Dirichlet series
| L(s) = 1 | − 6.77e5·5-s + 9.63e6·7-s + 6.03e8·11-s − 2.47e9·13-s − 7.42e9·17-s − 1.95e10·19-s − 4.58e11·23-s − 7.88e11·25-s + 8.44e11·29-s + 1.24e12·31-s − 6.52e12·35-s + 1.76e13·37-s + 3.31e13·41-s + 2.41e13·43-s + 1.19e13·47-s − 4.17e14·49-s − 2.56e14·53-s − 4.09e14·55-s − 9.05e14·59-s + 2.51e14·61-s + 1.67e15·65-s + 2.67e15·67-s − 4.18e15·71-s − 1.41e15·73-s + 5.81e15·77-s + 7.57e15·79-s − 1.48e16·83-s + ⋯ |
| L(s) = 1 | − 0.776·5-s + 0.631·7-s + 0.849·11-s − 0.841·13-s − 0.258·17-s − 0.264·19-s − 1.22·23-s − 1.03·25-s + 0.313·29-s + 0.262·31-s − 0.490·35-s + 0.826·37-s + 0.647·41-s + 0.314·43-s + 0.0732·47-s − 1.79·49-s − 0.566·53-s − 0.658·55-s − 0.802·59-s + 0.167·61-s + 0.653·65-s + 0.803·67-s − 0.768·71-s − 0.204·73-s + 0.536·77-s + 0.561·79-s − 0.724·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+17/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(26873856\) = \(2^{12} \cdot 3^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.02859\times 10^{8}\) |
| Root analytic conductor: | \(11.4856\) |
| Motivic weight: | \(17\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 26873856,\ (\ :17/2, 17/2, 17/2, 17/2),\ 1)\) |
Particular Values
| \(L(9)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{19}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | ||
| good | 5 | $C_2 \wr S_4$ | \( 1 + 677824 T + 249566118148 p T^{2} + 2377297831999552 p^{4} T^{3} + \)\(36\!\cdots\!22\)\( p^{5} T^{4} + 2377297831999552 p^{21} T^{5} + 249566118148 p^{35} T^{6} + 677824 p^{51} T^{7} + p^{68} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 9632880 T + 72930471741604 p T^{2} - \)\(13\!\cdots\!60\)\( p^{2} T^{3} + \)\(61\!\cdots\!58\)\( p^{4} T^{4} - \)\(13\!\cdots\!60\)\( p^{19} T^{5} + 72930471741604 p^{35} T^{6} - 9632880 p^{51} T^{7} + p^{68} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 603687680 T + 1093698267630025004 T^{2} - \)\(52\!\cdots\!40\)\( p T^{3} + \)\(60\!\cdots\!50\)\( p^{2} T^{4} - \)\(52\!\cdots\!40\)\( p^{18} T^{5} + 1093698267630025004 p^{34} T^{6} - 603687680 p^{51} T^{7} + p^{68} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 2476119800 T + 1553297207541801340 p T^{2} + \)\(23\!\cdots\!00\)\( p^{2} T^{3} + \)\(10\!\cdots\!74\)\( p^{3} T^{4} + \)\(23\!\cdots\!00\)\( p^{19} T^{5} + 1553297207541801340 p^{35} T^{6} + 2476119800 p^{51} T^{7} + p^{68} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 7423708544 T + \)\(16\!\cdots\!28\)\( p T^{2} + \)\(19\!\cdots\!12\)\( T^{3} + \)\(33\!\cdots\!10\)\( T^{4} + \)\(19\!\cdots\!12\)\( p^{17} T^{5} + \)\(16\!\cdots\!28\)\( p^{35} T^{6} + 7423708544 p^{51} T^{7} + p^{68} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 19551988960 T + \)\(19\!\cdots\!04\)\( T^{2} + \)\(26\!\cdots\!60\)\( T^{3} + \)\(15\!\cdots\!46\)\( T^{4} + \)\(26\!\cdots\!60\)\( p^{17} T^{5} + \)\(19\!\cdots\!04\)\( p^{34} T^{6} + 19551988960 p^{51} T^{7} + p^{68} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 458239426048 T + \)\(19\!\cdots\!04\)\( T^{2} + \)\(82\!\cdots\!16\)\( T^{3} + \)\(47\!\cdots\!70\)\( T^{4} + \)\(82\!\cdots\!16\)\( p^{17} T^{5} + \)\(19\!\cdots\!04\)\( p^{34} T^{6} + 458239426048 p^{51} T^{7} + p^{68} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 844849583040 T + \)\(14\!\cdots\!40\)\( T^{2} - \)\(76\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!58\)\( T^{4} - \)\(76\!\cdots\!60\)\( p^{17} T^{5} + \)\(14\!\cdots\!40\)\( p^{34} T^{6} - 844849583040 p^{51} T^{7} + p^{68} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 1246685956208 T + \)\(25\!\cdots\!40\)\( T^{2} - \)\(42\!\cdots\!84\)\( T^{3} + \)\(11\!\cdots\!38\)\( T^{4} - \)\(42\!\cdots\!84\)\( p^{17} T^{5} + \)\(25\!\cdots\!40\)\( p^{34} T^{6} - 1246685956208 p^{51} T^{7} + p^{68} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 17661470383320 T + \)\(60\!\cdots\!32\)\( T^{2} - \)\(50\!\cdots\!60\)\( T^{3} + \)\(49\!\cdots\!10\)\( T^{4} - \)\(50\!\cdots\!60\)\( p^{17} T^{5} + \)\(60\!\cdots\!32\)\( p^{34} T^{6} - 17661470383320 p^{51} T^{7} + p^{68} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 33100198696320 T + \)\(77\!\cdots\!00\)\( T^{2} - \)\(20\!\cdots\!40\)\( T^{3} + \)\(28\!\cdots\!78\)\( T^{4} - \)\(20\!\cdots\!40\)\( p^{17} T^{5} + \)\(77\!\cdots\!00\)\( p^{34} T^{6} - 33100198696320 p^{51} T^{7} + p^{68} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 24133870623520 T + \)\(99\!\cdots\!68\)\( T^{2} + \)\(15\!\cdots\!60\)\( T^{3} + \)\(57\!\cdots\!50\)\( T^{4} + \)\(15\!\cdots\!60\)\( p^{17} T^{5} + \)\(99\!\cdots\!68\)\( p^{34} T^{6} - 24133870623520 p^{51} T^{7} + p^{68} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 11964304009728 T + \)\(86\!\cdots\!92\)\( T^{2} - \)\(13\!\cdots\!80\)\( T^{3} + \)\(31\!\cdots\!46\)\( T^{4} - \)\(13\!\cdots\!80\)\( p^{17} T^{5} + \)\(86\!\cdots\!92\)\( p^{34} T^{6} - 11964304009728 p^{51} T^{7} + p^{68} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 256855748281408 T + \)\(42\!\cdots\!04\)\( T^{2} + \)\(65\!\cdots\!56\)\( T^{3} + \)\(85\!\cdots\!90\)\( T^{4} + \)\(65\!\cdots\!56\)\( p^{17} T^{5} + \)\(42\!\cdots\!04\)\( p^{34} T^{6} + 256855748281408 p^{51} T^{7} + p^{68} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 905361180505600 T + \)\(34\!\cdots\!76\)\( T^{2} + \)\(32\!\cdots\!00\)\( T^{3} + \)\(56\!\cdots\!70\)\( T^{4} + \)\(32\!\cdots\!00\)\( p^{17} T^{5} + \)\(34\!\cdots\!76\)\( p^{34} T^{6} + 905361180505600 p^{51} T^{7} + p^{68} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 251141035416376 T + \)\(76\!\cdots\!12\)\( T^{2} - \)\(11\!\cdots\!80\)\( T^{3} + \)\(24\!\cdots\!86\)\( T^{4} - \)\(11\!\cdots\!80\)\( p^{17} T^{5} + \)\(76\!\cdots\!12\)\( p^{34} T^{6} - 251141035416376 p^{51} T^{7} + p^{68} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 2671944825444800 T + \)\(35\!\cdots\!88\)\( T^{2} - \)\(91\!\cdots\!00\)\( T^{3} + \)\(79\!\cdots\!94\)\( p T^{4} - \)\(91\!\cdots\!00\)\( p^{17} T^{5} + \)\(35\!\cdots\!88\)\( p^{34} T^{6} - 2671944825444800 p^{51} T^{7} + p^{68} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 4181227521935360 T + \)\(40\!\cdots\!56\)\( T^{2} + \)\(18\!\cdots\!40\)\( T^{3} + \)\(54\!\cdots\!46\)\( T^{4} + \)\(18\!\cdots\!40\)\( p^{17} T^{5} + \)\(40\!\cdots\!56\)\( p^{34} T^{6} + 4181227521935360 p^{51} T^{7} + p^{68} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 1410879602368040 T + \)\(10\!\cdots\!40\)\( T^{2} + \)\(88\!\cdots\!20\)\( T^{3} + \)\(60\!\cdots\!18\)\( T^{4} + \)\(88\!\cdots\!20\)\( p^{17} T^{5} + \)\(10\!\cdots\!40\)\( p^{34} T^{6} + 1410879602368040 p^{51} T^{7} + p^{68} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 7574345714454640 T + \)\(49\!\cdots\!08\)\( T^{2} - \)\(13\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!78\)\( T^{4} - \)\(13\!\cdots\!20\)\( p^{17} T^{5} + \)\(49\!\cdots\!08\)\( p^{34} T^{6} - 7574345714454640 p^{51} T^{7} + p^{68} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 14868236752232192 T + \)\(12\!\cdots\!24\)\( T^{2} + \)\(11\!\cdots\!84\)\( T^{3} + \)\(69\!\cdots\!10\)\( T^{4} + \)\(11\!\cdots\!84\)\( p^{17} T^{5} + \)\(12\!\cdots\!24\)\( p^{34} T^{6} + 14868236752232192 p^{51} T^{7} + p^{68} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 31667797088981760 T + \)\(54\!\cdots\!52\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!34\)\( T^{4} + \)\(12\!\cdots\!00\)\( p^{17} T^{5} + \)\(54\!\cdots\!52\)\( p^{34} T^{6} + 31667797088981760 p^{51} T^{7} + p^{68} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 17206349902780360 T + \)\(17\!\cdots\!88\)\( T^{2} + \)\(25\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!98\)\( T^{4} + \)\(25\!\cdots\!60\)\( p^{17} T^{5} + \)\(17\!\cdots\!88\)\( p^{34} T^{6} + 17206349902780360 p^{51} T^{7} + p^{68} T^{8} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.282996319562217289039364472503, −7.57369324891136154792904210173, −7.57181292894512491928924198900, −7.48796670990780743528739089816, −7.14863145629770852112281855630, −6.45490702913947093439399254190, −6.26368008227098025480248913993, −6.23184228319600439546055968796, −6.05414702621249044626227615911, −5.19268113527127004871188492738, −5.09153984609252122213412666754, −5.04995417639832854648566286499, −4.54515268748018690357055143963, −4.10342743618663769645584142607, −3.97811691446190021758524156730, −3.71393919152435647059314566218, −3.68093008969360506113504322740, −2.76095673401052783909451639059, −2.70456383227081586460046639227, −2.51397482161795222059842562847, −2.15942383954409066875157646101, −1.62608560331461520054326281571, −1.37486022927554556647545331914, −1.13796528162232074369558293659, −1.05791257486672405466881167816, 0, 0, 0, 0, 1.05791257486672405466881167816, 1.13796528162232074369558293659, 1.37486022927554556647545331914, 1.62608560331461520054326281571, 2.15942383954409066875157646101, 2.51397482161795222059842562847, 2.70456383227081586460046639227, 2.76095673401052783909451639059, 3.68093008969360506113504322740, 3.71393919152435647059314566218, 3.97811691446190021758524156730, 4.10342743618663769645584142607, 4.54515268748018690357055143963, 5.04995417639832854648566286499, 5.09153984609252122213412666754, 5.19268113527127004871188492738, 6.05414702621249044626227615911, 6.23184228319600439546055968796, 6.26368008227098025480248913993, 6.45490702913947093439399254190, 7.14863145629770852112281855630, 7.48796670990780743528739089816, 7.57181292894512491928924198900, 7.57369324891136154792904210173, 8.282996319562217289039364472503