Dirichlet series
| L(s) = 1 | + 2.88e3·7-s − 2.91e3·9-s + 2.28e4·17-s + 2.07e5·23-s + 2.10e5·25-s − 1.78e4·31-s + 6.87e5·41-s − 1.98e6·47-s + 3.26e6·49-s − 8.39e6·63-s − 6.33e6·71-s + 8.92e6·73-s + 1.25e6·79-s + 5.31e6·81-s + 4.44e6·89-s − 1.41e7·97-s − 3.18e7·103-s − 3.59e7·113-s + 6.59e7·119-s + 7.26e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 6.67e7·153-s + ⋯ |
| L(s) = 1 | + 3.17·7-s − 4/3·9-s + 1.13·17-s + 3.55·23-s + 2.68·25-s − 0.107·31-s + 1.55·41-s − 2.79·47-s + 3.95·49-s − 4.23·63-s − 2.10·71-s + 2.68·73-s + 0.285·79-s + 10/9·81-s + 0.668·89-s − 1.57·97-s − 2.87·103-s − 2.34·113-s + 3.58·119-s + 3.72·121-s − 1.50·153-s + 11.2·161-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{56} \cdot 3^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.28717\times 10^{16}\) |
| Root analytic conductor: | \(10.9524\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [7/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(0.8034565307\) |
| \(L(\frac12)\) | \(\approx\) | \(0.8034565307\) |
| \(L(\frac{9}{2})\) | 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 | \( ( 1 + p^{6} T^{2} )^{4} \) | |
| good | 5 | \( 1 - 210152 T^{2} + 18035115644 T^{4} - 37183502711256 p^{2} T^{6} + 3451278010941686 p^{6} T^{8} - 37183502711256 p^{16} T^{10} + 18035115644 p^{28} T^{12} - 210152 p^{42} T^{14} + p^{56} T^{16} \) |
| 7 | \( ( 1 - 1440 T + 30204 p^{2} T^{2} - 1933104672 T^{3} + 2357036209702 T^{4} - 1933104672 p^{7} T^{5} + 30204 p^{16} T^{6} - 1440 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 11 | \( 1 - 72660440 T^{2} + 3083374758221948 T^{4} - \)\(91\!\cdots\!36\)\( T^{6} + \)\(20\!\cdots\!90\)\( T^{8} - \)\(91\!\cdots\!36\)\( p^{14} T^{10} + 3083374758221948 p^{28} T^{12} - 72660440 p^{42} T^{14} + p^{56} T^{16} \) | |
| 13 | \( 1 - 232491496 T^{2} + 32941533643236412 T^{4} - \)\(31\!\cdots\!64\)\( T^{6} + \)\(22\!\cdots\!94\)\( T^{8} - \)\(31\!\cdots\!64\)\( p^{14} T^{10} + 32941533643236412 p^{28} T^{12} - 232491496 p^{42} T^{14} + p^{56} T^{16} \) | |
| 17 | \( ( 1 - 11448 T + 675462492 T^{2} - 18823419349128 T^{3} + 276096649249682822 T^{4} - 18823419349128 p^{7} T^{5} + 675462492 p^{14} T^{6} - 11448 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 19 | \( 1 - 6272279448 T^{2} + 17918977415529239612 T^{4} - \)\(30\!\cdots\!88\)\( T^{6} + \)\(33\!\cdots\!06\)\( T^{8} - \)\(30\!\cdots\!88\)\( p^{14} T^{10} + 17918977415529239612 p^{28} T^{12} - 6272279448 p^{42} T^{14} + p^{56} T^{16} \) | |
| 23 | \( ( 1 - 103680 T + 9423592732 T^{2} - 713887013378304 T^{3} + 52230530957828619686 T^{4} - 713887013378304 p^{7} T^{5} + 9423592732 p^{14} T^{6} - 103680 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 29 | \( 1 - 68451270696 T^{2} + \)\(28\!\cdots\!08\)\( T^{4} - \)\(78\!\cdots\!28\)\( T^{6} + \)\(15\!\cdots\!18\)\( T^{8} - \)\(78\!\cdots\!28\)\( p^{14} T^{10} + \)\(28\!\cdots\!08\)\( p^{28} T^{12} - 68451270696 p^{42} T^{14} + p^{56} T^{16} \) | |
| 31 | \( ( 1 + 288 p T + 31981062108 T^{2} - 8136257043579552 T^{3} + \)\(19\!\cdots\!42\)\( T^{4} - 8136257043579552 p^{7} T^{5} + 31981062108 p^{14} T^{6} + 288 p^{22} T^{7} + p^{28} T^{8} )^{2} \) | |
| 37 | \( 1 - 112648835112 T^{2} + \)\(21\!\cdots\!40\)\( T^{4} - \)\(31\!\cdots\!48\)\( T^{6} + \)\(23\!\cdots\!10\)\( T^{8} - \)\(31\!\cdots\!48\)\( p^{14} T^{10} + \)\(21\!\cdots\!40\)\( p^{28} T^{12} - 112648835112 p^{42} T^{14} + p^{56} T^{16} \) | |
| 41 | \( ( 1 - 343528 T + 365749435452 T^{2} + 35483236220764776 T^{3} + \)\(34\!\cdots\!10\)\( T^{4} + 35483236220764776 p^{7} T^{5} + 365749435452 p^{14} T^{6} - 343528 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 43 | \( 1 - 34201218696 p T^{2} + \)\(10\!\cdots\!24\)\( T^{4} - \)\(51\!\cdots\!60\)\( T^{6} + \)\(16\!\cdots\!90\)\( T^{8} - \)\(51\!\cdots\!60\)\( p^{14} T^{10} + \)\(10\!\cdots\!24\)\( p^{28} T^{12} - 34201218696 p^{43} T^{14} + p^{56} T^{16} \) | |
| 47 | \( ( 1 + 993600 T + 1878320705212 T^{2} + 1254938173705429056 T^{3} + \)\(13\!\cdots\!34\)\( T^{4} + 1254938173705429056 p^{7} T^{5} + 1878320705212 p^{14} T^{6} + 993600 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 53 | \( 1 - 4648954961256 T^{2} + \)\(22\!\cdots\!20\)\( p T^{4} - \)\(21\!\cdots\!92\)\( T^{6} + \)\(28\!\cdots\!58\)\( T^{8} - \)\(21\!\cdots\!92\)\( p^{14} T^{10} + \)\(22\!\cdots\!20\)\( p^{29} T^{12} - 4648954961256 p^{42} T^{14} + p^{56} T^{16} \) | |
| 59 | \( 1 - 9083206385752 T^{2} + \)\(51\!\cdots\!36\)\( T^{4} - \)\(19\!\cdots\!32\)\( T^{6} + \)\(56\!\cdots\!14\)\( T^{8} - \)\(19\!\cdots\!32\)\( p^{14} T^{10} + \)\(51\!\cdots\!36\)\( p^{28} T^{12} - 9083206385752 p^{42} T^{14} + p^{56} T^{16} \) | |
| 61 | \( 1 - 12317465640296 T^{2} + \)\(81\!\cdots\!36\)\( T^{4} - \)\(38\!\cdots\!68\)\( T^{6} + \)\(13\!\cdots\!98\)\( T^{8} - \)\(38\!\cdots\!68\)\( p^{14} T^{10} + \)\(81\!\cdots\!36\)\( p^{28} T^{12} - 12317465640296 p^{42} T^{14} + p^{56} T^{16} \) | |
| 67 | \( 1 - 12595380351512 T^{2} + \)\(86\!\cdots\!04\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{6} - \)\(25\!\cdots\!50\)\( T^{8} - \)\(12\!\cdots\!00\)\( p^{14} T^{10} + \)\(86\!\cdots\!04\)\( p^{28} T^{12} - 12595380351512 p^{42} T^{14} + p^{56} T^{16} \) | |
| 71 | \( ( 1 + 3168000 T + 30793182554716 T^{2} + 85783814145563284224 T^{3} + \)\(39\!\cdots\!98\)\( T^{4} + 85783814145563284224 p^{7} T^{5} + 30793182554716 p^{14} T^{6} + 3168000 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 73 | \( ( 1 - 4460376 T + 45247419174460 T^{2} - \)\(13\!\cdots\!64\)\( T^{3} + \)\(74\!\cdots\!14\)\( T^{4} - \)\(13\!\cdots\!64\)\( p^{7} T^{5} + 45247419174460 p^{14} T^{6} - 4460376 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 79 | \( ( 1 - 625824 T + 24699015020316 T^{2} + 47958880344498060768 T^{3} + \)\(19\!\cdots\!62\)\( T^{4} + 47958880344498060768 p^{7} T^{5} + 24699015020316 p^{14} T^{6} - 625824 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 83 | \( 1 - 123395401699736 T^{2} + \)\(80\!\cdots\!28\)\( T^{4} - \)\(35\!\cdots\!72\)\( T^{6} + \)\(11\!\cdots\!42\)\( T^{8} - \)\(35\!\cdots\!72\)\( p^{14} T^{10} + \)\(80\!\cdots\!28\)\( p^{28} T^{12} - 123395401699736 p^{42} T^{14} + p^{56} T^{16} \) | |
| 89 | \( ( 1 - 2223704 T + 82674152988156 T^{2} - 35737168093538692776 T^{3} + \)\(40\!\cdots\!18\)\( T^{4} - 35737168093538692776 p^{7} T^{5} + 82674152988156 p^{14} T^{6} - 2223704 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 97 | \( ( 1 + 7078792 T + 170684496494492 T^{2} + \)\(18\!\cdots\!24\)\( T^{3} + \)\(11\!\cdots\!90\)\( T^{4} + \)\(18\!\cdots\!24\)\( p^{7} T^{5} + 170684496494492 p^{14} T^{6} + 7078792 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
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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
−3.71167811640470043758541590768, −3.53733364759281153737703917591, −3.52902455088443271614249950111, −3.44327635699956473402348654368, −3.32458043925607657050604824846, −2.95683447553892169693828823473, −2.73887222241377493490262531550, −2.66367915932821032618084381600, −2.62683894835042969262415584923, −2.52471891590517805469820950729, −2.51664713737584135192155192956, −2.23274680215755362895878004662, −2.02446455628715034609537460342, −1.63557579436524281887523390688, −1.62655948264052987946366071035, −1.37777400558786488045057717935, −1.36557729240289961601153599207, −1.20807926282437696209435889529, −1.15744366707027802686405694912, −1.08945284774274772324724056478, −0.867955422876507878198037407735, −0.61949442224616038518966907070, −0.48171075196552471783209751037, −0.23967245070304361214897837582, −0.03414949023601577846083792027, 0.03414949023601577846083792027, 0.23967245070304361214897837582, 0.48171075196552471783209751037, 0.61949442224616038518966907070, 0.867955422876507878198037407735, 1.08945284774274772324724056478, 1.15744366707027802686405694912, 1.20807926282437696209435889529, 1.36557729240289961601153599207, 1.37777400558786488045057717935, 1.62655948264052987946366071035, 1.63557579436524281887523390688, 2.02446455628715034609537460342, 2.23274680215755362895878004662, 2.51664713737584135192155192956, 2.52471891590517805469820950729, 2.62683894835042969262415584923, 2.66367915932821032618084381600, 2.73887222241377493490262531550, 2.95683447553892169693828823473, 3.32458043925607657050604824846, 3.44327635699956473402348654368, 3.52902455088443271614249950111, 3.53733364759281153737703917591, 3.71167811640470043758541590768
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.