Dirichlet series
| L(s) = 1 | + 7·2-s − 3·3-s + 28·4-s − 7·5-s − 21·6-s + 4·7-s + 84·8-s − 4·9-s − 49·10-s − 10·11-s − 84·12-s − 7·13-s + 28·14-s + 21·15-s + 210·16-s − 6·17-s − 28·18-s − 5·19-s − 196·20-s − 12·21-s − 70·22-s − 11·23-s − 252·24-s + 28·25-s − 49·26-s + 17·27-s + 112·28-s + ⋯ |
| L(s) = 1 | + 4.94·2-s − 1.73·3-s + 14·4-s − 3.13·5-s − 8.57·6-s + 1.51·7-s + 29.6·8-s − 4/3·9-s − 15.4·10-s − 3.01·11-s − 24.2·12-s − 1.94·13-s + 7.48·14-s + 5.42·15-s + 52.5·16-s − 1.45·17-s − 6.59·18-s − 1.14·19-s − 43.8·20-s − 2.61·21-s − 14.9·22-s − 2.29·23-s − 51.4·24-s + 28/5·25-s − 9.60·26-s + 3.27·27-s + 21.1·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 5^{7} \cdot 13^{7} \cdot 31^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 5^{7} \cdot 13^{7} \cdot 31^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{7} \cdot 5^{7} \cdot 13^{7} \cdot 31^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(3.57334\times 10^{10}\) |
| Root analytic conductor: | \(5.67271\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 2^{7} \cdot 5^{7} \cdot 13^{7} \cdot 31^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
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 | \( ( 1 - T )^{7} \) |
| 5 | \( ( 1 + T )^{7} \) | |
| 13 | \( ( 1 + T )^{7} \) | |
| 31 | \( ( 1 - T )^{7} \) | |
| good | 3 | \( 1 + p T + 13 T^{2} + 34 T^{3} + 91 T^{4} + 7 p^{3} T^{5} + 392 T^{6} + 692 T^{7} + 392 p T^{8} + 7 p^{5} T^{9} + 91 p^{3} T^{10} + 34 p^{4} T^{11} + 13 p^{5} T^{12} + p^{7} T^{13} + p^{7} T^{14} \) |
| 7 | \( 1 - 4 T + 29 T^{2} - 104 T^{3} + 471 T^{4} - 1377 T^{5} + 4826 T^{6} - 11838 T^{7} + 4826 p T^{8} - 1377 p^{2} T^{9} + 471 p^{3} T^{10} - 104 p^{4} T^{11} + 29 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 10 T + 86 T^{2} + 502 T^{3} + 2696 T^{4} + 11740 T^{5} + 47836 T^{6} + 164146 T^{7} + 47836 p T^{8} + 11740 p^{2} T^{9} + 2696 p^{3} T^{10} + 502 p^{4} T^{11} + 86 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 6 T + 66 T^{2} + 364 T^{3} + 2625 T^{4} + 695 p T^{5} + 64633 T^{6} + 248494 T^{7} + 64633 p T^{8} + 695 p^{3} T^{9} + 2625 p^{3} T^{10} + 364 p^{4} T^{11} + 66 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 5 T + 74 T^{2} + 188 T^{3} + 2017 T^{4} + 1491 T^{5} + 34545 T^{6} - 9320 T^{7} + 34545 p T^{8} + 1491 p^{2} T^{9} + 2017 p^{3} T^{10} + 188 p^{4} T^{11} + 74 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 11 T + 129 T^{2} + 964 T^{3} + 7284 T^{4} + 42762 T^{5} + 251715 T^{6} + 1201916 T^{7} + 251715 p T^{8} + 42762 p^{2} T^{9} + 7284 p^{3} T^{10} + 964 p^{4} T^{11} + 129 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 18 T + 218 T^{2} + 1848 T^{3} + 13145 T^{4} + 78201 T^{5} + 437425 T^{6} + 2322370 T^{7} + 437425 p T^{8} + 78201 p^{2} T^{9} + 13145 p^{3} T^{10} + 1848 p^{4} T^{11} + 218 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 8 T + 220 T^{2} + 1496 T^{3} + 22132 T^{4} + 124992 T^{5} + 1301652 T^{6} + 5963090 T^{7} + 1301652 p T^{8} + 124992 p^{2} T^{9} + 22132 p^{3} T^{10} + 1496 p^{4} T^{11} + 220 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 12 T + 132 T^{2} + 833 T^{3} + 6312 T^{4} + 29669 T^{5} + 161332 T^{6} + 558734 T^{7} + 161332 p T^{8} + 29669 p^{2} T^{9} + 6312 p^{3} T^{10} + 833 p^{4} T^{11} + 132 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 5 T + 212 T^{2} + 1174 T^{3} + 21429 T^{4} + 115467 T^{5} + 1362449 T^{6} + 6384180 T^{7} + 1362449 p T^{8} + 115467 p^{2} T^{9} + 21429 p^{3} T^{10} + 1174 p^{4} T^{11} + 212 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 10 T + 165 T^{2} + 1620 T^{3} + 16223 T^{4} + 141935 T^{5} + 23230 p T^{6} + 7965130 T^{7} + 23230 p^{2} T^{8} + 141935 p^{2} T^{9} + 16223 p^{3} T^{10} + 1620 p^{4} T^{11} + 165 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 18 T + 292 T^{2} + 3273 T^{3} + 33794 T^{4} + 282473 T^{5} + 2337524 T^{6} + 16912446 T^{7} + 2337524 p T^{8} + 282473 p^{2} T^{9} + 33794 p^{3} T^{10} + 3273 p^{4} T^{11} + 292 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 11 T + 317 T^{2} + 2800 T^{3} + 44866 T^{4} + 326072 T^{5} + 3862883 T^{6} + 23489802 T^{7} + 3862883 p T^{8} + 326072 p^{2} T^{9} + 44866 p^{3} T^{10} + 2800 p^{4} T^{11} + 317 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 25 T + 418 T^{2} + 5294 T^{3} + 57532 T^{4} + 563390 T^{5} + 4999956 T^{6} + 40854450 T^{7} + 4999956 p T^{8} + 563390 p^{2} T^{9} + 57532 p^{3} T^{10} + 5294 p^{4} T^{11} + 418 p^{5} T^{12} + 25 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 22 T + 403 T^{2} + 4582 T^{3} + 53107 T^{4} + 506645 T^{5} + 5229834 T^{6} + 43072338 T^{7} + 5229834 p T^{8} + 506645 p^{2} T^{9} + 53107 p^{3} T^{10} + 4582 p^{4} T^{11} + 403 p^{5} T^{12} + 22 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 22 T + 532 T^{2} + 7832 T^{3} + 115141 T^{4} + 1265120 T^{5} + 13620619 T^{6} + 116285580 T^{7} + 13620619 p T^{8} + 1265120 p^{2} T^{9} + 115141 p^{3} T^{10} + 7832 p^{4} T^{11} + 532 p^{5} T^{12} + 22 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 19 T + 4 p T^{2} - 2734 T^{3} + 30172 T^{4} - 282766 T^{5} + 3316898 T^{6} - 28472058 T^{7} + 3316898 p T^{8} - 282766 p^{2} T^{9} + 30172 p^{3} T^{10} - 2734 p^{4} T^{11} + 4 p^{6} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 20 T + 384 T^{2} + 4906 T^{3} + 63766 T^{4} + 684288 T^{5} + 7370184 T^{6} + 66752736 T^{7} + 7370184 p T^{8} + 684288 p^{2} T^{9} + 63766 p^{3} T^{10} + 4906 p^{4} T^{11} + 384 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 8 T + 193 T^{2} + 8 p T^{3} + 19953 T^{4} + 42807 T^{5} + 2151206 T^{6} + 6017110 T^{7} + 2151206 p T^{8} + 42807 p^{2} T^{9} + 19953 p^{3} T^{10} + 8 p^{5} T^{11} + 193 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 4 T + 222 T^{2} + 1248 T^{3} + 37507 T^{4} + 193795 T^{5} + 4294939 T^{6} + 21271948 T^{7} + 4294939 p T^{8} + 193795 p^{2} T^{9} + 37507 p^{3} T^{10} + 1248 p^{4} T^{11} + 222 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 13 T + 447 T^{2} - 4379 T^{3} + 90218 T^{4} - 725437 T^{5} + 11663493 T^{6} - 81093224 T^{7} + 11663493 p T^{8} - 725437 p^{2} T^{9} + 90218 p^{3} T^{10} - 4379 p^{4} T^{11} + 447 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−4.32522323653892797695422517834, −4.30028702920232674825224933605, −4.19181895865006803017737526830, −4.00319539667116745460095259567, −3.90695109708215111764972312066, −3.61154686543084605408675153908, −3.49323030402840912035117358700, −3.48401599782395734335930531003, −3.47422783410218608540524557055, −3.27395502478517877399459117605, −3.23124823779547509277882140163, −2.86416701139254670662242669819, −2.79422178037703051817540858948, −2.72310347420052371002465858277, −2.68170552120062259073956686256, −2.53902005028705099244245624358, −2.46149878626283249864179728974, −2.32144356043289382363375317343, −1.99940205976104445564109963545, −1.88353098180462333334840426895, −1.67125960773575846618900742645, −1.64305091193420845285226855776, −1.46041542421377256089717123979, −1.33428833246104454857483595092, −1.20454128906389793665785039463, 0, 0, 0, 0, 0, 0, 0, 1.20454128906389793665785039463, 1.33428833246104454857483595092, 1.46041542421377256089717123979, 1.64305091193420845285226855776, 1.67125960773575846618900742645, 1.88353098180462333334840426895, 1.99940205976104445564109963545, 2.32144356043289382363375317343, 2.46149878626283249864179728974, 2.53902005028705099244245624358, 2.68170552120062259073956686256, 2.72310347420052371002465858277, 2.79422178037703051817540858948, 2.86416701139254670662242669819, 3.23124823779547509277882140163, 3.27395502478517877399459117605, 3.47422783410218608540524557055, 3.48401599782395734335930531003, 3.49323030402840912035117358700, 3.61154686543084605408675153908, 3.90695109708215111764972312066, 4.00319539667116745460095259567, 4.19181895865006803017737526830, 4.30028702920232674825224933605, 4.32522323653892797695422517834
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.