Dirichlet series
| L(s) = 1 | + 2-s + 3·3-s − 3·4-s − 7·5-s + 3·6-s − 7-s − 3·8-s − 5·9-s − 7·10-s + 7·11-s − 9·12-s + 13-s − 14-s − 21·15-s + 2·16-s + 17-s − 5·18-s − 7·19-s + 21·20-s − 3·21-s + 7·22-s − 8·23-s − 9·24-s + 28·25-s + 26-s − 23·27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s − 3/2·4-s − 3.13·5-s + 1.22·6-s − 0.377·7-s − 1.06·8-s − 5/3·9-s − 2.21·10-s + 2.11·11-s − 2.59·12-s + 0.277·13-s − 0.267·14-s − 5.42·15-s + 1/2·16-s + 0.242·17-s − 1.17·18-s − 1.60·19-s + 4.69·20-s − 0.654·21-s + 1.49·22-s − 1.66·23-s − 1.83·24-s + 28/5·25-s + 0.196·26-s − 4.42·27-s + 0.566·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{7} \cdot 11^{7} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{7} \cdot 11^{7} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(5^{7} \cdot 11^{7} \cdot 19^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.81678\times 10^{6}\) |
| Root analytic conductor: | \(2.88866\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 5^{7} \cdot 11^{7} \cdot 19^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(3.447741339\) |
| \(L(\frac12)\) | \(\approx\) | \(3.447741339\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + T )^{7} \) |
| 11 | \( ( 1 - T )^{7} \) | |
| 19 | \( ( 1 + T )^{7} \) | |
| good | 2 | \( 1 - T + p^{2} T^{2} - p^{2} T^{3} + 11 T^{4} - 3 p^{2} T^{5} + 3 p^{3} T^{6} - 21 T^{7} + 3 p^{4} T^{8} - 3 p^{4} T^{9} + 11 p^{3} T^{10} - p^{6} T^{11} + p^{7} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 - p T + 14 T^{2} - 34 T^{3} + 101 T^{4} - 196 T^{5} + 448 T^{6} - 722 T^{7} + 448 p T^{8} - 196 p^{2} T^{9} + 101 p^{3} T^{10} - 34 p^{4} T^{11} + 14 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 + T + 22 T^{2} + 24 T^{3} + 289 T^{4} + 234 T^{5} + 2620 T^{6} + 1906 T^{7} + 2620 p T^{8} + 234 p^{2} T^{9} + 289 p^{3} T^{10} + 24 p^{4} T^{11} + 22 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - T + 42 T^{2} - 133 T^{3} + 890 T^{4} - 4262 T^{5} + 14415 T^{6} - 71436 T^{7} + 14415 p T^{8} - 4262 p^{2} T^{9} + 890 p^{3} T^{10} - 133 p^{4} T^{11} + 42 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - T + 58 T^{2} + 39 T^{3} + 1500 T^{4} + 3424 T^{5} + 27509 T^{6} + 85144 T^{7} + 27509 p T^{8} + 3424 p^{2} T^{9} + 1500 p^{3} T^{10} + 39 p^{4} T^{11} + 58 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 8 T + 123 T^{2} + 763 T^{3} + 7065 T^{4} + 35557 T^{5} + 245811 T^{6} + 1015316 T^{7} + 245811 p T^{8} + 35557 p^{2} T^{9} + 7065 p^{3} T^{10} + 763 p^{4} T^{11} + 123 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 11 T + 204 T^{2} - 1661 T^{3} + 17386 T^{4} - 3788 p T^{5} + 826445 T^{6} - 4114540 T^{7} + 826445 p T^{8} - 3788 p^{3} T^{9} + 17386 p^{3} T^{10} - 1661 p^{4} T^{11} + 204 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 7 T + 134 T^{2} - 755 T^{3} + 8790 T^{4} - 40338 T^{5} + 368637 T^{6} - 1458620 T^{7} + 368637 p T^{8} - 40338 p^{2} T^{9} + 8790 p^{3} T^{10} - 755 p^{4} T^{11} + 134 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 17 T + 249 T^{2} + 2604 T^{3} + 24900 T^{4} + 194048 T^{5} + 1422282 T^{6} + 8931106 T^{7} + 1422282 p T^{8} + 194048 p^{2} T^{9} + 24900 p^{3} T^{10} + 2604 p^{4} T^{11} + 249 p^{5} T^{12} + 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 17 T + 367 T^{2} - 4217 T^{3} + 51158 T^{4} - 433498 T^{5} + 3690766 T^{6} - 23713924 T^{7} + 3690766 p T^{8} - 433498 p^{2} T^{9} + 51158 p^{3} T^{10} - 4217 p^{4} T^{11} + 367 p^{5} T^{12} - 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 3 T + 142 T^{2} + 264 T^{3} + 7987 T^{4} + 2058 T^{5} + 257910 T^{6} - 319370 T^{7} + 257910 p T^{8} + 2058 p^{2} T^{9} + 7987 p^{3} T^{10} + 264 p^{4} T^{11} + 142 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 14 T + 247 T^{2} - 2709 T^{3} + 657 p T^{4} - 262093 T^{5} + 2257181 T^{6} - 15539452 T^{7} + 2257181 p T^{8} - 262093 p^{2} T^{9} + 657 p^{4} T^{10} - 2709 p^{4} T^{11} + 247 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 7 T + 108 T^{2} - 675 T^{3} + 9530 T^{4} - 52204 T^{5} + 547277 T^{6} - 2614264 T^{7} + 547277 p T^{8} - 52204 p^{2} T^{9} + 9530 p^{3} T^{10} - 675 p^{4} T^{11} + 108 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 35 T + 770 T^{2} - 12271 T^{3} + 160142 T^{4} - 1748514 T^{5} + 16520853 T^{6} - 135376868 T^{7} + 16520853 p T^{8} - 1748514 p^{2} T^{9} + 160142 p^{3} T^{10} - 12271 p^{4} T^{11} + 770 p^{5} T^{12} - 35 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 17 T + 324 T^{2} - 3187 T^{3} + 30270 T^{4} - 180228 T^{5} + 1088085 T^{6} - 5986464 T^{7} + 1088085 p T^{8} - 180228 p^{2} T^{9} + 30270 p^{3} T^{10} - 3187 p^{4} T^{11} + 324 p^{5} T^{12} - 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 4 T + 336 T^{2} - 1066 T^{3} + 54224 T^{4} - 141621 T^{5} + 5450933 T^{6} - 11814518 T^{7} + 5450933 p T^{8} - 141621 p^{2} T^{9} + 54224 p^{3} T^{10} - 1066 p^{4} T^{11} + 336 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 10 T + 387 T^{2} - 3125 T^{3} + 68793 T^{4} - 453699 T^{5} + 7368323 T^{6} - 40062736 T^{7} + 7368323 p T^{8} - 453699 p^{2} T^{9} + 68793 p^{3} T^{10} - 3125 p^{4} T^{11} + 387 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 22 T + 580 T^{2} - 7786 T^{3} + 116824 T^{4} - 1135719 T^{5} + 12708239 T^{6} - 99863438 T^{7} + 12708239 p T^{8} - 1135719 p^{2} T^{9} + 116824 p^{3} T^{10} - 7786 p^{4} T^{11} + 580 p^{5} T^{12} - 22 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 11 T + 447 T^{2} - 4568 T^{3} + 92208 T^{4} - 834952 T^{5} + 11329598 T^{6} - 85731834 T^{7} + 11329598 p T^{8} - 834952 p^{2} T^{9} + 92208 p^{3} T^{10} - 4568 p^{4} T^{11} + 447 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 39 T + 1114 T^{2} - 22419 T^{3} + 371662 T^{4} - 5021958 T^{5} + 58158901 T^{6} - 569353216 T^{7} + 58158901 p T^{8} - 5021958 p^{2} T^{9} + 371662 p^{3} T^{10} - 22419 p^{4} T^{11} + 1114 p^{5} T^{12} - 39 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 18 T + 542 T^{2} - 7984 T^{3} + 130960 T^{4} - 1578961 T^{5} + 18395813 T^{6} - 179972130 T^{7} + 18395813 p T^{8} - 1578961 p^{2} T^{9} + 130960 p^{3} T^{10} - 7984 p^{4} T^{11} + 542 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 4 T + 264 T^{2} + 934 T^{3} + 42502 T^{4} + 225679 T^{5} + 5033241 T^{6} + 27046986 T^{7} + 5033241 p T^{8} + 225679 p^{2} T^{9} + 42502 p^{3} T^{10} + 934 p^{4} T^{11} + 264 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−4.62359158142365317075068531266, −4.44856510659346713751510685672, −4.39405183658322571128944608155, −4.03609477160869133708738860338, −3.93389707846361486050084934348, −3.87670641826391346368223502769, −3.85205684256680292528431862841, −3.70216848163064425812907354282, −3.66930388689064444549099590087, −3.52463813550576053830771804103, −3.33891798969482724267950015168, −3.07104096336660036267224358907, −2.91090536982010738435443149877, −2.86605906921975882889489991889, −2.68337922048314293816474941490, −2.44791247922315919286203093124, −2.14389531229314839874012979233, −2.10698632993113475245190427506, −2.06747234374865495005375723467, −1.78669165550340696671943807640, −0.989063738569412530378773142968, −0.951375051304025515462569909213, −0.75391757395635986766845305494, −0.45573796087617507196452614855, −0.42528458992795971717384144059, 0.42528458992795971717384144059, 0.45573796087617507196452614855, 0.75391757395635986766845305494, 0.951375051304025515462569909213, 0.989063738569412530378773142968, 1.78669165550340696671943807640, 2.06747234374865495005375723467, 2.10698632993113475245190427506, 2.14389531229314839874012979233, 2.44791247922315919286203093124, 2.68337922048314293816474941490, 2.86605906921975882889489991889, 2.91090536982010738435443149877, 3.07104096336660036267224358907, 3.33891798969482724267950015168, 3.52463813550576053830771804103, 3.66930388689064444549099590087, 3.70216848163064425812907354282, 3.85205684256680292528431862841, 3.87670641826391346368223502769, 3.93389707846361486050084934348, 4.03609477160869133708738860338, 4.39405183658322571128944608155, 4.44856510659346713751510685672, 4.62359158142365317075068531266
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.