Dirichlet series
| L(s) = 1 | − 4·2-s − 8·3-s + 5·4-s + 7·5-s + 32·6-s − 16·7-s + 27·9-s − 28·10-s − 10·11-s − 40·12-s − 7·13-s + 64·14-s − 56·15-s − 5·16-s − 13·17-s − 108·18-s − 7·19-s + 35·20-s + 128·21-s + 40·22-s − 13·23-s + 28·25-s + 28·26-s − 45·27-s − 80·28-s − 4·29-s + 224·30-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 4.61·3-s + 5/2·4-s + 3.13·5-s + 13.0·6-s − 6.04·7-s + 9·9-s − 8.85·10-s − 3.01·11-s − 11.5·12-s − 1.94·13-s + 17.1·14-s − 14.4·15-s − 5/4·16-s − 3.15·17-s − 25.4·18-s − 1.60·19-s + 7.82·20-s + 27.9·21-s + 8.52·22-s − 2.71·23-s + 28/5·25-s + 5.49·26-s − 8.66·27-s − 15.1·28-s − 0.742·29-s + 40.8·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{7} \cdot 89^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{7} \cdot 89^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(5^{7} \cdot 89^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(7152.53\) |
| Root analytic conductor: | \(1.88503\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 5^{7} \cdot 89^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 - T )^{7} \) |
| 89 | \( ( 1 - T )^{7} \) | |
| good | 2 | \( 1 + p^{2} T + 11 T^{2} + 3 p^{3} T^{3} + 23 p T^{4} + 77 T^{5} + 59 p T^{6} + 171 T^{7} + 59 p^{2} T^{8} + 77 p^{2} T^{9} + 23 p^{4} T^{10} + 3 p^{7} T^{11} + 11 p^{5} T^{12} + p^{8} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 + 8 T + 37 T^{2} + 125 T^{3} + 340 T^{4} + 260 p T^{5} + 1576 T^{6} + 2866 T^{7} + 1576 p T^{8} + 260 p^{3} T^{9} + 340 p^{3} T^{10} + 125 p^{4} T^{11} + 37 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 + 16 T + 143 T^{2} + 908 T^{3} + 92 p^{2} T^{4} + 18272 T^{5} + 61958 T^{6} + 177804 T^{7} + 61958 p T^{8} + 18272 p^{2} T^{9} + 92 p^{5} T^{10} + 908 p^{4} T^{11} + 143 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 10 T + 91 T^{2} + 511 T^{3} + 2672 T^{4} + 966 p T^{5} + 41830 T^{6} + 136602 T^{7} + 41830 p T^{8} + 966 p^{3} T^{9} + 2672 p^{3} T^{10} + 511 p^{4} T^{11} + 91 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 7 T + 47 T^{2} + 168 T^{3} + 622 T^{4} + 2167 T^{5} + 9118 T^{6} + 36132 T^{7} + 9118 p T^{8} + 2167 p^{2} T^{9} + 622 p^{3} T^{10} + 168 p^{4} T^{11} + 47 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 13 T + 108 T^{2} + 591 T^{3} + 3017 T^{4} + 14845 T^{5} + 78122 T^{6} + 344702 T^{7} + 78122 p T^{8} + 14845 p^{2} T^{9} + 3017 p^{3} T^{10} + 591 p^{4} T^{11} + 108 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 7 T + 81 T^{2} + 366 T^{3} + 2784 T^{4} + 9257 T^{5} + 61132 T^{6} + 176376 T^{7} + 61132 p T^{8} + 9257 p^{2} T^{9} + 2784 p^{3} T^{10} + 366 p^{4} T^{11} + 81 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 13 T + 144 T^{2} + 1105 T^{3} + 7943 T^{4} + 47855 T^{5} + 274740 T^{6} + 1359354 T^{7} + 274740 p T^{8} + 47855 p^{2} T^{9} + 7943 p^{3} T^{10} + 1105 p^{4} T^{11} + 144 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 4 T + 113 T^{2} + 625 T^{3} + 6740 T^{4} + 40902 T^{5} + 266814 T^{6} + 1538922 T^{7} + 266814 p T^{8} + 40902 p^{2} T^{9} + 6740 p^{3} T^{10} + 625 p^{4} T^{11} + 113 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - T + 77 T^{2} - 34 T^{3} + 3570 T^{4} - 2367 T^{5} + 134894 T^{6} - 138904 T^{7} + 134894 p T^{8} - 2367 p^{2} T^{9} + 3570 p^{3} T^{10} - 34 p^{4} T^{11} + 77 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 5 T + 168 T^{2} + 803 T^{3} + 14161 T^{4} + 59989 T^{5} + 771494 T^{6} + 2748646 T^{7} + 771494 p T^{8} + 59989 p^{2} T^{9} + 14161 p^{3} T^{10} + 803 p^{4} T^{11} + 168 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 5 T + 92 T^{2} - 673 T^{3} + 7237 T^{4} - 46413 T^{5} + 387726 T^{6} - 2290138 T^{7} + 387726 p T^{8} - 46413 p^{2} T^{9} + 7237 p^{3} T^{10} - 673 p^{4} T^{11} + 92 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 31 T + 687 T^{2} + 10436 T^{3} + 129854 T^{4} + 1291529 T^{5} + 10962460 T^{6} + 77409324 T^{7} + 10962460 p T^{8} + 1291529 p^{2} T^{9} + 129854 p^{3} T^{10} + 10436 p^{4} T^{11} + 687 p^{5} T^{12} + 31 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 14 T + 271 T^{2} + 2621 T^{3} + 31508 T^{4} + 240678 T^{5} + 2207362 T^{6} + 13822254 T^{7} + 2207362 p T^{8} + 240678 p^{2} T^{9} + 31508 p^{3} T^{10} + 2621 p^{4} T^{11} + 271 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 13 T + 269 T^{2} + 2568 T^{3} + 34060 T^{4} + 265121 T^{5} + 2673394 T^{6} + 17174660 T^{7} + 2673394 p T^{8} + 265121 p^{2} T^{9} + 34060 p^{3} T^{10} + 2568 p^{4} T^{11} + 269 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 14 T + 253 T^{2} + 2446 T^{3} + 31472 T^{4} + 260310 T^{5} + 2563464 T^{6} + 17624256 T^{7} + 2563464 p T^{8} + 260310 p^{2} T^{9} + 31472 p^{3} T^{10} + 2446 p^{4} T^{11} + 253 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 3 T + 222 T^{2} - 759 T^{3} + 29403 T^{4} - 83651 T^{5} + 2524030 T^{6} - 6527094 T^{7} + 2524030 p T^{8} - 83651 p^{2} T^{9} + 29403 p^{3} T^{10} - 759 p^{4} T^{11} + 222 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - T + 328 T^{2} - 279 T^{3} + 53091 T^{4} - 39287 T^{5} + 5339384 T^{6} - 3265170 T^{7} + 5339384 p T^{8} - 39287 p^{2} T^{9} + 53091 p^{3} T^{10} - 279 p^{4} T^{11} + 328 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 8 T + 187 T^{2} + 1328 T^{3} + 22776 T^{4} + 168048 T^{5} + 2047314 T^{6} + 13028752 T^{7} + 2047314 p T^{8} + 168048 p^{2} T^{9} + 22776 p^{3} T^{10} + 1328 p^{4} T^{11} + 187 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 9 T + 286 T^{2} - 1613 T^{3} + 37567 T^{4} - 166989 T^{5} + 3689378 T^{6} - 14466938 T^{7} + 3689378 p T^{8} - 166989 p^{2} T^{9} + 37567 p^{3} T^{10} - 1613 p^{4} T^{11} + 286 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 9 T + 430 T^{2} - 2867 T^{3} + 81347 T^{4} - 413191 T^{5} + 9305818 T^{6} - 38366618 T^{7} + 9305818 p T^{8} - 413191 p^{2} T^{9} + 81347 p^{3} T^{10} - 2867 p^{4} T^{11} + 430 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 42 T + 1008 T^{2} + 16390 T^{3} + 2485 p T^{4} + 2129142 T^{5} + 19880600 T^{6} + 179268272 T^{7} + 19880600 p T^{8} + 2129142 p^{2} T^{9} + 2485 p^{4} T^{10} + 16390 p^{4} T^{11} + 1008 p^{5} T^{12} + 42 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 7 T + 609 T^{2} + 3724 T^{3} + 164856 T^{4} + 856227 T^{5} + 25710150 T^{6} + 108804964 T^{7} + 25710150 p T^{8} + 856227 p^{2} T^{9} + 164856 p^{3} T^{10} + 3724 p^{4} T^{11} + 609 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−6.04179844967899344415925700148, −6.01957630914516778122045869529, −5.76538843896948282923103507555, −5.65340940362821267541754816365, −5.44832777227399333657953202112, −5.25319731601321216044861769518, −5.03565720502203011302796914355, −5.02484126305243650496672456040, −4.89880680809120967132129460765, −4.85190281950692521821296613566, −4.74642929351272300982901740042, −4.16220540712002498661950244258, −4.09106058440639987689007269483, −3.86899877953181198275381479500, −3.56544450618704653806895186022, −3.45587790298020185856735092222, −2.98832092242241861645588382726, −2.93780312850244441451367290352, −2.64264400794348778617203117093, −2.63148264583210902853824114277, −2.47212589396977697230581201243, −2.38440209038555509919050674647, −1.83296779380478863816478260892, −1.59911430237976937126415057322, −1.38471060990701707155474720672, 0, 0, 0, 0, 0, 0, 0, 1.38471060990701707155474720672, 1.59911430237976937126415057322, 1.83296779380478863816478260892, 2.38440209038555509919050674647, 2.47212589396977697230581201243, 2.63148264583210902853824114277, 2.64264400794348778617203117093, 2.93780312850244441451367290352, 2.98832092242241861645588382726, 3.45587790298020185856735092222, 3.56544450618704653806895186022, 3.86899877953181198275381479500, 4.09106058440639987689007269483, 4.16220540712002498661950244258, 4.74642929351272300982901740042, 4.85190281950692521821296613566, 4.89880680809120967132129460765, 5.02484126305243650496672456040, 5.03565720502203011302796914355, 5.25319731601321216044861769518, 5.44832777227399333657953202112, 5.65340940362821267541754816365, 5.76538843896948282923103507555, 6.01957630914516778122045869529, 6.04179844967899344415925700148
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.