Dirichlet series
| L(s) = 1 | + 7·3-s + 7·5-s + 7-s + 28·9-s + 5·11-s + 9·13-s + 49·15-s + 11·17-s + 2·19-s + 7·21-s + 7·23-s + 28·25-s + 84·27-s + 8·29-s + 9·31-s + 35·33-s + 7·35-s + 7·37-s + 63·39-s + 9·41-s + 5·43-s + 196·45-s + 2·47-s − 18·49-s + 77·51-s + 15·53-s + 35·55-s + ⋯ |
| L(s) = 1 | + 4.04·3-s + 3.13·5-s + 0.377·7-s + 28/3·9-s + 1.50·11-s + 2.49·13-s + 12.6·15-s + 2.66·17-s + 0.458·19-s + 1.52·21-s + 1.45·23-s + 28/5·25-s + 16.1·27-s + 1.48·29-s + 1.61·31-s + 6.09·33-s + 1.18·35-s + 1.15·37-s + 10.0·39-s + 1.40·41-s + 0.762·43-s + 29.2·45-s + 0.291·47-s − 2.57·49-s + 10.7·51-s + 2.06·53-s + 4.71·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{7} \cdot 5^{7} \cdot 67^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{7} \cdot 5^{7} \cdot 67^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{14} \cdot 3^{7} \cdot 5^{7} \cdot 67^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.51173\times 10^{10}\) |
| Root analytic conductor: | \(5.66567\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{14} \cdot 3^{7} \cdot 5^{7} \cdot 67^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1389.908515\) |
| \(L(\frac12)\) | \(\approx\) | \(1389.908515\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{7} \) | |
| 5 | \( ( 1 - T )^{7} \) | |
| 67 | \( ( 1 + T )^{7} \) | |
| good | 7 | \( 1 - T + 19 T^{2} - 8 T^{3} + 128 T^{4} - 41 T^{5} + 498 T^{6} - 444 T^{7} + 498 p T^{8} - 41 p^{2} T^{9} + 128 p^{3} T^{10} - 8 p^{4} T^{11} + 19 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) |
| 11 | \( 1 - 5 T + 47 T^{2} - 124 T^{3} + 782 T^{4} - 1205 T^{5} + 8984 T^{6} - 10948 T^{7} + 8984 p T^{8} - 1205 p^{2} T^{9} + 782 p^{3} T^{10} - 124 p^{4} T^{11} + 47 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 9 T + 82 T^{2} - 501 T^{3} + 2853 T^{4} - 13419 T^{5} + 57830 T^{6} - 218514 T^{7} + 57830 p T^{8} - 13419 p^{2} T^{9} + 2853 p^{3} T^{10} - 501 p^{4} T^{11} + 82 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 11 T + 131 T^{2} - 919 T^{3} + 6497 T^{4} - 34076 T^{5} + 177839 T^{6} - 735646 T^{7} + 177839 p T^{8} - 34076 p^{2} T^{9} + 6497 p^{3} T^{10} - 919 p^{4} T^{11} + 131 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 2 T + 73 T^{2} - 22 T^{3} + 2165 T^{4} + 3821 T^{5} + 39189 T^{6} + 134574 T^{7} + 39189 p T^{8} + 3821 p^{2} T^{9} + 2165 p^{3} T^{10} - 22 p^{4} T^{11} + 73 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 7 T + 119 T^{2} - 704 T^{3} + 6512 T^{4} - 32704 T^{5} + 219182 T^{6} - 929666 T^{7} + 219182 p T^{8} - 32704 p^{2} T^{9} + 6512 p^{3} T^{10} - 704 p^{4} T^{11} + 119 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 8 T + 173 T^{2} - 1024 T^{3} + 12881 T^{4} - 60359 T^{5} + 566477 T^{6} - 2164546 T^{7} + 566477 p T^{8} - 60359 p^{2} T^{9} + 12881 p^{3} T^{10} - 1024 p^{4} T^{11} + 173 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 9 T + 157 T^{2} - 684 T^{3} + 6900 T^{4} - 2043 T^{5} + 96962 T^{6} + 651768 T^{7} + 96962 p T^{8} - 2043 p^{2} T^{9} + 6900 p^{3} T^{10} - 684 p^{4} T^{11} + 157 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 7 T + 97 T^{2} - 224 T^{3} + 3260 T^{4} + 3352 T^{5} + 85122 T^{6} + 362766 T^{7} + 85122 p T^{8} + 3352 p^{2} T^{9} + 3260 p^{3} T^{10} - 224 p^{4} T^{11} + 97 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 9 T + 119 T^{2} - 696 T^{3} + 6948 T^{4} - 31269 T^{5} + 313624 T^{6} - 1370892 T^{7} + 313624 p T^{8} - 31269 p^{2} T^{9} + 6948 p^{3} T^{10} - 696 p^{4} T^{11} + 119 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 5 T + 151 T^{2} - 622 T^{3} + 10436 T^{4} - 32797 T^{5} + 484086 T^{6} - 1283088 T^{7} + 484086 p T^{8} - 32797 p^{2} T^{9} + 10436 p^{3} T^{10} - 622 p^{4} T^{11} + 151 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 2 T + 203 T^{2} - 856 T^{3} + 18173 T^{4} - 117683 T^{5} + 1046795 T^{6} - 7697422 T^{7} + 1046795 p T^{8} - 117683 p^{2} T^{9} + 18173 p^{3} T^{10} - 856 p^{4} T^{11} + 203 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 15 T + 374 T^{2} - 75 p T^{3} + 57039 T^{4} - 470469 T^{5} + 4901962 T^{6} - 32054298 T^{7} + 4901962 p T^{8} - 470469 p^{2} T^{9} + 57039 p^{3} T^{10} - 75 p^{5} T^{11} + 374 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 197 T^{2} + 108 T^{3} + 22893 T^{4} + 15603 T^{5} + 1834513 T^{6} + 1268472 T^{7} + 1834513 p T^{8} + 15603 p^{2} T^{9} + 22893 p^{3} T^{10} + 108 p^{4} T^{11} + 197 p^{5} T^{12} + p^{7} T^{14} \) | |
| 61 | \( 1 - 11 T + 214 T^{2} - 883 T^{3} + 12641 T^{4} + 24563 T^{5} + 347856 T^{6} + 4536294 T^{7} + 347856 p T^{8} + 24563 p^{2} T^{9} + 12641 p^{3} T^{10} - 883 p^{4} T^{11} + 214 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 18 T + 527 T^{2} - 6351 T^{3} + 105570 T^{4} - 953004 T^{5} + 11665102 T^{6} - 84162318 T^{7} + 11665102 p T^{8} - 953004 p^{2} T^{9} + 105570 p^{3} T^{10} - 6351 p^{4} T^{11} + 527 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 21 T + 7 p T^{2} - 6561 T^{3} + 92751 T^{4} - 882216 T^{5} + 9543917 T^{6} - 75489564 T^{7} + 9543917 p T^{8} - 882216 p^{2} T^{9} + 92751 p^{3} T^{10} - 6561 p^{4} T^{11} + 7 p^{6} T^{12} - 21 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 5 T + 385 T^{2} - 628 T^{3} + 60698 T^{4} + 73721 T^{5} + 5848464 T^{6} + 14581032 T^{7} + 5848464 p T^{8} + 73721 p^{2} T^{9} + 60698 p^{3} T^{10} - 628 p^{4} T^{11} + 385 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 6 T + 431 T^{2} - 2205 T^{3} + 82446 T^{4} - 361914 T^{5} + 9684928 T^{6} - 36558774 T^{7} + 9684928 p T^{8} - 361914 p^{2} T^{9} + 82446 p^{3} T^{10} - 2205 p^{4} T^{11} + 431 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 7 T + 365 T^{2} - 1685 T^{3} + 61205 T^{4} - 158422 T^{5} + 6655073 T^{6} - 11367164 T^{7} + 6655073 p T^{8} - 158422 p^{2} T^{9} + 61205 p^{3} T^{10} - 1685 p^{4} T^{11} + 365 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 9 T + 394 T^{2} + 3615 T^{3} + 86565 T^{4} + 702723 T^{5} + 12173822 T^{6} + 85552134 T^{7} + 12173822 p T^{8} + 702723 p^{2} T^{9} + 86565 p^{3} T^{10} + 3615 p^{4} T^{11} + 394 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.76823816926161184924745730374, −3.66522395301175037066704208289, −3.59000046800083051986132785353, −3.57036480508435668868599142864, −3.23014573216680997480781041233, −3.01146049298271198026223079526, −2.96412321183185493433033126292, −2.90577685248693583417071702984, −2.85961979279554001326932442421, −2.84947636762531458057586867505, −2.76080747749366082121384861344, −2.31344606191308829215848706860, −2.14237067573081077624282783612, −2.08236890162550031217679510828, −2.01822772513790315274726660076, −1.93332321155591937100575075267, −1.85669906538231883078903290778, −1.78172177507753736377305033964, −1.20829014664377172560016926980, −1.15224035206312284303481680784, −1.07069515272113199371367906169, −1.01479507601889089896201613233, −1.01233301282380661302842048901, −0.822889663965966532829509629722, −0.75502098428600611087460402646, 0.75502098428600611087460402646, 0.822889663965966532829509629722, 1.01233301282380661302842048901, 1.01479507601889089896201613233, 1.07069515272113199371367906169, 1.15224035206312284303481680784, 1.20829014664377172560016926980, 1.78172177507753736377305033964, 1.85669906538231883078903290778, 1.93332321155591937100575075267, 2.01822772513790315274726660076, 2.08236890162550031217679510828, 2.14237067573081077624282783612, 2.31344606191308829215848706860, 2.76080747749366082121384861344, 2.84947636762531458057586867505, 2.85961979279554001326932442421, 2.90577685248693583417071702984, 2.96412321183185493433033126292, 3.01146049298271198026223079526, 3.23014573216680997480781041233, 3.57036480508435668868599142864, 3.59000046800083051986132785353, 3.66522395301175037066704208289, 3.76823816926161184924745730374
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.