Dirichlet series
| L(s) = 1 | + 2-s + 3·3-s − 3·4-s + 7·5-s + 3·6-s + 7-s − 4·8-s − 3·9-s + 7·10-s + 2·11-s − 9·12-s + 20·13-s + 14-s + 21·15-s + 16-s + 7·17-s − 3·18-s − 7·19-s − 21·20-s + 3·21-s + 2·22-s − 3·23-s − 12·24-s + 10·25-s + 20·26-s − 20·27-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 − 1.41·8-s − 9-s + 2.21·10-s + 0.603·11-s − 2.59·12-s + 5.54·13-s + 0.267·14-s + 5.42·15-s + 1/4·16-s + 1.69·17-s − 0.707·18-s − 1.60·19-s − 4.69·20-s + 0.654·21-s + 0.426·22-s − 0.625·23-s − 2.44·24-s + 2·25-s + 3.92·26-s − 3.84·27-s − 0.566·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{7} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{7} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(17^{7} \cdot 19^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(759.202\) |
| Root analytic conductor: | \(1.60597\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 17^{7} \cdot 19^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(14.40870077\) |
| \(L(\frac12)\) | \(\approx\) | \(14.40870077\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 17 | \( ( 1 - T )^{7} \) |
| 19 | \( ( 1 + T )^{7} \) | |
| good | 2 | \( 1 - T + p^{2} T^{2} - 3 T^{3} + 5 p T^{4} - 7 T^{5} + 3 p^{3} T^{6} - 3 p^{2} T^{7} + 3 p^{4} T^{8} - 7 p^{2} T^{9} + 5 p^{4} T^{10} - 3 p^{4} T^{11} + p^{7} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 - p T + 4 p T^{2} - 25 T^{3} + 71 T^{4} - 125 T^{5} + 295 T^{6} - 454 T^{7} + 295 p T^{8} - 125 p^{2} T^{9} + 71 p^{3} T^{10} - 25 p^{4} T^{11} + 4 p^{6} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) | |
| 5 | \( 1 - 7 T + 39 T^{2} - 156 T^{3} + 552 T^{4} - 327 p T^{5} + 4404 T^{6} - 10308 T^{7} + 4404 p T^{8} - 327 p^{3} T^{9} + 552 p^{3} T^{10} - 156 p^{4} T^{11} + 39 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 - T + 22 T^{2} - 13 T^{3} + 311 T^{4} - 179 T^{5} + 2974 T^{6} - 1310 T^{7} + 2974 p T^{8} - 179 p^{2} T^{9} + 311 p^{3} T^{10} - 13 p^{4} T^{11} + 22 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 2 T + 31 T^{2} - 125 T^{3} + 570 T^{4} - 2730 T^{5} + 8856 T^{6} - 35422 T^{7} + 8856 p T^{8} - 2730 p^{2} T^{9} + 570 p^{3} T^{10} - 125 p^{4} T^{11} + 31 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 20 T + 232 T^{2} - 1902 T^{3} + 12321 T^{4} - 65750 T^{5} + 297982 T^{6} - 1156848 T^{7} + 297982 p T^{8} - 65750 p^{2} T^{9} + 12321 p^{3} T^{10} - 1902 p^{4} T^{11} + 232 p^{5} T^{12} - 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 3 T + 77 T^{2} + 328 T^{3} + 3018 T^{4} + 15905 T^{5} + 81986 T^{6} + 464840 T^{7} + 81986 p T^{8} + 15905 p^{2} T^{9} + 3018 p^{3} T^{10} + 328 p^{4} T^{11} + 77 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 10 T + 223 T^{2} - 1638 T^{3} + 20035 T^{4} - 113484 T^{5} + 975616 T^{6} - 4314746 T^{7} + 975616 p T^{8} - 113484 p^{2} T^{9} + 20035 p^{3} T^{10} - 1638 p^{4} T^{11} + 223 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 21 T + 313 T^{2} - 3358 T^{3} + 30961 T^{4} - 237834 T^{5} + 1621826 T^{6} - 9530866 T^{7} + 1621826 p T^{8} - 237834 p^{2} T^{9} + 30961 p^{3} T^{10} - 3358 p^{4} T^{11} + 313 p^{5} T^{12} - 21 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 21 T + 348 T^{2} - 3987 T^{3} + 39329 T^{4} - 322397 T^{5} + 2361358 T^{6} - 15122894 T^{7} + 2361358 p T^{8} - 322397 p^{2} T^{9} + 39329 p^{3} T^{10} - 3987 p^{4} T^{11} + 348 p^{5} T^{12} - 21 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 11 T + 138 T^{2} - 1097 T^{3} + 11259 T^{4} - 82435 T^{5} + 629750 T^{6} - 3732010 T^{7} + 629750 p T^{8} - 82435 p^{2} T^{9} + 11259 p^{3} T^{10} - 1097 p^{4} T^{11} + 138 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 3 T + 161 T^{2} - 648 T^{3} + 12187 T^{4} - 67254 T^{5} + 634886 T^{6} - 3857906 T^{7} + 634886 p T^{8} - 67254 p^{2} T^{9} + 12187 p^{3} T^{10} - 648 p^{4} T^{11} + 161 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 13 T + 301 T^{2} + 2940 T^{3} + 39583 T^{4} + 305102 T^{5} + 2988258 T^{6} + 18360402 T^{7} + 2988258 p T^{8} + 305102 p^{2} T^{9} + 39583 p^{3} T^{10} + 2940 p^{4} T^{11} + 301 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 15 T + 304 T^{2} + 3213 T^{3} + 39447 T^{4} + 326419 T^{5} + 3069312 T^{6} + 21005586 T^{7} + 3069312 p T^{8} + 326419 p^{2} T^{9} + 39447 p^{3} T^{10} + 3213 p^{4} T^{11} + 304 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 6 T + 251 T^{2} + 925 T^{3} + 28302 T^{4} + 53198 T^{5} + 2051024 T^{6} + 2286782 T^{7} + 2051024 p T^{8} + 53198 p^{2} T^{9} + 28302 p^{3} T^{10} + 925 p^{4} T^{11} + 251 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 15 T + 230 T^{2} - 2493 T^{3} + 28029 T^{4} - 252859 T^{5} + 2411732 T^{6} - 18647890 T^{7} + 2411732 p T^{8} - 252859 p^{2} T^{9} + 28029 p^{3} T^{10} - 2493 p^{4} T^{11} + 230 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 21 T + 358 T^{2} - 3899 T^{3} + 40297 T^{4} - 339267 T^{5} + 3062852 T^{6} - 24066794 T^{7} + 3062852 p T^{8} - 339267 p^{2} T^{9} + 40297 p^{3} T^{10} - 3899 p^{4} T^{11} + 358 p^{5} T^{12} - 21 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 9 T + 445 T^{2} + 3336 T^{3} + 87892 T^{4} + 545123 T^{5} + 10018048 T^{6} + 50218312 T^{7} + 10018048 p T^{8} + 545123 p^{2} T^{9} + 87892 p^{3} T^{10} + 3336 p^{4} T^{11} + 445 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 23 T + 415 T^{2} - 4532 T^{3} + 48602 T^{4} - 373171 T^{5} + 3280150 T^{6} - 23044500 T^{7} + 3280150 p T^{8} - 373171 p^{2} T^{9} + 48602 p^{3} T^{10} - 4532 p^{4} T^{11} + 415 p^{5} T^{12} - 23 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 25 T + 501 T^{2} + 6128 T^{3} + 70600 T^{4} + 637995 T^{5} + 6345156 T^{6} + 53434664 T^{7} + 6345156 p T^{8} + 637995 p^{2} T^{9} + 70600 p^{3} T^{10} + 6128 p^{4} T^{11} + 501 p^{5} T^{12} + 25 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 14 T + 407 T^{2} + 4450 T^{3} + 68873 T^{4} + 645956 T^{5} + 7253584 T^{6} + 62016844 T^{7} + 7253584 p T^{8} + 645956 p^{2} T^{9} + 68873 p^{3} T^{10} + 4450 p^{4} T^{11} + 407 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 17 T + 595 T^{2} + 7732 T^{3} + 152454 T^{4} + 1566021 T^{5} + 22098310 T^{6} + 180097820 T^{7} + 22098310 p T^{8} + 1566021 p^{2} T^{9} + 152454 p^{3} T^{10} + 7732 p^{4} T^{11} + 595 p^{5} T^{12} + 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 27 T + 564 T^{2} - 7109 T^{3} + 71853 T^{4} - 365667 T^{5} + 791113 T^{6} + 13550416 T^{7} + 791113 p T^{8} - 365667 p^{2} T^{9} + 71853 p^{3} T^{10} - 7109 p^{4} T^{11} + 564 p^{5} T^{12} - 27 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−5.78851025162154605257910375673, −5.70747202619034653414693428547, −5.47204398531710419123799277303, −5.28843341142705050822729444311, −4.93515814222493605554397807962, −4.89970937610852446352712496890, −4.68169301000484973839336020941, −4.44245358208540390875392524766, −4.28616655393729286265449800379, −4.13192865388636432238053451282, −3.98140354271025277025342487271, −3.92735724961411583325493942985, −3.39976974111471357821582548320, −3.39905975746586249345932343462, −3.27786832014925784606546104227, −3.17148579333967928318265859029, −2.76613967054234989471837492521, −2.56952266006570223128664125046, −2.47398245171048799158223626287, −2.32053917969984827176938895830, −1.82659392169528452301009649698, −1.56958359979448346997332811603, −1.47416222141666986275783048351, −1.02270515125810914990495121873, −0.905272130326029169237146707088, 0.905272130326029169237146707088, 1.02270515125810914990495121873, 1.47416222141666986275783048351, 1.56958359979448346997332811603, 1.82659392169528452301009649698, 2.32053917969984827176938895830, 2.47398245171048799158223626287, 2.56952266006570223128664125046, 2.76613967054234989471837492521, 3.17148579333967928318265859029, 3.27786832014925784606546104227, 3.39905975746586249345932343462, 3.39976974111471357821582548320, 3.92735724961411583325493942985, 3.98140354271025277025342487271, 4.13192865388636432238053451282, 4.28616655393729286265449800379, 4.44245358208540390875392524766, 4.68169301000484973839336020941, 4.89970937610852446352712496890, 4.93515814222493605554397807962, 5.28843341142705050822729444311, 5.47204398531710419123799277303, 5.70747202619034653414693428547, 5.78851025162154605257910375673
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.