Dirichlet series
| L(s) = 1 | + 7·2-s − 3-s + 28·4-s + 3·5-s − 7·6-s + 4·7-s + 84·8-s − 3·9-s + 21·10-s − 11-s − 28·12-s + 7·13-s + 28·14-s − 3·15-s + 210·16-s + 4·17-s − 21·18-s + 19-s + 84·20-s − 4·21-s − 7·22-s − 84·24-s − 7·25-s + 49·26-s − 5·27-s + 112·28-s + 5·29-s + ⋯ |
| L(s) = 1 | + 4.94·2-s − 0.577·3-s + 14·4-s + 1.34·5-s − 2.85·6-s + 1.51·7-s + 29.6·8-s − 9-s + 6.64·10-s − 0.301·11-s − 8.08·12-s + 1.94·13-s + 7.48·14-s − 0.774·15-s + 52.5·16-s + 0.970·17-s − 4.94·18-s + 0.229·19-s + 18.7·20-s − 0.872·21-s − 1.49·22-s − 17.1·24-s − 7/5·25-s + 9.60·26-s − 0.962·27-s + 21.1·28-s + 0.928·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 157^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 157^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{7} \cdot 157^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(622.941\) |
| Root analytic conductor: | \(1.58344\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{7} \cdot 157^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(112.6262885\) |
| \(L(\frac12)\) | \(\approx\) | \(112.6262885\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{7} \) |
| 157 | \( ( 1 + T )^{7} \) | |
| good | 3 | \( 1 + T + 4 T^{2} + 4 p T^{3} + 2 p^{2} T^{4} + 44 T^{5} + 98 T^{6} + 106 T^{7} + 98 p T^{8} + 44 p^{2} T^{9} + 2 p^{5} T^{10} + 4 p^{5} T^{11} + 4 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) |
| 5 | \( 1 - 3 T + 16 T^{2} - 32 T^{3} + 26 p T^{4} - 202 T^{5} + 142 p T^{6} - 938 T^{7} + 142 p^{2} T^{8} - 202 p^{2} T^{9} + 26 p^{4} T^{10} - 32 p^{4} T^{11} + 16 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 - 4 T + 25 T^{2} - 81 T^{3} + 325 T^{4} - 929 T^{5} + 461 p T^{6} - 7807 T^{7} + 461 p^{2} T^{8} - 929 p^{2} T^{9} + 325 p^{3} T^{10} - 81 p^{4} T^{11} + 25 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + T + 16 T^{2} + 6 p T^{3} + 174 T^{4} + 82 p T^{5} + 4390 T^{6} + 6624 T^{7} + 4390 p T^{8} + 82 p^{3} T^{9} + 174 p^{3} T^{10} + 6 p^{5} T^{11} + 16 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 7 T + 47 T^{2} - 202 T^{3} + 1153 T^{4} - 4785 T^{5} + 20983 T^{6} - 70060 T^{7} + 20983 p T^{8} - 4785 p^{2} T^{9} + 1153 p^{3} T^{10} - 202 p^{4} T^{11} + 47 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 4 T + 81 T^{2} - 241 T^{3} + 3045 T^{4} - 7193 T^{5} + 72993 T^{6} - 143949 T^{7} + 72993 p T^{8} - 7193 p^{2} T^{9} + 3045 p^{3} T^{10} - 241 p^{4} T^{11} + 81 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - T + 62 T^{2} - 34 T^{3} + 1742 T^{4} + 58 p T^{5} + 34508 T^{6} + 52118 T^{7} + 34508 p T^{8} + 58 p^{3} T^{9} + 1742 p^{3} T^{10} - 34 p^{4} T^{11} + 62 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 89 T^{2} + 27 T^{3} + 4125 T^{4} + 1441 T^{5} + 133827 T^{6} + 38173 T^{7} + 133827 p T^{8} + 1441 p^{2} T^{9} + 4125 p^{3} T^{10} + 27 p^{4} T^{11} + 89 p^{5} T^{12} + p^{7} T^{14} \) | |
| 29 | \( 1 - 5 T + 90 T^{2} - 34 T^{3} + 2838 T^{4} + 5446 T^{5} + 124704 T^{6} + 48166 T^{7} + 124704 p T^{8} + 5446 p^{2} T^{9} + 2838 p^{3} T^{10} - 34 p^{4} T^{11} + 90 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 117 T^{2} - 64 T^{3} + 7049 T^{4} - 5696 T^{5} + 293013 T^{6} - 229760 T^{7} + 293013 p T^{8} - 5696 p^{2} T^{9} + 7049 p^{3} T^{10} - 64 p^{4} T^{11} + 117 p^{5} T^{12} + p^{7} T^{14} \) | |
| 37 | \( 1 + 3 T + 65 T^{2} - 134 T^{3} + 1547 T^{4} - 7467 T^{5} + 162803 T^{6} + 161452 T^{7} + 162803 p T^{8} - 7467 p^{2} T^{9} + 1547 p^{3} T^{10} - 134 p^{4} T^{11} + 65 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 8 T + 99 T^{2} + 280 T^{3} + 2489 T^{4} + 11192 T^{5} + 182355 T^{6} + 1186384 T^{7} + 182355 p T^{8} + 11192 p^{2} T^{9} + 2489 p^{3} T^{10} + 280 p^{4} T^{11} + 99 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 7 T + 243 T^{2} + 1566 T^{3} + 27575 T^{4} + 153617 T^{5} + 1864213 T^{6} + 8531236 T^{7} + 1864213 p T^{8} + 153617 p^{2} T^{9} + 27575 p^{3} T^{10} + 1566 p^{4} T^{11} + 243 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 12 T + 125 T^{2} + 712 T^{3} + 6849 T^{4} + 53716 T^{5} + 515045 T^{6} + 3333296 T^{7} + 515045 p T^{8} + 53716 p^{2} T^{9} + 6849 p^{3} T^{10} + 712 p^{4} T^{11} + 125 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 13 T + 354 T^{2} - 3424 T^{3} + 52612 T^{4} - 399054 T^{5} + 4439602 T^{6} - 26935094 T^{7} + 4439602 p T^{8} - 399054 p^{2} T^{9} + 52612 p^{3} T^{10} - 3424 p^{4} T^{11} + 354 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 15 T + 271 T^{2} + 3274 T^{3} + 41211 T^{4} + 374057 T^{5} + 3632045 T^{6} + 27828108 T^{7} + 3632045 p T^{8} + 374057 p^{2} T^{9} + 41211 p^{3} T^{10} + 3274 p^{4} T^{11} + 271 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 9 T + 318 T^{2} - 2478 T^{3} + 47590 T^{4} - 315808 T^{5} + 4394360 T^{6} - 24176298 T^{7} + 4394360 p T^{8} - 315808 p^{2} T^{9} + 47590 p^{3} T^{10} - 2478 p^{4} T^{11} + 318 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 11 T + 376 T^{2} + 3564 T^{3} + 66726 T^{4} + 527660 T^{5} + 7042042 T^{6} + 45207584 T^{7} + 7042042 p T^{8} + 527660 p^{2} T^{9} + 66726 p^{3} T^{10} + 3564 p^{4} T^{11} + 376 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 18 T + 361 T^{2} - 3972 T^{3} + 53229 T^{4} - 474894 T^{5} + 5140693 T^{6} - 38845944 T^{7} + 5140693 p T^{8} - 474894 p^{2} T^{9} + 53229 p^{3} T^{10} - 3972 p^{4} T^{11} + 361 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 16 T + 455 T^{2} - 5568 T^{3} + 91613 T^{4} - 886256 T^{5} + 10629291 T^{6} - 82485376 T^{7} + 10629291 p T^{8} - 886256 p^{2} T^{9} + 91613 p^{3} T^{10} - 5568 p^{4} T^{11} + 455 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 9 T + 470 T^{2} + 3400 T^{3} + 99226 T^{4} + 590200 T^{5} + 12332928 T^{6} + 59650606 T^{7} + 12332928 p T^{8} + 590200 p^{2} T^{9} + 99226 p^{3} T^{10} + 3400 p^{4} T^{11} + 470 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 24 T + 469 T^{2} + 5112 T^{3} + 49517 T^{4} + 257736 T^{5} + 1412161 T^{6} + 1784400 T^{7} + 1412161 p T^{8} + 257736 p^{2} T^{9} + 49517 p^{3} T^{10} + 5112 p^{4} T^{11} + 469 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 10 T + 387 T^{2} + 1925 T^{3} + 57671 T^{4} + 81509 T^{5} + 5455843 T^{6} - 1396719 T^{7} + 5455843 p T^{8} + 81509 p^{2} T^{9} + 57671 p^{3} T^{10} + 1925 p^{4} T^{11} + 387 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 8 T + 187 T^{2} - 1472 T^{3} + 19465 T^{4} - 199480 T^{5} + 1819195 T^{6} - 24317312 T^{7} + 1819195 p T^{8} - 199480 p^{2} T^{9} + 19465 p^{3} T^{10} - 1472 p^{4} T^{11} + 187 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.55087812644635764858319172764, −5.54540390500821730732408681595, −5.52272183343897971826001207422, −5.36758246918698202927921056045, −5.09969065975059562239713157152, −5.09562369933902695618193975260, −4.55497636245761492870713584245, −4.49391221967447325652283025961, −4.46208552532287274214277702123, −4.38191522131422711512760171450, −4.18000326799861117486150129967, −3.97530130453165384301587969948, −3.47928562915086222096227812335, −3.40313301601609700467061442519, −3.35171038210063327409849970252, −3.29054215199891102431981302837, −3.14007080891049684634933901691, −2.93750813045119411201409592417, −2.38643376136929883071421310486, −2.23808371418401553285857264310, −1.99626800842551403942868665284, −1.83243317403571995030226300056, −1.82857314649572091146034358964, −1.35875074774183946298790161624, −1.16261258244543578025671592030, 1.16261258244543578025671592030, 1.35875074774183946298790161624, 1.82857314649572091146034358964, 1.83243317403571995030226300056, 1.99626800842551403942868665284, 2.23808371418401553285857264310, 2.38643376136929883071421310486, 2.93750813045119411201409592417, 3.14007080891049684634933901691, 3.29054215199891102431981302837, 3.35171038210063327409849970252, 3.40313301601609700467061442519, 3.47928562915086222096227812335, 3.97530130453165384301587969948, 4.18000326799861117486150129967, 4.38191522131422711512760171450, 4.46208552532287274214277702123, 4.49391221967447325652283025961, 4.55497636245761492870713584245, 5.09562369933902695618193975260, 5.09969065975059562239713157152, 5.36758246918698202927921056045, 5.52272183343897971826001207422, 5.54540390500821730732408681595, 5.55087812644635764858319172764
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.