Dirichlet series
| L(s) = 1 | + 7·2-s + 3-s + 28·4-s + 7·5-s + 7·6-s + 84·8-s − 6·9-s + 49·10-s + 28·12-s + 6·13-s + 7·15-s + 210·16-s + 7·17-s − 42·18-s + 10·19-s + 196·20-s − 23-s + 84·24-s + 28·25-s + 42·26-s − 27-s − 9·29-s + 49·30-s + 18·31-s + 462·32-s + 49·34-s − 168·36-s + ⋯ |
| L(s) = 1 | + 4.94·2-s + 0.577·3-s + 14·4-s + 3.13·5-s + 2.85·6-s + 29.6·8-s − 2·9-s + 15.4·10-s + 8.08·12-s + 1.66·13-s + 1.80·15-s + 52.5·16-s + 1.69·17-s − 9.89·18-s + 2.29·19-s + 43.8·20-s − 0.208·23-s + 17.1·24-s + 28/5·25-s + 8.23·26-s − 0.192·27-s − 1.67·29-s + 8.94·30-s + 3.23·31-s + 81.6·32-s + 8.40·34-s − 28·36-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 5^{7} \cdot 7^{14} \cdot 17^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 5^{7} \cdot 7^{14} \cdot 17^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{7} \cdot 5^{7} \cdot 7^{14} \cdot 17^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.76042\times 10^{12}\) |
| Root analytic conductor: | \(8.15569\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{7} \cdot 5^{7} \cdot 7^{14} \cdot 17^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(9300.203676\) |
| \(L(\frac12)\) | \(\approx\) | \(9300.203676\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{7} \) |
| 5 | \( ( 1 - T )^{7} \) | |
| 7 | \( 1 \) | |
| 17 | \( ( 1 - T )^{7} \) | |
| good | 3 | \( 1 - T + 7 T^{2} - 4 p T^{3} + 34 T^{4} - 56 T^{5} + 16 p^{2} T^{6} - 188 T^{7} + 16 p^{3} T^{8} - 56 p^{2} T^{9} + 34 p^{3} T^{10} - 4 p^{5} T^{11} + 7 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) |
| 11 | \( 1 + 36 T^{2} - 28 T^{3} + 753 T^{4} - 634 T^{5} + 11475 T^{6} - 8154 T^{7} + 11475 p T^{8} - 634 p^{2} T^{9} + 753 p^{3} T^{10} - 28 p^{4} T^{11} + 36 p^{5} T^{12} + p^{7} T^{14} \) | |
| 13 | \( 1 - 6 T + 68 T^{2} - 352 T^{3} + 2033 T^{4} - 9380 T^{5} + 37337 T^{6} - 151272 T^{7} + 37337 p T^{8} - 9380 p^{2} T^{9} + 2033 p^{3} T^{10} - 352 p^{4} T^{11} + 68 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 10 T + 78 T^{2} - 334 T^{3} + 820 T^{4} + 2866 T^{5} - 36757 T^{6} + 209484 T^{7} - 36757 p T^{8} + 2866 p^{2} T^{9} + 820 p^{3} T^{10} - 334 p^{4} T^{11} + 78 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + T + 100 T^{2} + 247 T^{3} + 4532 T^{4} + 17067 T^{5} + 133261 T^{6} + 23834 p T^{7} + 133261 p T^{8} + 17067 p^{2} T^{9} + 4532 p^{3} T^{10} + 247 p^{4} T^{11} + 100 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 9 T + 83 T^{2} + 708 T^{3} + 5341 T^{4} + 35595 T^{5} + 221247 T^{6} + 1159192 T^{7} + 221247 p T^{8} + 35595 p^{2} T^{9} + 5341 p^{3} T^{10} + 708 p^{4} T^{11} + 83 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 18 T + 291 T^{2} - 2996 T^{3} + 28199 T^{4} - 207054 T^{5} + 1425613 T^{6} - 8150728 T^{7} + 1425613 p T^{8} - 207054 p^{2} T^{9} + 28199 p^{3} T^{10} - 2996 p^{4} T^{11} + 291 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 8 T + 244 T^{2} - 1610 T^{3} + 26082 T^{4} - 140772 T^{5} + 1578945 T^{6} - 6817852 T^{7} + 1578945 p T^{8} - 140772 p^{2} T^{9} + 26082 p^{3} T^{10} - 1610 p^{4} T^{11} + 244 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 15 T + 252 T^{2} - 2297 T^{3} + 22474 T^{4} - 149997 T^{5} + 1147825 T^{6} - 6647062 T^{7} + 1147825 p T^{8} - 149997 p^{2} T^{9} + 22474 p^{3} T^{10} - 2297 p^{4} T^{11} + 252 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 9 T + 143 T^{2} + 722 T^{3} + 8071 T^{4} + 18099 T^{5} + 252521 T^{6} + 94660 T^{7} + 252521 p T^{8} + 18099 p^{2} T^{9} + 8071 p^{3} T^{10} + 722 p^{4} T^{11} + 143 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 10 T + 198 T^{2} - 1716 T^{3} + 19764 T^{4} - 151830 T^{5} + 1329055 T^{6} - 8692760 T^{7} + 1329055 p T^{8} - 151830 p^{2} T^{9} + 19764 p^{3} T^{10} - 1716 p^{4} T^{11} + 198 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 10 T + 132 T^{2} - 492 T^{3} + 6909 T^{4} - 29724 T^{5} + 664897 T^{6} - 3554996 T^{7} + 664897 p T^{8} - 29724 p^{2} T^{9} + 6909 p^{3} T^{10} - 492 p^{4} T^{11} + 132 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 6 T + 221 T^{2} + 1500 T^{3} + 25417 T^{4} + 186954 T^{5} + 2010885 T^{6} + 14056904 T^{7} + 2010885 p T^{8} + 186954 p^{2} T^{9} + 25417 p^{3} T^{10} + 1500 p^{4} T^{11} + 221 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 23 T + 423 T^{2} - 4940 T^{3} + 49705 T^{4} - 384505 T^{5} + 2922263 T^{6} - 20886744 T^{7} + 2922263 p T^{8} - 384505 p^{2} T^{9} + 49705 p^{3} T^{10} - 4940 p^{4} T^{11} + 423 p^{5} T^{12} - 23 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 13 T + 297 T^{2} - 3794 T^{3} + 48569 T^{4} - 512987 T^{5} + 4961929 T^{6} - 42815516 T^{7} + 4961929 p T^{8} - 512987 p^{2} T^{9} + 48569 p^{3} T^{10} - 3794 p^{4} T^{11} + 297 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 20 T + 581 T^{2} + 7612 T^{3} + 124476 T^{4} + 1205714 T^{5} + 14277028 T^{6} + 108708332 T^{7} + 14277028 p T^{8} + 1205714 p^{2} T^{9} + 124476 p^{3} T^{10} + 7612 p^{4} T^{11} + 581 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 22 T + 545 T^{2} - 7472 T^{3} + 109323 T^{4} - 1116346 T^{5} + 12252107 T^{6} - 100622752 T^{7} + 12252107 p T^{8} - 1116346 p^{2} T^{9} + 109323 p^{3} T^{10} - 7472 p^{4} T^{11} + 545 p^{5} T^{12} - 22 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 8 T + 283 T^{2} + 3094 T^{3} + 48150 T^{4} + 472612 T^{5} + 5823170 T^{6} + 44559736 T^{7} + 5823170 p T^{8} + 472612 p^{2} T^{9} + 48150 p^{3} T^{10} + 3094 p^{4} T^{11} + 283 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 3 T + 295 T^{2} - 1360 T^{3} + 49051 T^{4} - 246209 T^{5} + 5677653 T^{6} - 25060880 T^{7} + 5677653 p T^{8} - 246209 p^{2} T^{9} + 49051 p^{3} T^{10} - 1360 p^{4} T^{11} + 295 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 21 T + 361 T^{2} - 5024 T^{3} + 68388 T^{4} - 735144 T^{5} + 8037514 T^{6} - 76968690 T^{7} + 8037514 p T^{8} - 735144 p^{2} T^{9} + 68388 p^{3} T^{10} - 5024 p^{4} T^{11} + 361 p^{5} T^{12} - 21 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 28 T + 851 T^{2} - 15032 T^{3} + 267617 T^{4} - 3465188 T^{5} + 44444811 T^{6} - 440700560 T^{7} + 44444811 p T^{8} - 3465188 p^{2} T^{9} + 267617 p^{3} T^{10} - 15032 p^{4} T^{11} + 851 p^{5} T^{12} - 28 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.49406214162718500514754951563, −3.45996117691649647535407248196, −3.40389184658083201945999559909, −3.29701133891456062682308426930, −2.85707008640029671389878372166, −2.85113397726318174473717139119, −2.83952617904833647782119797605, −2.72802848166138935533874490981, −2.71956680609085223876885575767, −2.65650963661467564271238356649, −2.58864909714357481852868882326, −2.26387705693069143729583165993, −2.03611418374349098267822096136, −2.02496424430961586107701620333, −1.92288890175591333964954299163, −1.92170314625718849419566381351, −1.87516415860657704705067911302, −1.48747881299257218169856559492, −1.28578881530607944523284929191, −1.12634575808581476626584991234, −1.01868270455161250506509704467, −0.871621713487686043670445588525, −0.819115899544722775035350056498, −0.69296581011903057508094802349, −0.55395024770409649981180663370, 0.55395024770409649981180663370, 0.69296581011903057508094802349, 0.819115899544722775035350056498, 0.871621713487686043670445588525, 1.01868270455161250506509704467, 1.12634575808581476626584991234, 1.28578881530607944523284929191, 1.48747881299257218169856559492, 1.87516415860657704705067911302, 1.92170314625718849419566381351, 1.92288890175591333964954299163, 2.02496424430961586107701620333, 2.03611418374349098267822096136, 2.26387705693069143729583165993, 2.58864909714357481852868882326, 2.65650963661467564271238356649, 2.71956680609085223876885575767, 2.72802848166138935533874490981, 2.83952617904833647782119797605, 2.85113397726318174473717139119, 2.85707008640029671389878372166, 3.29701133891456062682308426930, 3.40389184658083201945999559909, 3.45996117691649647535407248196, 3.49406214162718500514754951563
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.