Dirichlet series
| L(s) = 1 | − 7·2-s + 7·3-s + 28·4-s − 5·5-s − 49·6-s + 9·7-s − 84·8-s + 28·9-s + 35·10-s + 7·11-s + 196·12-s + 4·13-s − 63·14-s − 35·15-s + 210·16-s − 6·17-s − 196·18-s + 9·19-s − 140·20-s + 63·21-s − 49·22-s − 8·23-s − 588·24-s − 3·25-s − 28·26-s + 84·27-s + 252·28-s + ⋯ |
| L(s) = 1 | − 4.94·2-s + 4.04·3-s + 14·4-s − 2.23·5-s − 20.0·6-s + 3.40·7-s − 29.6·8-s + 28/3·9-s + 11.0·10-s + 2.11·11-s + 56.5·12-s + 1.10·13-s − 16.8·14-s − 9.03·15-s + 52.5·16-s − 1.45·17-s − 46.1·18-s + 2.06·19-s − 31.3·20-s + 13.7·21-s − 10.4·22-s − 1.66·23-s − 120.·24-s − 3/5·25-s − 5.49·26-s + 16.1·27-s + 47.6·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{7} \cdot 11^{7} \cdot 61^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{7} \cdot 11^{7} \cdot 61^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{7} \cdot 3^{7} \cdot 11^{7} \cdot 61^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.54859\times 10^{10}\) |
| Root analytic conductor: | \(5.66990\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{7} \cdot 3^{7} \cdot 11^{7} \cdot 61^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(28.37057572\) |
| \(L(\frac12)\) | \(\approx\) | \(28.37057572\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{7} \) |
| 3 | \( ( 1 - T )^{7} \) | |
| 11 | \( ( 1 - T )^{7} \) | |
| 61 | \( ( 1 - T )^{7} \) | |
| good | 5 | \( 1 + p T + 28 T^{2} + 94 T^{3} + 346 T^{4} + 932 T^{5} + 2603 T^{6} + 1138 p T^{7} + 2603 p T^{8} + 932 p^{2} T^{9} + 346 p^{3} T^{10} + 94 p^{4} T^{11} + 28 p^{5} T^{12} + p^{7} T^{13} + p^{7} T^{14} \) |
| 7 | \( 1 - 9 T + 53 T^{2} - 218 T^{3} + 787 T^{4} - 348 p T^{5} + 1021 p T^{6} - 19062 T^{7} + 1021 p^{2} T^{8} - 348 p^{3} T^{9} + 787 p^{3} T^{10} - 218 p^{4} T^{11} + 53 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 4 T + 28 T^{2} - 14 T^{3} + 319 T^{4} - 81 T^{5} + 6379 T^{6} - 305 p T^{7} + 6379 p T^{8} - 81 p^{2} T^{9} + 319 p^{3} T^{10} - 14 p^{4} T^{11} + 28 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 6 T + 56 T^{2} + 245 T^{3} + 972 T^{4} + 1603 T^{5} + 2739 T^{6} - 28128 T^{7} + 2739 p T^{8} + 1603 p^{2} T^{9} + 972 p^{3} T^{10} + 245 p^{4} T^{11} + 56 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 9 T + 111 T^{2} - 742 T^{3} + 5599 T^{4} - 30012 T^{5} + 166841 T^{6} - 718890 T^{7} + 166841 p T^{8} - 30012 p^{2} T^{9} + 5599 p^{3} T^{10} - 742 p^{4} T^{11} + 111 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 8 T + 98 T^{2} + 441 T^{3} + 3648 T^{4} + 11744 T^{5} + 92871 T^{6} + 256318 T^{7} + 92871 p T^{8} + 11744 p^{2} T^{9} + 3648 p^{3} T^{10} + 441 p^{4} T^{11} + 98 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 4 T + 105 T^{2} + 485 T^{3} + 6144 T^{4} + 26521 T^{5} + 250044 T^{6} + 917740 T^{7} + 250044 p T^{8} + 26521 p^{2} T^{9} + 6144 p^{3} T^{10} + 485 p^{4} T^{11} + 105 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 17 T + 230 T^{2} - 2084 T^{3} + 520 p T^{4} - 3270 p T^{5} + 601321 T^{6} - 3294182 T^{7} + 601321 p T^{8} - 3270 p^{3} T^{9} + 520 p^{4} T^{10} - 2084 p^{4} T^{11} + 230 p^{5} T^{12} - 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 19 T + 357 T^{2} - 4142 T^{3} + 45451 T^{4} - 377480 T^{5} + 2956223 T^{6} - 500482 p T^{7} + 2956223 p T^{8} - 377480 p^{2} T^{9} + 45451 p^{3} T^{10} - 4142 p^{4} T^{11} + 357 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 5 T + 243 T^{2} - 970 T^{3} + 26508 T^{4} - 86478 T^{5} + 1705372 T^{6} - 4521438 T^{7} + 1705372 p T^{8} - 86478 p^{2} T^{9} + 26508 p^{3} T^{10} - 970 p^{4} T^{11} + 243 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 24 T + 458 T^{2} - 6071 T^{3} + 69138 T^{4} - 640436 T^{5} + 5252333 T^{6} - 36460442 T^{7} + 5252333 p T^{8} - 640436 p^{2} T^{9} + 69138 p^{3} T^{10} - 6071 p^{4} T^{11} + 458 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 6 T + 209 T^{2} - 1150 T^{3} + 22680 T^{4} - 107719 T^{5} + 1559756 T^{6} - 6275246 T^{7} + 1559756 p T^{8} - 107719 p^{2} T^{9} + 22680 p^{3} T^{10} - 1150 p^{4} T^{11} + 209 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 3 T + 197 T^{2} - 1022 T^{3} + 20085 T^{4} - 128036 T^{5} + 1416341 T^{6} - 8781154 T^{7} + 1416341 p T^{8} - 128036 p^{2} T^{9} + 20085 p^{3} T^{10} - 1022 p^{4} T^{11} + 197 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 10 T + 320 T^{2} - 2568 T^{3} + 45501 T^{4} - 306445 T^{5} + 3948285 T^{6} - 22395551 T^{7} + 3948285 p T^{8} - 306445 p^{2} T^{9} + 45501 p^{3} T^{10} - 2568 p^{4} T^{11} + 320 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 22 T + 421 T^{2} - 4788 T^{3} + 50240 T^{4} - 374709 T^{5} + 2994564 T^{6} - 20687394 T^{7} + 2994564 p T^{8} - 374709 p^{2} T^{9} + 50240 p^{3} T^{10} - 4788 p^{4} T^{11} + 421 p^{5} T^{12} - 22 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - T + 305 T^{2} - 211 T^{3} + 46407 T^{4} - 10643 T^{5} + 4574967 T^{6} - 345826 T^{7} + 4574967 p T^{8} - 10643 p^{2} T^{9} + 46407 p^{3} T^{10} - 211 p^{4} T^{11} + 305 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 32 T + 738 T^{2} - 12977 T^{3} + 185644 T^{4} - 2261288 T^{5} + 23779529 T^{6} - 216674702 T^{7} + 23779529 p T^{8} - 2261288 p^{2} T^{9} + 185644 p^{3} T^{10} - 12977 p^{4} T^{11} + 738 p^{5} T^{12} - 32 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 8 T + 261 T^{2} + 1286 T^{3} + 29164 T^{4} + 75261 T^{5} + 2431460 T^{6} + 3805370 T^{7} + 2431460 p T^{8} + 75261 p^{2} T^{9} + 29164 p^{3} T^{10} + 1286 p^{4} T^{11} + 261 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 30 T + 636 T^{2} + 10091 T^{3} + 132970 T^{4} + 1523538 T^{5} + 15733803 T^{6} + 148515002 T^{7} + 15733803 p T^{8} + 1523538 p^{2} T^{9} + 132970 p^{3} T^{10} + 10091 p^{4} T^{11} + 636 p^{5} T^{12} + 30 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 5 T + 472 T^{2} - 2248 T^{3} + 104236 T^{4} - 455178 T^{5} + 14046421 T^{6} - 52446858 T^{7} + 14046421 p T^{8} - 455178 p^{2} T^{9} + 104236 p^{3} T^{10} - 2248 p^{4} T^{11} + 472 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 14 T + 564 T^{2} - 5698 T^{3} + 133631 T^{4} - 1045409 T^{5} + 18848719 T^{6} - 121183741 T^{7} + 18848719 p T^{8} - 1045409 p^{2} T^{9} + 133631 p^{3} T^{10} - 5698 p^{4} T^{11} + 564 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.74335561068927480959165720323, −3.72920078987322313857671599229, −3.72164738315245774857934273277, −3.47400914304975983914297070880, −3.07125575441440522717902581703, −3.06245008037940278603170491294, −2.98625083268455318659426363941, −2.70657597138877180397386652621, −2.70135392310973632177050323602, −2.60312481903230128713310795477, −2.47360455665368736441343339290, −2.09429342548682513773491397522, −2.03052996581321614699337247080, −2.01059842869371222407079023696, −2.00713108192013960301448737916, −1.77415317025255983950731578988, −1.71324500774833716198037264899, −1.66585520809192549930232856215, −1.23765001500018875281274284957, −1.06658186562614224117357043294, −0.921289868558479632337230529784, −0.803024804123774092556996641787, −0.798959122639955287919416697658, −0.62017909927224501615766842410, −0.50638669457012463507033856750, 0.50638669457012463507033856750, 0.62017909927224501615766842410, 0.798959122639955287919416697658, 0.803024804123774092556996641787, 0.921289868558479632337230529784, 1.06658186562614224117357043294, 1.23765001500018875281274284957, 1.66585520809192549930232856215, 1.71324500774833716198037264899, 1.77415317025255983950731578988, 2.00713108192013960301448737916, 2.01059842869371222407079023696, 2.03052996581321614699337247080, 2.09429342548682513773491397522, 2.47360455665368736441343339290, 2.60312481903230128713310795477, 2.70135392310973632177050323602, 2.70657597138877180397386652621, 2.98625083268455318659426363941, 3.06245008037940278603170491294, 3.07125575441440522717902581703, 3.47400914304975983914297070880, 3.72164738315245774857934273277, 3.72920078987322313857671599229, 3.74335561068927480959165720323
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.