Dirichlet series
| L(s) = 1 | + 3·5-s + 26·7-s + 65·11-s + 17·13-s + 119·17-s + 125·19-s − 107·23-s − 345·25-s + 6·29-s + 44·31-s + 78·35-s + 254·37-s − 219·41-s + 419·43-s − 300·47-s − 277·49-s − 284·53-s + 195·55-s + 46·59-s + 638·61-s + 51·65-s + 796·67-s + 1.47e3·71-s + 1.71e3·73-s + 1.69e3·77-s + 1.13e3·79-s + 596·83-s + ⋯ |
| L(s) = 1 | + 0.268·5-s + 1.40·7-s + 1.78·11-s + 0.362·13-s + 1.69·17-s + 1.50·19-s − 0.970·23-s − 2.75·25-s + 0.0384·29-s + 0.254·31-s + 0.376·35-s + 1.12·37-s − 0.834·41-s + 1.48·43-s − 0.931·47-s − 0.807·49-s − 0.736·53-s + 0.478·55-s + 0.101·59-s + 1.33·61-s + 0.0973·65-s + 1.45·67-s + 2.46·71-s + 2.74·73-s + 2.50·77-s + 1.61·79-s + 0.788·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 3^{14} \cdot 17^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 3^{14} \cdot 17^{7}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{21} \cdot 3^{14} \cdot 17^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.02454\times 10^{13}\) |
| Root analytic conductor: | \(8.49813\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{21} \cdot 3^{14} \cdot 17^{7} ,\ ( \ : [3/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(53.87211781\) |
| \(L(\frac12)\) | \(\approx\) | \(53.87211781\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 17 | \( ( 1 - p T )^{7} \) | |
| good | 5 | \( 1 - 3 T + 354 T^{2} - 107 p T^{3} + 81988 T^{4} - 120457 T^{5} + 2699037 p T^{6} - 88678 p^{3} T^{7} + 2699037 p^{4} T^{8} - 120457 p^{6} T^{9} + 81988 p^{9} T^{10} - 107 p^{13} T^{11} + 354 p^{15} T^{12} - 3 p^{18} T^{13} + p^{21} T^{14} \) |
| 7 | \( 1 - 26 T + 953 T^{2} - 15508 T^{3} + 36471 p T^{4} - 947054 T^{5} - 38567559 T^{6} + 1035613336 T^{7} - 38567559 p^{3} T^{8} - 947054 p^{6} T^{9} + 36471 p^{10} T^{10} - 15508 p^{12} T^{11} + 953 p^{15} T^{12} - 26 p^{18} T^{13} + p^{21} T^{14} \) | |
| 11 | \( 1 - 65 T + 5460 T^{2} - 199745 T^{3} + 11724094 T^{4} - 396579979 T^{5} + 22768443575 T^{6} - 700532534822 T^{7} + 22768443575 p^{3} T^{8} - 396579979 p^{6} T^{9} + 11724094 p^{9} T^{10} - 199745 p^{12} T^{11} + 5460 p^{15} T^{12} - 65 p^{18} T^{13} + p^{21} T^{14} \) | |
| 13 | \( 1 - 17 T + 3566 T^{2} - 155977 T^{3} + 13169472 T^{4} - 445736523 T^{5} + 31584320725 T^{6} - 1462127060726 T^{7} + 31584320725 p^{3} T^{8} - 445736523 p^{6} T^{9} + 13169472 p^{9} T^{10} - 155977 p^{12} T^{11} + 3566 p^{15} T^{12} - 17 p^{18} T^{13} + p^{21} T^{14} \) | |
| 19 | \( 1 - 125 T + 32564 T^{2} - 3925317 T^{3} + 536291318 T^{4} - 57835084551 T^{5} + 5510555058279 T^{6} - 503257770414814 T^{7} + 5510555058279 p^{3} T^{8} - 57835084551 p^{6} T^{9} + 536291318 p^{9} T^{10} - 3925317 p^{12} T^{11} + 32564 p^{15} T^{12} - 125 p^{18} T^{13} + p^{21} T^{14} \) | |
| 23 | \( 1 + 107 T + 53940 T^{2} + 4243039 T^{3} + 1375470422 T^{4} + 87462702521 T^{5} + 23069079907751 T^{6} + 1240155430385582 T^{7} + 23069079907751 p^{3} T^{8} + 87462702521 p^{6} T^{9} + 1375470422 p^{9} T^{10} + 4243039 p^{12} T^{11} + 53940 p^{15} T^{12} + 107 p^{18} T^{13} + p^{21} T^{14} \) | |
| 29 | \( 1 - 6 T + 76371 T^{2} - 5151988 T^{3} + 2925956401 T^{4} - 304123775298 T^{5} + 90874281998963 T^{6} - 8870279562105832 T^{7} + 90874281998963 p^{3} T^{8} - 304123775298 p^{6} T^{9} + 2925956401 p^{9} T^{10} - 5151988 p^{12} T^{11} + 76371 p^{15} T^{12} - 6 p^{18} T^{13} + p^{21} T^{14} \) | |
| 31 | \( 1 - 44 T + 49205 T^{2} - 1421992 T^{3} + 1232792589 T^{4} - 133784265156 T^{5} + 36888446996425 T^{6} - 8534897219439056 T^{7} + 36888446996425 p^{3} T^{8} - 133784265156 p^{6} T^{9} + 1232792589 p^{9} T^{10} - 1421992 p^{12} T^{11} + 49205 p^{15} T^{12} - 44 p^{18} T^{13} + p^{21} T^{14} \) | |
| 37 | \( 1 - 254 T + 152571 T^{2} - 26631556 T^{3} + 12380828641 T^{4} - 1512128030218 T^{5} + 667910390260651 T^{6} - 66410541275692936 T^{7} + 667910390260651 p^{3} T^{8} - 1512128030218 p^{6} T^{9} + 12380828641 p^{9} T^{10} - 26631556 p^{12} T^{11} + 152571 p^{15} T^{12} - 254 p^{18} T^{13} + p^{21} T^{14} \) | |
| 41 | \( 1 + 219 T + 185186 T^{2} + 55111939 T^{3} + 21969995724 T^{4} + 6592395348305 T^{5} + 2189011091591209 T^{6} + 516453459461183394 T^{7} + 2189011091591209 p^{3} T^{8} + 6592395348305 p^{6} T^{9} + 21969995724 p^{9} T^{10} + 55111939 p^{12} T^{11} + 185186 p^{15} T^{12} + 219 p^{18} T^{13} + p^{21} T^{14} \) | |
| 43 | \( 1 - 419 T + 538868 T^{2} - 171088339 T^{3} + 121736298046 T^{4} - 30470845203665 T^{5} + 15529153543662711 T^{6} - 3103403003686933362 T^{7} + 15529153543662711 p^{3} T^{8} - 30470845203665 p^{6} T^{9} + 121736298046 p^{9} T^{10} - 171088339 p^{12} T^{11} + 538868 p^{15} T^{12} - 419 p^{18} T^{13} + p^{21} T^{14} \) | |
| 47 | \( 1 + 300 T + 340969 T^{2} + 35379512 T^{3} + 44599263157 T^{4} + 1380783473428 T^{5} + 5820010719291693 T^{6} + 319026518445721104 T^{7} + 5820010719291693 p^{3} T^{8} + 1380783473428 p^{6} T^{9} + 44599263157 p^{9} T^{10} + 35379512 p^{12} T^{11} + 340969 p^{15} T^{12} + 300 p^{18} T^{13} + p^{21} T^{14} \) | |
| 53 | \( 1 + 284 T + 338771 T^{2} - 16036792 T^{3} + 45743266021 T^{4} - 10691740383356 T^{5} + 11466639834495487 T^{6} - 466766187276585232 T^{7} + 11466639834495487 p^{3} T^{8} - 10691740383356 p^{6} T^{9} + 45743266021 p^{9} T^{10} - 16036792 p^{12} T^{11} + 338771 p^{15} T^{12} + 284 p^{18} T^{13} + p^{21} T^{14} \) | |
| 59 | \( 1 - 46 T + 661433 T^{2} - 14945332 T^{3} + 220177235209 T^{4} + 2689413526270 T^{5} + 55620054516264137 T^{6} + 1563781907253732712 T^{7} + 55620054516264137 p^{3} T^{8} + 2689413526270 p^{6} T^{9} + 220177235209 p^{9} T^{10} - 14945332 p^{12} T^{11} + 661433 p^{15} T^{12} - 46 p^{18} T^{13} + p^{21} T^{14} \) | |
| 61 | \( 1 - 638 T + 796371 T^{2} - 259635300 T^{3} + 284706012689 T^{4} - 83685460081450 T^{5} + 91585740973986003 T^{6} - 24629039672906966472 T^{7} + 91585740973986003 p^{3} T^{8} - 83685460081450 p^{6} T^{9} + 284706012689 p^{9} T^{10} - 259635300 p^{12} T^{11} + 796371 p^{15} T^{12} - 638 p^{18} T^{13} + p^{21} T^{14} \) | |
| 67 | \( 1 - 796 T + 843445 T^{2} - 474463576 T^{3} + 434669122405 T^{4} - 239244171464260 T^{5} + 180700031996011913 T^{6} - 84885796738878461648 T^{7} + 180700031996011913 p^{3} T^{8} - 239244171464260 p^{6} T^{9} + 434669122405 p^{9} T^{10} - 474463576 p^{12} T^{11} + 843445 p^{15} T^{12} - 796 p^{18} T^{13} + p^{21} T^{14} \) | |
| 71 | \( 1 - 1472 T + 1895245 T^{2} - 1489924640 T^{3} + 1058612301997 T^{4} - 531767222771328 T^{5} + 281647459552971457 T^{6} - \)\(13\!\cdots\!40\)\( T^{7} + 281647459552971457 p^{3} T^{8} - 531767222771328 p^{6} T^{9} + 1058612301997 p^{9} T^{10} - 1489924640 p^{12} T^{11} + 1895245 p^{15} T^{12} - 1472 p^{18} T^{13} + p^{21} T^{14} \) | |
| 73 | \( 1 - 1712 T + 2844231 T^{2} - 2853363072 T^{3} + 2641240205581 T^{4} - 1911511071335504 T^{5} + 1324276635663873627 T^{6} - \)\(82\!\cdots\!32\)\( T^{7} + 1324276635663873627 p^{3} T^{8} - 1911511071335504 p^{6} T^{9} + 2641240205581 p^{9} T^{10} - 2853363072 p^{12} T^{11} + 2844231 p^{15} T^{12} - 1712 p^{18} T^{13} + p^{21} T^{14} \) | |
| 79 | \( 1 - 1134 T + 1652497 T^{2} - 995418940 T^{3} + 1245056154849 T^{4} - 761460794061514 T^{5} + 864315482858125905 T^{6} - \)\(45\!\cdots\!84\)\( T^{7} + 864315482858125905 p^{3} T^{8} - 761460794061514 p^{6} T^{9} + 1245056154849 p^{9} T^{10} - 995418940 p^{12} T^{11} + 1652497 p^{15} T^{12} - 1134 p^{18} T^{13} + p^{21} T^{14} \) | |
| 83 | \( 1 - 596 T + 1446597 T^{2} + 59437368 T^{3} + 846468515397 T^{4} + 243263035026356 T^{5} + 803333180124238777 T^{6} + 12916505499420261648 T^{7} + 803333180124238777 p^{3} T^{8} + 243263035026356 p^{6} T^{9} + 846468515397 p^{9} T^{10} + 59437368 p^{12} T^{11} + 1446597 p^{15} T^{12} - 596 p^{18} T^{13} + p^{21} T^{14} \) | |
| 89 | \( 1 + 694 T + 2622227 T^{2} + 1063809756 T^{3} + 3483368838233 T^{4} + 1079225771839434 T^{5} + 3386971173433246083 T^{6} + \)\(91\!\cdots\!00\)\( T^{7} + 3386971173433246083 p^{3} T^{8} + 1079225771839434 p^{6} T^{9} + 3483368838233 p^{9} T^{10} + 1063809756 p^{12} T^{11} + 2622227 p^{15} T^{12} + 694 p^{18} T^{13} + p^{21} T^{14} \) | |
| 97 | \( 1 - 2536 T + 7635023 T^{2} - 12963867984 T^{3} + 22468525613293 T^{4} - 28310220541550360 T^{5} + 35083068932946523139 T^{6} - \)\(33\!\cdots\!12\)\( T^{7} + 35083068932946523139 p^{3} T^{8} - 28310220541550360 p^{6} T^{9} + 22468525613293 p^{9} T^{10} - 12963867984 p^{12} T^{11} + 7635023 p^{15} T^{12} - 2536 p^{18} T^{13} + p^{21} T^{14} \) | |
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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
−4.21595960043040141022716850372, −4.06740319057999964056686889548, −3.87245850712892067336706079619, −3.83519101110307237682952681424, −3.50077139789920848953201241817, −3.38123459280745821858195512929, −3.34877416734750132792952413243, −3.18934545597494939537925709794, −3.00752112668854588758544391002, −2.90533768314707613129636335503, −2.90508832094880162166990744212, −2.16053694530025144304246942149, −2.15284170326904767680885823863, −2.05365447469452497049999579601, −2.02694975754609744388706509557, −1.87300580938177089986096016440, −1.81629275648447273797289424441, −1.58232496752584933811990733923, −1.35341730592656442427451010955, −0.894630577938067830945599931172, −0.862499973278605173857352109564, −0.74555441366124507945067814487, −0.71337030059210514093489454299, −0.59037598130494062200921180040, −0.26262498320072904688911710252, 0.26262498320072904688911710252, 0.59037598130494062200921180040, 0.71337030059210514093489454299, 0.74555441366124507945067814487, 0.862499973278605173857352109564, 0.894630577938067830945599931172, 1.35341730592656442427451010955, 1.58232496752584933811990733923, 1.81629275648447273797289424441, 1.87300580938177089986096016440, 2.02694975754609744388706509557, 2.05365447469452497049999579601, 2.15284170326904767680885823863, 2.16053694530025144304246942149, 2.90508832094880162166990744212, 2.90533768314707613129636335503, 3.00752112668854588758544391002, 3.18934545597494939537925709794, 3.34877416734750132792952413243, 3.38123459280745821858195512929, 3.50077139789920848953201241817, 3.83519101110307237682952681424, 3.87245850712892067336706079619, 4.06740319057999964056686889548, 4.21595960043040141022716850372
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.