Dirichlet series
| L(s) = 1 | − 3·3-s − 4·7-s − 5·9-s + 7·11-s − 7·13-s + 7·19-s + 12·21-s + 7·23-s + 27·27-s − 11·29-s + 10·31-s − 21·33-s − 19·37-s + 21·39-s − 16·41-s − 6·43-s − 6·47-s − 25·49-s − 15·53-s − 21·57-s + 11·59-s + 5·61-s + 20·63-s − 9·67-s − 21·69-s + 14·71-s − 10·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.51·7-s − 5/3·9-s + 2.11·11-s − 1.94·13-s + 1.60·19-s + 2.61·21-s + 1.45·23-s + 5.19·27-s − 2.04·29-s + 1.79·31-s − 3.65·33-s − 3.12·37-s + 3.36·39-s − 2.49·41-s − 0.914·43-s − 0.875·47-s − 3.57·49-s − 2.06·53-s − 2.78·57-s + 1.43·59-s + 0.640·61-s + 2.51·63-s − 1.09·67-s − 2.52·69-s + 1.66·71-s − 1.17·73-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{14} \cdot 23^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{14} \cdot 23^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{28} \cdot 5^{14} \cdot 23^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.15466\times 10^{13}\) |
| Root analytic conductor: | \(8.57101\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 2^{28} \cdot 5^{14} \cdot 23^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
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 | 2 | \( 1 \) |
| 5 | \( 1 \) | |
| 23 | \( ( 1 - T )^{7} \) | |
| good | 3 | \( 1 + p T + 14 T^{2} + 10 p T^{3} + 85 T^{4} + 152 T^{5} + 341 T^{6} + 536 T^{7} + 341 p T^{8} + 152 p^{2} T^{9} + 85 p^{3} T^{10} + 10 p^{5} T^{11} + 14 p^{5} T^{12} + p^{7} T^{13} + p^{7} T^{14} \) |
| 7 | \( 1 + 4 T + 41 T^{2} + 127 T^{3} + 759 T^{4} + 1908 T^{5} + 8307 T^{6} + 16918 T^{7} + 8307 p T^{8} + 1908 p^{2} T^{9} + 759 p^{3} T^{10} + 127 p^{4} T^{11} + 41 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 7 T + 50 T^{2} - 24 p T^{3} + 1304 T^{4} - 5217 T^{5} + 20319 T^{6} - 69648 T^{7} + 20319 p T^{8} - 5217 p^{2} T^{9} + 1304 p^{3} T^{10} - 24 p^{5} T^{11} + 50 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 7 T + 6 p T^{2} + 454 T^{3} + 2833 T^{4} + 13096 T^{5} + 59795 T^{6} + 217800 T^{7} + 59795 p T^{8} + 13096 p^{2} T^{9} + 2833 p^{3} T^{10} + 454 p^{4} T^{11} + 6 p^{6} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 61 T^{2} + 105 T^{3} + 1641 T^{4} + 6090 T^{5} + 29753 T^{6} + 146422 T^{7} + 29753 p T^{8} + 6090 p^{2} T^{9} + 1641 p^{3} T^{10} + 105 p^{4} T^{11} + 61 p^{5} T^{12} + p^{7} T^{14} \) | |
| 19 | \( 1 - 7 T + 80 T^{2} - 560 T^{3} + 3576 T^{4} - 20921 T^{5} + 103617 T^{6} - 488640 T^{7} + 103617 p T^{8} - 20921 p^{2} T^{9} + 3576 p^{3} T^{10} - 560 p^{4} T^{11} + 80 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 11 T + 147 T^{2} + 984 T^{3} + 8570 T^{4} + 46128 T^{5} + 336468 T^{6} + 1566138 T^{7} + 336468 p T^{8} + 46128 p^{2} T^{9} + 8570 p^{3} T^{10} + 984 p^{4} T^{11} + 147 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 10 T + 107 T^{2} - 617 T^{3} + 5296 T^{4} - 28550 T^{5} + 208730 T^{6} - 954795 T^{7} + 208730 p T^{8} - 28550 p^{2} T^{9} + 5296 p^{3} T^{10} - 617 p^{4} T^{11} + 107 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 19 T + 327 T^{2} + 3518 T^{3} + 35497 T^{4} + 275617 T^{5} + 2069855 T^{6} + 12731020 T^{7} + 2069855 p T^{8} + 275617 p^{2} T^{9} + 35497 p^{3} T^{10} + 3518 p^{4} T^{11} + 327 p^{5} T^{12} + 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 16 T + 299 T^{2} + 2981 T^{3} + 33474 T^{4} + 254854 T^{5} + 2163504 T^{6} + 13191531 T^{7} + 2163504 p T^{8} + 254854 p^{2} T^{9} + 33474 p^{3} T^{10} + 2981 p^{4} T^{11} + 299 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 6 T + 271 T^{2} + 1484 T^{3} + 32567 T^{4} + 155570 T^{5} + 2251961 T^{6} + 8845208 T^{7} + 2251961 p T^{8} + 155570 p^{2} T^{9} + 32567 p^{3} T^{10} + 1484 p^{4} T^{11} + 271 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 6 T + 151 T^{2} + 562 T^{3} + 11256 T^{4} + 558 p T^{5} + 624278 T^{6} + 1222452 T^{7} + 624278 p T^{8} + 558 p^{3} T^{9} + 11256 p^{3} T^{10} + 562 p^{4} T^{11} + 151 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 15 T + 311 T^{2} + 3174 T^{3} + 40385 T^{4} + 331769 T^{5} + 3249663 T^{6} + 21958644 T^{7} + 3249663 p T^{8} + 331769 p^{2} T^{9} + 40385 p^{3} T^{10} + 3174 p^{4} T^{11} + 311 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 11 T + 161 T^{2} - 1606 T^{3} + 15241 T^{4} - 104989 T^{5} + 902577 T^{6} - 6212660 T^{7} + 902577 p T^{8} - 104989 p^{2} T^{9} + 15241 p^{3} T^{10} - 1606 p^{4} T^{11} + 161 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 5 T + 112 T^{2} + 218 T^{3} + 7820 T^{4} + 11169 T^{5} + 779059 T^{6} + 139788 T^{7} + 779059 p T^{8} + 11169 p^{2} T^{9} + 7820 p^{3} T^{10} + 218 p^{4} T^{11} + 112 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 9 T + 225 T^{2} + 1886 T^{3} + 24401 T^{4} + 213987 T^{5} + 1845009 T^{6} + 16692652 T^{7} + 1845009 p T^{8} + 213987 p^{2} T^{9} + 24401 p^{3} T^{10} + 1886 p^{4} T^{11} + 225 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 14 T + 341 T^{2} - 3253 T^{3} + 50270 T^{4} - 415786 T^{5} + 5172636 T^{6} - 36898691 T^{7} + 5172636 p T^{8} - 415786 p^{2} T^{9} + 50270 p^{3} T^{10} - 3253 p^{4} T^{11} + 341 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 10 T + 481 T^{2} + 3914 T^{3} + 100568 T^{4} + 665794 T^{5} + 163422 p T^{6} + 63270868 T^{7} + 163422 p^{2} T^{8} + 665794 p^{2} T^{9} + 100568 p^{3} T^{10} + 3914 p^{4} T^{11} + 481 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 32 T + 719 T^{2} - 12292 T^{3} + 173907 T^{4} - 2116176 T^{5} + 22653621 T^{6} - 212954872 T^{7} + 22653621 p T^{8} - 2116176 p^{2} T^{9} + 173907 p^{3} T^{10} - 12292 p^{4} T^{11} + 719 p^{5} T^{12} - 32 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + T + 359 T^{2} - 572 T^{3} + 58259 T^{4} - 231357 T^{5} + 6074581 T^{6} - 29368688 T^{7} + 6074581 p T^{8} - 231357 p^{2} T^{9} + 58259 p^{3} T^{10} - 572 p^{4} T^{11} + 359 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 24 T + 669 T^{2} + 10556 T^{3} + 175975 T^{4} + 2095656 T^{5} + 25712747 T^{6} + 239948584 T^{7} + 25712747 p T^{8} + 2095656 p^{2} T^{9} + 175975 p^{3} T^{10} + 10556 p^{4} T^{11} + 669 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 7 T + 310 T^{2} - 3110 T^{3} + 64344 T^{4} - 560137 T^{5} + 9041429 T^{6} - 66958876 T^{7} + 9041429 p T^{8} - 560137 p^{2} T^{9} + 64344 p^{3} T^{10} - 3110 p^{4} T^{11} + 310 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.85938064255510724539618704741, −3.75759168588257445022578100305, −3.50307768730777689048054548151, −3.47802356393914532343553868567, −3.32835797241995564675683093834, −3.20609155288029770278052061937, −3.15270945341380659116902216944, −3.06842808519690024442715866295, −3.04328229474573710620995575932, −3.02552367857664762426613466920, −2.86652249668780638245163216080, −2.50253527919052222855117869258, −2.44127617153368722668186768334, −2.21202423241496865937589462652, −2.19119933756943432345763921704, −2.15894586901009680548294061818, −2.06027542982969031802288120685, −1.80927896420159866372613206838, −1.56772275560774391378887998603, −1.28322911796741988139679975805, −1.25113431791562657510432028267, −1.23955647603215255479049019914, −1.10271555919447599237331867835, −1.03107224770745423223291503661, −0.854884482121790281599985143079, 0, 0, 0, 0, 0, 0, 0, 0.854884482121790281599985143079, 1.03107224770745423223291503661, 1.10271555919447599237331867835, 1.23955647603215255479049019914, 1.25113431791562657510432028267, 1.28322911796741988139679975805, 1.56772275560774391378887998603, 1.80927896420159866372613206838, 2.06027542982969031802288120685, 2.15894586901009680548294061818, 2.19119933756943432345763921704, 2.21202423241496865937589462652, 2.44127617153368722668186768334, 2.50253527919052222855117869258, 2.86652249668780638245163216080, 3.02552367857664762426613466920, 3.04328229474573710620995575932, 3.06842808519690024442715866295, 3.15270945341380659116902216944, 3.20609155288029770278052061937, 3.32835797241995564675683093834, 3.47802356393914532343553868567, 3.50307768730777689048054548151, 3.75759168588257445022578100305, 3.85938064255510724539618704741
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.