Dirichlet series
| L(s) = 1 | + 7·2-s + 28·4-s + 7·5-s + 5·7-s + 84·8-s + 49·10-s + 7·11-s + 5·13-s + 35·14-s + 210·16-s + 3·17-s + 11·19-s + 196·20-s + 49·22-s + 3·23-s + 28·25-s + 35·26-s + 140·28-s + 12·29-s + 5·31-s + 462·32-s + 21·34-s + 35·35-s + 2·37-s + 77·38-s + 588·40-s + 6·41-s + ⋯ |
| L(s) = 1 | + 4.94·2-s + 14·4-s + 3.13·5-s + 1.88·7-s + 29.6·8-s + 15.4·10-s + 2.11·11-s + 1.38·13-s + 9.35·14-s + 52.5·16-s + 0.727·17-s + 2.52·19-s + 43.8·20-s + 10.4·22-s + 0.625·23-s + 28/5·25-s + 6.86·26-s + 26.4·28-s + 2.22·29-s + 0.898·31-s + 81.6·32-s + 3.60·34-s + 5.91·35-s + 0.328·37-s + 12.4·38-s + 92.9·40-s + 0.937·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{28} \cdot 5^{7} \cdot 11^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{28} \cdot 5^{7} \cdot 11^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{7} \cdot 3^{28} \cdot 5^{7} \cdot 11^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.22749\times 10^{12}\) |
| Root analytic conductor: | \(8.43485\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{7} \cdot 3^{28} \cdot 5^{7} \cdot 11^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(21605.38666\) |
| \(L(\frac12)\) | \(\approx\) | \(21605.38666\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{7} \) |
| 3 | \( 1 \) | |
| 5 | \( ( 1 - T )^{7} \) | |
| 11 | \( ( 1 - T )^{7} \) | |
| good | 7 | \( 1 - 5 T + 31 T^{2} - 96 T^{3} + 318 T^{4} - 624 T^{5} + 1529 T^{6} - 2731 T^{7} + 1529 p T^{8} - 624 p^{2} T^{9} + 318 p^{3} T^{10} - 96 p^{4} T^{11} + 31 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) |
| 13 | \( 1 - 5 T + 49 T^{2} - 105 T^{3} + 765 T^{4} - 327 T^{5} + 8777 T^{6} + 2426 T^{7} + 8777 p T^{8} - 327 p^{2} T^{9} + 765 p^{3} T^{10} - 105 p^{4} T^{11} + 49 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 3 T + 50 T^{2} - 157 T^{3} + 1233 T^{4} - 4652 T^{5} + 24641 T^{6} - 96876 T^{7} + 24641 p T^{8} - 4652 p^{2} T^{9} + 1233 p^{3} T^{10} - 157 p^{4} T^{11} + 50 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 11 T + 106 T^{2} - 595 T^{3} + 3545 T^{4} - 16960 T^{5} + 94155 T^{6} - 405966 T^{7} + 94155 p T^{8} - 16960 p^{2} T^{9} + 3545 p^{3} T^{10} - 595 p^{4} T^{11} + 106 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 3 T + 104 T^{2} - 354 T^{3} + 5601 T^{4} - 17655 T^{5} + 193102 T^{6} - 512544 T^{7} + 193102 p T^{8} - 17655 p^{2} T^{9} + 5601 p^{3} T^{10} - 354 p^{4} T^{11} + 104 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 12 T + 191 T^{2} - 1684 T^{3} + 15471 T^{4} - 104990 T^{5} + 714167 T^{6} - 3842349 T^{7} + 714167 p T^{8} - 104990 p^{2} T^{9} + 15471 p^{3} T^{10} - 1684 p^{4} T^{11} + 191 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 5 T + 64 T^{2} - 423 T^{3} + 4203 T^{4} - 21966 T^{5} + 5249 p T^{6} - 841990 T^{7} + 5249 p^{2} T^{8} - 21966 p^{2} T^{9} + 4203 p^{3} T^{10} - 423 p^{4} T^{11} + 64 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 2 T + 109 T^{2} - 402 T^{3} + 6690 T^{4} - 25986 T^{5} + 333659 T^{6} - 1020256 T^{7} + 333659 p T^{8} - 25986 p^{2} T^{9} + 6690 p^{3} T^{10} - 402 p^{4} T^{11} + 109 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 6 T + 200 T^{2} - 937 T^{3} + 17907 T^{4} - 67328 T^{5} + 1003736 T^{6} - 3187938 T^{7} + 1003736 p T^{8} - 67328 p^{2} T^{9} + 17907 p^{3} T^{10} - 937 p^{4} T^{11} + 200 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 17 T + 241 T^{2} - 1865 T^{3} + 15967 T^{4} - 104027 T^{5} + 931503 T^{6} - 5765814 T^{7} + 931503 p T^{8} - 104027 p^{2} T^{9} + 15967 p^{3} T^{10} - 1865 p^{4} T^{11} + 241 p^{5} T^{12} - 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 3 T + 230 T^{2} + 472 T^{3} + 25191 T^{4} + 39725 T^{5} + 1753622 T^{6} + 2278896 T^{7} + 1753622 p T^{8} + 39725 p^{2} T^{9} + 25191 p^{3} T^{10} + 472 p^{4} T^{11} + 230 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 3 T + 194 T^{2} - 631 T^{3} + 20109 T^{4} - 61886 T^{5} + 1459631 T^{6} - 3886476 T^{7} + 1459631 p T^{8} - 61886 p^{2} T^{9} + 20109 p^{3} T^{10} - 631 p^{4} T^{11} + 194 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 18 T + 170 T^{2} - 1381 T^{3} + 9570 T^{4} - 51878 T^{5} + 116765 T^{6} + 680010 T^{7} + 116765 p T^{8} - 51878 p^{2} T^{9} + 9570 p^{3} T^{10} - 1381 p^{4} T^{11} + 170 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 11 T + 97 T^{2} - 578 T^{3} + 4786 T^{4} - 12974 T^{5} + 243471 T^{6} - 2292741 T^{7} + 243471 p T^{8} - 12974 p^{2} T^{9} + 4786 p^{3} T^{10} - 578 p^{4} T^{11} + 97 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 20 T + 124 T^{2} - 337 T^{3} + 251 p T^{4} - 268078 T^{5} + 1383039 T^{6} - 3014943 T^{7} + 1383039 p T^{8} - 268078 p^{2} T^{9} + 251 p^{4} T^{10} - 337 p^{4} T^{11} + 124 p^{5} T^{12} - 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 12 T + 356 T^{2} - 3095 T^{3} + 56580 T^{4} - 411844 T^{5} + 5840825 T^{6} - 36031986 T^{7} + 5840825 p T^{8} - 411844 p^{2} T^{9} + 56580 p^{3} T^{10} - 3095 p^{4} T^{11} + 356 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 2 T + 151 T^{2} + 197 T^{3} + 7427 T^{4} + 7151 T^{5} + 432006 T^{6} - 2588760 T^{7} + 432006 p T^{8} + 7151 p^{2} T^{9} + 7427 p^{3} T^{10} + 197 p^{4} T^{11} + 151 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 7 T + 415 T^{2} + 2527 T^{3} + 80893 T^{4} + 421381 T^{5} + 9638439 T^{6} + 41871786 T^{7} + 9638439 p T^{8} + 421381 p^{2} T^{9} + 80893 p^{3} T^{10} + 2527 p^{4} T^{11} + 415 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 362 T^{2} + 165 T^{3} + 58809 T^{4} + 58884 T^{5} + 6165520 T^{6} + 7492950 T^{7} + 6165520 p T^{8} + 58884 p^{2} T^{9} + 58809 p^{3} T^{10} + 165 p^{4} T^{11} + 362 p^{5} T^{12} + p^{7} T^{14} \) | |
| 89 | \( 1 + 3 T + 308 T^{2} + 2412 T^{3} + 51951 T^{4} + 454617 T^{5} + 7022404 T^{6} + 46713288 T^{7} + 7022404 p T^{8} + 454617 p^{2} T^{9} + 51951 p^{3} T^{10} + 2412 p^{4} T^{11} + 308 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 23 T + 607 T^{2} - 9915 T^{3} + 153441 T^{4} - 2013465 T^{5} + 23102855 T^{6} - 246842410 T^{7} + 23102855 p T^{8} - 2013465 p^{2} T^{9} + 153441 p^{3} T^{10} - 9915 p^{4} T^{11} + 607 p^{5} T^{12} - 23 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.52907780358046293334351675130, −3.46595175865199591619964725349, −3.45214977740080640189903609620, −3.33345865752333029670284210385, −2.84900487950041146465934606982, −2.82951440979272729273411192482, −2.81165260933788586477914805894, −2.75599214140743410581885709633, −2.65152416283174355763166740837, −2.58684170456655719916861941247, −2.48784580512635751140617313660, −2.02352494478604353826077693443, −1.94122285575140517555111212240, −1.92352727465941171635617744077, −1.88916629755480280784572217064, −1.85785313867962415033426808120, −1.71791578167033221688262349404, −1.70700076394549643847228557107, −1.08633382600023838607247204996, −1.05668137257888996879577313102, −1.03072153846747291921214618087, −1.02442238922405198054330720375, −0.838451005729669310109454297810, −0.803618828115592714039795009325, −0.74031354258288660622552707658, 0.74031354258288660622552707658, 0.803618828115592714039795009325, 0.838451005729669310109454297810, 1.02442238922405198054330720375, 1.03072153846747291921214618087, 1.05668137257888996879577313102, 1.08633382600023838607247204996, 1.70700076394549643847228557107, 1.71791578167033221688262349404, 1.85785313867962415033426808120, 1.88916629755480280784572217064, 1.92352727465941171635617744077, 1.94122285575140517555111212240, 2.02352494478604353826077693443, 2.48784580512635751140617313660, 2.58684170456655719916861941247, 2.65152416283174355763166740837, 2.75599214140743410581885709633, 2.81165260933788586477914805894, 2.82951440979272729273411192482, 2.84900487950041146465934606982, 3.33345865752333029670284210385, 3.45214977740080640189903609620, 3.46595175865199591619964725349, 3.52907780358046293334351675130
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.