Dirichlet series
| L(s) = 1 | + 10·2-s − 33·4-s − 1.02e3·8-s − 972·9-s + 2.14e3·11-s − 1.14e4·16-s − 9.72e3·18-s + 2.14e4·22-s + 3.04e4·23-s + 4.02e4·25-s + 3.25e4·29-s − 6.37e4·32-s + 3.20e4·36-s + 9.13e4·37-s − 4.45e5·43-s − 7.06e4·44-s + 3.04e5·46-s + 4.02e5·50-s + 2.60e4·53-s + 3.25e5·58-s + 2.24e5·64-s − 7.68e5·67-s + 2.25e5·71-s + 9.99e5·72-s + 9.13e5·74-s + 1.11e6·79-s + 5.90e5·81-s + ⋯ |
| L(s) = 1 | + 5/4·2-s − 0.515·4-s − 2.00·8-s − 4/3·9-s + 1.60·11-s − 2.78·16-s − 5/3·18-s + 2.00·22-s + 2.50·23-s + 2.57·25-s + 1.33·29-s − 1.94·32-s + 0.687·36-s + 1.80·37-s − 5.60·43-s − 0.829·44-s + 3.12·46-s + 3.21·50-s + 0.175·53-s + 1.66·58-s + 0.858·64-s − 2.55·67-s + 0.630·71-s + 2.67·72-s + 2.25·74-s + 2.26·79-s + 10/9·81-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.71071\times 10^{12}\) |
| Root analytic conductor: | \(5.81532\) |
| Motivic weight: | \(6\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 7^{16} ,\ ( \ : [3]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{7}{2})\) | \(\approx\) | \(5.160313801\) |
| \(L(\frac12)\) | \(\approx\) | \(5.160313801\) |
| \(L(4)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 7 | \( 1 \) | |
| good | 2 | \( ( 1 - 5 T + 27 p T^{2} - 23 p T^{3} + 265 p^{4} T^{4} - 23 p^{7} T^{5} + 27 p^{13} T^{6} - 5 p^{18} T^{7} + p^{24} T^{8} )^{2} \) |
| 5 | \( 1 - 40226 T^{2} + 1130825449 T^{4} - 945309374954 p^{2} T^{6} + 685993727134996 p^{4} T^{8} - 945309374954 p^{14} T^{10} + 1130825449 p^{24} T^{12} - 40226 p^{36} T^{14} + p^{48} T^{16} \) | |
| 11 | \( ( 1 - 1070 T + 6822033 T^{2} - 5373825550 T^{3} + 17953528514164 T^{4} - 5373825550 p^{6} T^{5} + 6822033 p^{12} T^{6} - 1070 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 13 | \( 1 - 16298786 T^{2} + 111852524426401 T^{4} - 37539779661680550026 p T^{6} + \)\(20\!\cdots\!88\)\( T^{8} - 37539779661680550026 p^{13} T^{10} + 111852524426401 p^{24} T^{12} - 16298786 p^{36} T^{14} + p^{48} T^{16} \) | |
| 17 | \( 1 - 23889824 T^{2} + 122207207828156 p T^{4} - \)\(38\!\cdots\!92\)\( T^{6} + \)\(17\!\cdots\!42\)\( T^{8} - \)\(38\!\cdots\!92\)\( p^{12} T^{10} + 122207207828156 p^{25} T^{12} - 23889824 p^{36} T^{14} + p^{48} T^{16} \) | |
| 19 | \( 1 - 120011066 T^{2} + 4558960827908065 T^{4} + \)\(74\!\cdots\!14\)\( p T^{6} - \)\(16\!\cdots\!56\)\( T^{8} + \)\(74\!\cdots\!14\)\( p^{13} T^{10} + 4558960827908065 p^{24} T^{12} - 120011066 p^{36} T^{14} + p^{48} T^{16} \) | |
| 23 | \( ( 1 - 15224 T + 303174876 T^{2} - 3933308321128 T^{3} + 71062389247076086 T^{4} - 3933308321128 p^{6} T^{5} + 303174876 p^{12} T^{6} - 15224 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 29 | \( ( 1 - 16262 T + 1936712745 T^{2} - 19588912728706 T^{3} + 1557131395176891808 T^{4} - 19588912728706 p^{6} T^{5} + 1936712745 p^{12} T^{6} - 16262 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 31 | \( 1 - 3071909540 T^{2} + 5609111825159464114 T^{4} - \)\(68\!\cdots\!20\)\( T^{6} + \)\(68\!\cdots\!91\)\( T^{8} - \)\(68\!\cdots\!20\)\( p^{12} T^{10} + 5609111825159464114 p^{24} T^{12} - 3071909540 p^{36} T^{14} + p^{48} T^{16} \) | |
| 37 | \( ( 1 - 45670 T + 7173593881 T^{2} - 327580206713990 T^{3} + 24148714940087622596 T^{4} - 327580206713990 p^{6} T^{5} + 7173593881 p^{12} T^{6} - 45670 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 41 | \( 1 - 26005847576 T^{2} + \)\(34\!\cdots\!60\)\( T^{4} - \)\(28\!\cdots\!64\)\( T^{6} + \)\(15\!\cdots\!14\)\( T^{8} - \)\(28\!\cdots\!64\)\( p^{12} T^{10} + \)\(34\!\cdots\!60\)\( p^{24} T^{12} - 26005847576 p^{36} T^{14} + p^{48} T^{16} \) | |
| 43 | \( ( 1 + 222830 T + 23708395369 T^{2} + 1380667882433470 T^{3} + 73296626267097035876 T^{4} + 1380667882433470 p^{6} T^{5} + 23708395369 p^{12} T^{6} + 222830 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 47 | \( 1 - 59192143520 T^{2} + \)\(16\!\cdots\!20\)\( T^{4} - \)\(27\!\cdots\!64\)\( T^{6} + \)\(34\!\cdots\!70\)\( T^{8} - \)\(27\!\cdots\!64\)\( p^{12} T^{10} + \)\(16\!\cdots\!20\)\( p^{24} T^{12} - 59192143520 p^{36} T^{14} + p^{48} T^{16} \) | |
| 53 | \( ( 1 - 13034 T + 38080896873 T^{2} - 63769766876950 T^{3} + \)\(11\!\cdots\!52\)\( T^{4} - 63769766876950 p^{6} T^{5} + 38080896873 p^{12} T^{6} - 13034 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 59 | \( 1 - 34928672738 T^{2} + \)\(31\!\cdots\!25\)\( T^{4} - \)\(27\!\cdots\!74\)\( T^{6} + \)\(40\!\cdots\!28\)\( T^{8} - \)\(27\!\cdots\!74\)\( p^{12} T^{10} + \)\(31\!\cdots\!25\)\( p^{24} T^{12} - 34928672738 p^{36} T^{14} + p^{48} T^{16} \) | |
| 61 | \( 1 - 353197951976 T^{2} + \)\(57\!\cdots\!92\)\( T^{4} - \)\(55\!\cdots\!08\)\( T^{6} + \)\(34\!\cdots\!42\)\( T^{8} - \)\(55\!\cdots\!08\)\( p^{12} T^{10} + \)\(57\!\cdots\!92\)\( p^{24} T^{12} - 353197951976 p^{36} T^{14} + p^{48} T^{16} \) | |
| 67 | \( ( 1 + 384094 T + 316936300861 T^{2} + 89519136577479610 T^{3} + \)\(41\!\cdots\!76\)\( T^{4} + 89519136577479610 p^{6} T^{5} + 316936300861 p^{12} T^{6} + 384094 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 71 | \( ( 1 - 112844 T + 324448495296 T^{2} + 1073384954931548 T^{3} + \)\(47\!\cdots\!38\)\( T^{4} + 1073384954931548 p^{6} T^{5} + 324448495296 p^{12} T^{6} - 112844 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 73 | \( 1 - 979051001426 T^{2} + \)\(44\!\cdots\!89\)\( T^{4} - \)\(12\!\cdots\!38\)\( T^{6} + \)\(22\!\cdots\!56\)\( T^{8} - \)\(12\!\cdots\!38\)\( p^{12} T^{10} + \)\(44\!\cdots\!89\)\( p^{24} T^{12} - 979051001426 p^{36} T^{14} + p^{48} T^{16} \) | |
| 79 | \( ( 1 - 559592 T + 506004632830 T^{2} - 161020277829667928 T^{3} + \)\(12\!\cdots\!43\)\( T^{4} - 161020277829667928 p^{6} T^{5} + 506004632830 p^{12} T^{6} - 559592 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 83 | \( 1 - 577456669706 T^{2} + \)\(27\!\cdots\!89\)\( T^{4} - \)\(12\!\cdots\!18\)\( T^{6} + \)\(42\!\cdots\!16\)\( T^{8} - \)\(12\!\cdots\!18\)\( p^{12} T^{10} + \)\(27\!\cdots\!89\)\( p^{24} T^{12} - 577456669706 p^{36} T^{14} + p^{48} T^{16} \) | |
| 89 | \( 1 - 3692886607664 T^{2} + \)\(60\!\cdots\!64\)\( T^{4} - \)\(58\!\cdots\!12\)\( T^{6} + \)\(36\!\cdots\!06\)\( T^{8} - \)\(58\!\cdots\!12\)\( p^{12} T^{10} + \)\(60\!\cdots\!64\)\( p^{24} T^{12} - 3692886607664 p^{36} T^{14} + p^{48} T^{16} \) | |
| 97 | \( 1 - 4315064900690 T^{2} + \)\(91\!\cdots\!41\)\( T^{4} - \)\(12\!\cdots\!58\)\( T^{6} + \)\(12\!\cdots\!60\)\( T^{8} - \)\(12\!\cdots\!58\)\( p^{12} T^{10} + \)\(91\!\cdots\!41\)\( p^{24} T^{12} - 4315064900690 p^{36} T^{14} + p^{48} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.83416744569482972080729885429, −4.72400577640711156408702265740, −4.46104801411165626716960703991, −4.33567592810404601831685662055, −4.21420277282316136860174770129, −3.87527670076996405496619331026, −3.74078594854239528490807629975, −3.44564574445009132289307818736, −3.37976624399437435148800980670, −3.22918040996731432977171231619, −3.21148236792840809570734551321, −2.86835903143948128891538171692, −2.68527419596476466105958216331, −2.67194511566084456211540819502, −2.35192085183571367859660229850, −2.16671173265970664414758182462, −1.73665056638617538339498545074, −1.72606567080294662441044475509, −1.27437456335060212881789862430, −1.18780453035651825346332890748, −1.11258457351436258812163674787, −0.71965927962071644149456285154, −0.54221659349303010969648061742, −0.28340171946312187923465840759, −0.18985786895099191975885683823, 0.18985786895099191975885683823, 0.28340171946312187923465840759, 0.54221659349303010969648061742, 0.71965927962071644149456285154, 1.11258457351436258812163674787, 1.18780453035651825346332890748, 1.27437456335060212881789862430, 1.72606567080294662441044475509, 1.73665056638617538339498545074, 2.16671173265970664414758182462, 2.35192085183571367859660229850, 2.67194511566084456211540819502, 2.68527419596476466105958216331, 2.86835903143948128891538171692, 3.21148236792840809570734551321, 3.22918040996731432977171231619, 3.37976624399437435148800980670, 3.44564574445009132289307818736, 3.74078594854239528490807629975, 3.87527670076996405496619331026, 4.21420277282316136860174770129, 4.33567592810404601831685662055, 4.46104801411165626716960703991, 4.72400577640711156408702265740, 4.83416744569482972080729885429
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.