Dirichlet series
| L(s) = 1 | − 3·3-s + 17·5-s + 42·7-s − 47·9-s + 33·11-s + 35·13-s − 51·15-s + 66·17-s + 133·19-s − 126·21-s + 389·23-s − 271·25-s + 190·27-s − 233·29-s + 158·31-s − 99·33-s + 714·35-s + 436·37-s − 105·39-s − 94·41-s − 645·43-s − 799·45-s + 1.45e3·47-s − 272·49-s − 198·51-s − 3·53-s + 561·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.52·5-s + 2.26·7-s − 1.74·9-s + 0.904·11-s + 0.746·13-s − 0.877·15-s + 0.941·17-s + 1.60·19-s − 1.30·21-s + 3.52·23-s − 2.16·25-s + 1.35·27-s − 1.49·29-s + 0.915·31-s − 0.522·33-s + 3.44·35-s + 1.93·37-s − 0.431·39-s − 0.358·41-s − 2.28·43-s − 2.64·45-s + 4.50·47-s − 0.793·49-s − 0.543·51-s − 0.00777·53-s + 1.37·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{42} \cdot 19^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.78582\times 10^{12}\) |
| Root analytic conductor: | \(8.47032\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{42} \cdot 19^{7} ,\ ( \ : [3/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(20.16957411\) |
| \(L(\frac12)\) | \(\approx\) | \(20.16957411\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 19 | \( ( 1 - p T )^{7} \) | |
| good | 3 | \( 1 + p T + 56 T^{2} + 119 T^{3} + 662 p T^{4} - 143 T^{5} + 559 p^{4} T^{6} - 8882 p T^{7} + 559 p^{7} T^{8} - 143 p^{6} T^{9} + 662 p^{10} T^{10} + 119 p^{12} T^{11} + 56 p^{15} T^{12} + p^{19} T^{13} + p^{21} T^{14} \) |
| 5 | \( 1 - 17 T + 112 p T^{2} - 8673 T^{3} + 6408 p^{2} T^{4} - 2162603 T^{5} + 29024683 T^{6} - 335285398 T^{7} + 29024683 p^{3} T^{8} - 2162603 p^{6} T^{9} + 6408 p^{11} T^{10} - 8673 p^{12} T^{11} + 112 p^{16} T^{12} - 17 p^{18} T^{13} + p^{21} T^{14} \) | |
| 7 | \( 1 - 6 p T + 2036 T^{2} - 49620 T^{3} + 1420333 T^{4} - 25233680 T^{5} + 588771899 T^{6} - 9142918448 T^{7} + 588771899 p^{3} T^{8} - 25233680 p^{6} T^{9} + 1420333 p^{9} T^{10} - 49620 p^{12} T^{11} + 2036 p^{15} T^{12} - 6 p^{19} T^{13} + p^{21} T^{14} \) | |
| 11 | \( 1 - 3 p T + 4854 T^{2} - 141953 T^{3} + 13381374 T^{4} - 371205303 T^{5} + 24817448453 T^{6} - 593463083726 T^{7} + 24817448453 p^{3} T^{8} - 371205303 p^{6} T^{9} + 13381374 p^{9} T^{10} - 141953 p^{12} T^{11} + 4854 p^{15} T^{12} - 3 p^{19} T^{13} + p^{21} T^{14} \) | |
| 13 | \( 1 - 35 T + 4400 T^{2} - 129331 T^{3} + 13612616 T^{4} - 457222273 T^{5} + 41068444195 T^{6} - 1141273899586 T^{7} + 41068444195 p^{3} T^{8} - 457222273 p^{6} T^{9} + 13612616 p^{9} T^{10} - 129331 p^{12} T^{11} + 4400 p^{15} T^{12} - 35 p^{18} T^{13} + p^{21} T^{14} \) | |
| 17 | \( 1 - 66 T + 18028 T^{2} - 872774 T^{3} + 149354625 T^{4} - 6043011244 T^{5} + 870935208165 T^{6} - 32077140868782 T^{7} + 870935208165 p^{3} T^{8} - 6043011244 p^{6} T^{9} + 149354625 p^{9} T^{10} - 872774 p^{12} T^{11} + 18028 p^{15} T^{12} - 66 p^{18} T^{13} + p^{21} T^{14} \) | |
| 23 | \( 1 - 389 T + 111488 T^{2} - 22667649 T^{3} + 4034003110 T^{4} - 597619955103 T^{5} + 80427758323711 T^{6} - 9307287155911094 T^{7} + 80427758323711 p^{3} T^{8} - 597619955103 p^{6} T^{9} + 4034003110 p^{9} T^{10} - 22667649 p^{12} T^{11} + 111488 p^{15} T^{12} - 389 p^{18} T^{13} + p^{21} T^{14} \) | |
| 29 | \( 1 + 233 T + 75664 T^{2} + 15845769 T^{3} + 3429508600 T^{4} + 620236699059 T^{5} + 116689153518819 T^{6} + 17404867500924038 T^{7} + 116689153518819 p^{3} T^{8} + 620236699059 p^{6} T^{9} + 3429508600 p^{9} T^{10} + 15845769 p^{12} T^{11} + 75664 p^{15} T^{12} + 233 p^{18} T^{13} + p^{21} T^{14} \) | |
| 31 | \( 1 - 158 T + 109549 T^{2} - 17535156 T^{3} + 6027562513 T^{4} - 1064191672482 T^{5} + 234567962311141 T^{6} - 40094028422788888 T^{7} + 234567962311141 p^{3} T^{8} - 1064191672482 p^{6} T^{9} + 6027562513 p^{9} T^{10} - 17535156 p^{12} T^{11} + 109549 p^{15} T^{12} - 158 p^{18} T^{13} + p^{21} T^{14} \) | |
| 37 | \( 1 - 436 T + 127043 T^{2} - 22891608 T^{3} + 6811304069 T^{4} - 2174820112940 T^{5} + 582967160257807 T^{6} - 133502761989042384 T^{7} + 582967160257807 p^{3} T^{8} - 2174820112940 p^{6} T^{9} + 6811304069 p^{9} T^{10} - 22891608 p^{12} T^{11} + 127043 p^{15} T^{12} - 436 p^{18} T^{13} + p^{21} T^{14} \) | |
| 41 | \( 1 + 94 T + 361903 T^{2} + 29339572 T^{3} + 61586843749 T^{4} + 4355840907362 T^{5} + 6415914747512667 T^{6} + 382349576376339928 T^{7} + 6415914747512667 p^{3} T^{8} + 4355840907362 p^{6} T^{9} + 61586843749 p^{9} T^{10} + 29339572 p^{12} T^{11} + 361903 p^{15} T^{12} + 94 p^{18} T^{13} + p^{21} T^{14} \) | |
| 43 | \( 1 + 15 p T + 379152 T^{2} + 123171217 T^{3} + 45943447178 T^{4} + 12310936528791 T^{5} + 4434191203351863 T^{6} + 1120117219327592038 T^{7} + 4434191203351863 p^{3} T^{8} + 12310936528791 p^{6} T^{9} + 45943447178 p^{9} T^{10} + 123171217 p^{12} T^{11} + 379152 p^{15} T^{12} + 15 p^{19} T^{13} + p^{21} T^{14} \) | |
| 47 | \( 1 - 1451 T + 1272192 T^{2} - 790637551 T^{3} + 8602913786 p T^{4} - 176734808991281 T^{5} + 68752320247270263 T^{6} - 23501715764875089002 T^{7} + 68752320247270263 p^{3} T^{8} - 176734808991281 p^{6} T^{9} + 8602913786 p^{10} T^{10} - 790637551 p^{12} T^{11} + 1272192 p^{15} T^{12} - 1451 p^{18} T^{13} + p^{21} T^{14} \) | |
| 53 | \( 1 + 3 T + 206162 T^{2} - 48876965 T^{3} + 71676727992 T^{4} + 1153092151961 T^{5} + 11237050286058193 T^{6} - 1791771123996145518 T^{7} + 11237050286058193 p^{3} T^{8} + 1153092151961 p^{6} T^{9} + 71676727992 p^{9} T^{10} - 48876965 p^{12} T^{11} + 206162 p^{15} T^{12} + 3 p^{18} T^{13} + p^{21} T^{14} \) | |
| 59 | \( 1 + 297 T + 672038 T^{2} + 289914217 T^{3} + 209151452110 T^{4} + 114112443834511 T^{5} + 47149312524993941 T^{6} + 27613574907610061630 T^{7} + 47149312524993941 p^{3} T^{8} + 114112443834511 p^{6} T^{9} + 209151452110 p^{9} T^{10} + 289914217 p^{12} T^{11} + 672038 p^{15} T^{12} + 297 p^{18} T^{13} + p^{21} T^{14} \) | |
| 61 | \( 1 + 93 T + 546422 T^{2} + 119943789 T^{3} + 150877142008 T^{4} + 23828601705631 T^{5} + 41040418332020549 T^{6} + 871276752600150190 T^{7} + 41040418332020549 p^{3} T^{8} + 23828601705631 p^{6} T^{9} + 150877142008 p^{9} T^{10} + 119943789 p^{12} T^{11} + 546422 p^{15} T^{12} + 93 p^{18} T^{13} + p^{21} T^{14} \) | |
| 67 | \( 1 + 1641 T + 2704786 T^{2} + 2913835345 T^{3} + 2759273004534 T^{4} + 2168039790697727 T^{5} + 1457159616204475993 T^{6} + \)\(86\!\cdots\!26\)\( T^{7} + 1457159616204475993 p^{3} T^{8} + 2168039790697727 p^{6} T^{9} + 2759273004534 p^{9} T^{10} + 2913835345 p^{12} T^{11} + 2704786 p^{15} T^{12} + 1641 p^{18} T^{13} + p^{21} T^{14} \) | |
| 71 | \( 1 - 2392 T + 3603813 T^{2} - 3476382448 T^{3} + 2526898154145 T^{4} - 1308977126629672 T^{5} + 573364990396721229 T^{6} - \)\(25\!\cdots\!52\)\( T^{7} + 573364990396721229 p^{3} T^{8} - 1308977126629672 p^{6} T^{9} + 2526898154145 p^{9} T^{10} - 3476382448 p^{12} T^{11} + 3603813 p^{15} T^{12} - 2392 p^{18} T^{13} + p^{21} T^{14} \) | |
| 73 | \( 1 - 324 T + 1257510 T^{2} - 185687394 T^{3} + 858614357711 T^{4} - 96335188311284 T^{5} + 461971660192002879 T^{6} - 54639150971309212142 T^{7} + 461971660192002879 p^{3} T^{8} - 96335188311284 p^{6} T^{9} + 858614357711 p^{9} T^{10} - 185687394 p^{12} T^{11} + 1257510 p^{15} T^{12} - 324 p^{18} T^{13} + p^{21} T^{14} \) | |
| 79 | \( 1 - 2492 T + 3988981 T^{2} - 4203439288 T^{3} + 3192281577225 T^{4} - 1606568277556804 T^{5} + 513888728362209077 T^{6} - \)\(12\!\cdots\!92\)\( T^{7} + 513888728362209077 p^{3} T^{8} - 1606568277556804 p^{6} T^{9} + 3192281577225 p^{9} T^{10} - 4203439288 p^{12} T^{11} + 3988981 p^{15} T^{12} - 2492 p^{18} T^{13} + p^{21} T^{14} \) | |
| 83 | \( 1 + 310 T + 2557441 T^{2} + 576340020 T^{3} + 3230831978617 T^{4} + 561170294948922 T^{5} + 2620520802921802401 T^{6} + \)\(36\!\cdots\!28\)\( T^{7} + 2620520802921802401 p^{3} T^{8} + 561170294948922 p^{6} T^{9} + 3230831978617 p^{9} T^{10} + 576340020 p^{12} T^{11} + 2557441 p^{15} T^{12} + 310 p^{18} T^{13} + p^{21} T^{14} \) | |
| 89 | \( 1 + 440 T + 2989175 T^{2} + 1524116688 T^{3} + 4588789464333 T^{4} + 2298015679540296 T^{5} + 4671297455663212683 T^{6} + \)\(20\!\cdots\!92\)\( T^{7} + 4671297455663212683 p^{3} T^{8} + 2298015679540296 p^{6} T^{9} + 4588789464333 p^{9} T^{10} + 1524116688 p^{12} T^{11} + 2989175 p^{15} T^{12} + 440 p^{18} T^{13} + p^{21} T^{14} \) | |
| 97 | \( 1 + 532 T + 4215647 T^{2} + 2818552824 T^{3} + 8999627369469 T^{4} + 5755843692439532 T^{5} + 12419748656877320931 T^{6} + \)\(66\!\cdots\!80\)\( T^{7} + 12419748656877320931 p^{3} T^{8} + 5755843692439532 p^{6} T^{9} + 8999627369469 p^{9} T^{10} + 2818552824 p^{12} T^{11} + 4215647 p^{15} T^{12} + 532 p^{18} T^{13} + p^{21} T^{14} \) | |
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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
−4.14409418923867690371511098578, −3.96897698890108846692571772082, −3.94397570536716689802066036358, −3.90983740664651767381825262610, −3.63070615107584179647156325834, −3.13388819011562433026443934618, −3.11282757821081639027236579686, −3.10960139536546827488759717742, −3.04253846671401368401143848391, −2.97622383705458258397142371936, −2.91357626661144613290441531327, −2.26895339818481155379658869529, −2.25674520788246749921148958058, −2.09143196595837462915225098542, −1.89954440896035589594155250104, −1.89159543193149983431121482090, −1.75218045256932573854477551237, −1.48992142102309469701161919417, −1.37397791007743505990008908620, −0.943845001286390821224885017792, −0.882292532397910647050880118356, −0.842822255065040824492624986176, −0.77181406495772627094092285779, −0.45831159233758565363220714060, −0.16974964023535318222082081477, 0.16974964023535318222082081477, 0.45831159233758565363220714060, 0.77181406495772627094092285779, 0.842822255065040824492624986176, 0.882292532397910647050880118356, 0.943845001286390821224885017792, 1.37397791007743505990008908620, 1.48992142102309469701161919417, 1.75218045256932573854477551237, 1.89159543193149983431121482090, 1.89954440896035589594155250104, 2.09143196595837462915225098542, 2.25674520788246749921148958058, 2.26895339818481155379658869529, 2.91357626661144613290441531327, 2.97622383705458258397142371936, 3.04253846671401368401143848391, 3.10960139536546827488759717742, 3.11282757821081639027236579686, 3.13388819011562433026443934618, 3.63070615107584179647156325834, 3.90983740664651767381825262610, 3.94397570536716689802066036358, 3.96897698890108846692571772082, 4.14409418923867690371511098578
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.