{5} T^{12} - 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 20 T + 529 T^{2} - 100 p T^{3} + 107685 T^{4} - 1088764 T^{5} + 12096037 T^{6} - 97336680 T^{7} + 12096037 p T^{8} - 1088764 p^{2} T^{9} + 107685 p^{3} T^{10} - 100 p^{5} T^{11} + 529 p^{5} T^{12} - 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 10 T + 331 T^{2} - 3088 T^{3} + 793 p T^{4} - 453094 T^{5} + 6297923 T^{6} - 41415872 T^{7} + 6297923 p T^{8} - 453094 p^{2} T^{9} + 793 p^{4} T^{10} - 3088 p^{4} T^{11} + 331 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 2 T + 293 T^{2} + 672 T^{3} + 48493 T^{4} + 101038 T^{5} + 5408089 T^{6} + 10050656 T^{7} + 5408089 p T^{8} + 101038 p^{2} T^{9} + 48493 p^{3} T^{10} + 672 p^{4} T^{11} + 293 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 7 T + 406 T^{2} - 2165 T^{3} + 78228 T^{4} - 332467 T^{5} + 9448267 T^{6} - 33037258 T^{7} + 9448267 p T^{8} - 332467 p^{2} T^{9} + 78228 p^{3} T^{10} - 2165 p^{4} T^{11} + 406 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 3 T + 142 T^{2} + 1563 T^{3} + 15208 T^{4} + 131465 T^{5} + 2000929 T^{6} + 8498354 T^{7} + 2000929 p T^{8} + 131465 p^{2} T^{9} + 15208 p^{3} T^{10} + 1563 p^{4} T^{11} + 142 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 3 T + 542 T^{2} + 1923 T^{3} + 132516 T^{4} + 492061 T^{5} + 19451309 T^{6} + 64748442 T^{7} + 19451309 p T^{8} + 492061 p^{2} T^{9} + 132516 p^{3} T^{10} + 1923 p^{4} T^{11} + 542 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.59538608658470482949112252068, −3.56830042587467960660392061062, −3.54284956108390413078263305609, −3.26653130815030063464036942751, −3.14673070550187541120814930590, −2.76975855024723041253531825101, −2.75956666300957620598338030066, −2.60312592740050388531021624681, −2.51696238575639443977404175826, −2.44331315239776095956037253440, −2.38714589708734632940836959426, −1.98370660883590060897780355504, −1.86154212507835864955608984028, −1.74819586699568419167983453558, −1.66856625613296726725585883110, −1.59852927481738803540598065497, −1.47559796828296859477509443992, −1.45574003914008345995836368491, −1.01538740292530783683791403304, −0.905430025623087782128210164746, −0.798007251593992177286473698104, −0.61880431403807868338950344151, −0.45672265397589110156383873435, −0.40957465925589964260463155393, −0.13800697998428567837679746669, 0.13800697998428567837679746669, 0.40957465925589964260463155393, 0.45672265397589110156383873435, 0.61880431403807868338950344151, 0.798007251593992177286473698104, 0.905430025623087782128210164746, 1.01538740292530783683791403304, 1.45574003914008345995836368491, 1.47559796828296859477509443992, 1.59852927481738803540598065497, 1.66856625613296726725585883110, 1.74819586699568419167983453558, 1.86154212507835864955608984028, 1.98370660883590060897780355504, 2.38714589708734632940836959426, 2.44331315239776095956037253440, 2.51696238575639443977404175826, 2.60312592740050388531021624681, 2.75956666300957620598338030066, 2.76975855024723041253531825101, 3.14673070550187541120814930590, 3.26653130815030063464036942751, 3.54284956108390413078263305609, 3.56830042587467960660392061062, 3.59538608658470482949112252068
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.