Dirichlet series
| L(s) = 1 | − 2-s − 7·5-s + 10·7-s + 8-s + 7·10-s + 3·11-s + 6·13-s − 10·14-s − 2·16-s − 8·17-s + 10·19-s − 3·22-s − 11·23-s + 28·25-s − 6·26-s + 7·29-s + 18·31-s + 32-s + 8·34-s − 70·35-s + 13·37-s − 10·38-s − 7·40-s + 13·41-s + 9·43-s + 11·46-s − 2·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 3.13·5-s + 3.77·7-s + 0.353·8-s + 2.21·10-s + 0.904·11-s + 1.66·13-s − 2.67·14-s − 1/2·16-s − 1.94·17-s + 2.29·19-s − 0.639·22-s − 2.29·23-s + 28/5·25-s − 1.17·26-s + 1.29·29-s + 3.23·31-s + 0.176·32-s + 1.37·34-s − 11.8·35-s + 2.13·37-s − 1.62·38-s − 1.10·40-s + 2.03·41-s + 1.37·43-s + 1.62·46-s − 0.291·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 5^{7} \cdot 29^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 5^{7} \cdot 29^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{14} \cdot 5^{7} \cdot 29^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.33417\times 10^{7}\) |
| Root analytic conductor: | \(3.22807\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 3^{14} \cdot 5^{7} \cdot 29^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(8.244409138\) |
| \(L(\frac12)\) | \(\approx\) | \(8.244409138\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + T )^{7} \) | |
| 29 | \( ( 1 - T )^{7} \) | |
| good | 2 | \( 1 + T + T^{2} + T^{4} + T^{5} + 7 T^{6} + p T^{7} + 7 p T^{8} + p^{2} T^{9} + p^{3} T^{10} + p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) |
| 7 | \( 1 - 10 T + 64 T^{2} - 306 T^{3} + 26 p^{2} T^{4} - 4558 T^{5} + 14431 T^{6} - 40164 T^{7} + 14431 p T^{8} - 4558 p^{2} T^{9} + 26 p^{5} T^{10} - 306 p^{4} T^{11} + 64 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 3 T + 36 T^{2} - 147 T^{3} + 766 T^{4} - 3369 T^{5} + 11227 T^{6} - 46518 T^{7} + 11227 p T^{8} - 3369 p^{2} T^{9} + 766 p^{3} T^{10} - 147 p^{4} T^{11} + 36 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 6 T + 56 T^{2} - 292 T^{3} + 1618 T^{4} - 7146 T^{5} + 30345 T^{6} - 112056 T^{7} + 30345 p T^{8} - 7146 p^{2} T^{9} + 1618 p^{3} T^{10} - 292 p^{4} T^{11} + 56 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 8 T + 96 T^{2} + 562 T^{3} + 4066 T^{4} + 1120 p T^{5} + 6125 p T^{6} + 400748 T^{7} + 6125 p^{2} T^{8} + 1120 p^{3} T^{9} + 4066 p^{3} T^{10} + 562 p^{4} T^{11} + 96 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 10 T + 113 T^{2} - 796 T^{3} + 295 p T^{4} - 31310 T^{5} + 165901 T^{6} - 745592 T^{7} + 165901 p T^{8} - 31310 p^{2} T^{9} + 295 p^{4} T^{10} - 796 p^{4} T^{11} + 113 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 11 T + 145 T^{2} + 902 T^{3} + 6585 T^{4} + 27009 T^{5} + 157561 T^{6} + 560524 T^{7} + 157561 p T^{8} + 27009 p^{2} T^{9} + 6585 p^{3} T^{10} + 902 p^{4} T^{11} + 145 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 18 T + 293 T^{2} - 3132 T^{3} + 979 p T^{4} - 231798 T^{5} + 52423 p T^{6} - 9425656 T^{7} + 52423 p^{2} T^{8} - 231798 p^{2} T^{9} + 979 p^{4} T^{10} - 3132 p^{4} T^{11} + 293 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 13 T + 227 T^{2} - 2174 T^{3} + 22269 T^{4} - 169083 T^{5} + 1283239 T^{6} - 7880036 T^{7} + 1283239 p T^{8} - 169083 p^{2} T^{9} + 22269 p^{3} T^{10} - 2174 p^{4} T^{11} + 227 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 13 T + 239 T^{2} - 2218 T^{3} + 23301 T^{4} - 172819 T^{5} + 1354395 T^{6} - 8503692 T^{7} + 1354395 p T^{8} - 172819 p^{2} T^{9} + 23301 p^{3} T^{10} - 2218 p^{4} T^{11} + 239 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 9 T + 137 T^{2} - 1214 T^{3} + 13361 T^{4} - 89759 T^{5} + 786225 T^{6} - 4832036 T^{7} + 786225 p T^{8} - 89759 p^{2} T^{9} + 13361 p^{3} T^{10} - 1214 p^{4} T^{11} + 137 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 2 T + 152 T^{2} + 198 T^{3} + 12442 T^{4} + 6294 T^{5} + 708291 T^{6} + 155204 T^{7} + 708291 p T^{8} + 6294 p^{2} T^{9} + 12442 p^{3} T^{10} + 198 p^{4} T^{11} + 152 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 5 T + 171 T^{2} + 622 T^{3} + 16981 T^{4} + 53411 T^{5} + 1164679 T^{6} + 2921732 T^{7} + 1164679 p T^{8} + 53411 p^{2} T^{9} + 16981 p^{3} T^{10} + 622 p^{4} T^{11} + 171 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 8 T + 329 T^{2} - 2384 T^{3} + 51393 T^{4} - 316088 T^{5} + 4767777 T^{6} - 24036192 T^{7} + 4767777 p T^{8} - 316088 p^{2} T^{9} + 51393 p^{3} T^{10} - 2384 p^{4} T^{11} + 329 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 14 T + 111 T^{2} - 1404 T^{3} + 18353 T^{4} - 144354 T^{5} + 1255111 T^{6} - 11095432 T^{7} + 1255111 p T^{8} - 144354 p^{2} T^{9} + 18353 p^{3} T^{10} - 1404 p^{4} T^{11} + 111 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 14 T + 352 T^{2} - 3898 T^{3} + 58042 T^{4} - 526282 T^{5} + 5848403 T^{6} - 43580372 T^{7} + 5848403 p T^{8} - 526282 p^{2} T^{9} + 58042 p^{3} T^{10} - 3898 p^{4} T^{11} + 352 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 8 T + 273 T^{2} - 2000 T^{3} + 39717 T^{4} - 242424 T^{5} + 3839397 T^{6} - 20232544 T^{7} + 3839397 p T^{8} - 242424 p^{2} T^{9} + 39717 p^{3} T^{10} - 2000 p^{4} T^{11} + 273 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 3 T + 135 T^{2} + 434 T^{3} + 12621 T^{4} + 45459 T^{5} + 1473787 T^{6} + 1699996 T^{7} + 1473787 p T^{8} + 45459 p^{2} T^{9} + 12621 p^{3} T^{10} + 434 p^{4} T^{11} + 135 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 4 T + 401 T^{2} - 1272 T^{3} + 74769 T^{4} - 191276 T^{5} + 8654401 T^{6} - 18261104 T^{7} + 8654401 p T^{8} - 191276 p^{2} T^{9} + 74769 p^{3} T^{10} - 1272 p^{4} T^{11} + 401 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 17 T + 297 T^{2} + 3726 T^{3} + 43661 T^{4} + 423899 T^{5} + 4325821 T^{6} + 38523964 T^{7} + 4325821 p T^{8} + 423899 p^{2} T^{9} + 43661 p^{3} T^{10} + 3726 p^{4} T^{11} + 297 p^{5} T^{12} + 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 20 T + 544 T^{2} - 8750 T^{3} + 137274 T^{4} - 1700244 T^{5} + 20021293 T^{6} - 192198292 T^{7} + 20021293 p T^{8} - 1700244 p^{2} T^{9} + 137274 p^{3} T^{10} - 8750 p^{4} T^{11} + 544 p^{5} T^{12} - 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 13 T + 495 T^{2} - 5566 T^{3} + 118909 T^{4} - 1122819 T^{5} + 17567251 T^{6} - 136565540 T^{7} + 17567251 p T^{8} - 1122819 p^{2} T^{9} + 118909 p^{3} T^{10} - 5566 p^{4} T^{11} + 495 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−4.45938398245230190343925036505, −4.44372285388301507122802994511, −4.42450979705355023791013292884, −4.29465463002541131876381669413, −4.06864450285337550693451246209, −3.70237346913367078031509253429, −3.67766041552856972600998491292, −3.67434634179096979262001656158, −3.60822218332953945998330611961, −3.32115702740899757113288754156, −2.98815246567583561681664837086, −2.98320112722973558432814709772, −2.65737882212744162530078298861, −2.52039460895309857228438416730, −2.45050030337405982599065202299, −2.20945019288462792112643587398, −2.01912639071219725067105579814, −1.78536370583207551993209285268, −1.65259694094452858371360365222, −1.32865209688490805257829647409, −1.09860485634575322802576802578, −0.991939427866663918337482067132, −0.853950483066230731222759549205, −0.60210035958858252656005897022, −0.50227223507754740629165732016, 0.50227223507754740629165732016, 0.60210035958858252656005897022, 0.853950483066230731222759549205, 0.991939427866663918337482067132, 1.09860485634575322802576802578, 1.32865209688490805257829647409, 1.65259694094452858371360365222, 1.78536370583207551993209285268, 2.01912639071219725067105579814, 2.20945019288462792112643587398, 2.45050030337405982599065202299, 2.52039460895309857228438416730, 2.65737882212744162530078298861, 2.98320112722973558432814709772, 2.98815246567583561681664837086, 3.32115702740899757113288754156, 3.60822218332953945998330611961, 3.67434634179096979262001656158, 3.67766041552856972600998491292, 3.70237346913367078031509253429, 4.06864450285337550693451246209, 4.29465463002541131876381669413, 4.42450979705355023791013292884, 4.44372285388301507122802994511, 4.45938398245230190343925036505
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.