Dirichlet series
| L(s) = 1 | − 7·5-s − 6·7-s + 2·11-s − 6·13-s + 4·17-s − 8·19-s − 7·23-s + 28·25-s − 2·29-s − 8·31-s + 42·35-s − 16·37-s − 2·41-s − 10·43-s + 8·47-s + 3·49-s + 10·53-s − 14·55-s + 24·59-s + 8·61-s + 42·65-s − 20·67-s + 8·71-s + 2·73-s − 12·77-s − 2·79-s + 22·83-s + ⋯ |
| L(s) = 1 | − 3.13·5-s − 2.26·7-s + 0.603·11-s − 1.66·13-s + 0.970·17-s − 1.83·19-s − 1.45·23-s + 28/5·25-s − 0.371·29-s − 1.43·31-s + 7.09·35-s − 2.63·37-s − 0.312·41-s − 1.52·43-s + 1.16·47-s + 3/7·49-s + 1.37·53-s − 1.88·55-s + 3.12·59-s + 1.02·61-s + 5.20·65-s − 2.44·67-s + 0.949·71-s + 0.234·73-s − 1.36·77-s − 0.225·79-s + 2.41·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 3^{14} \cdot 5^{7} \cdot 23^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 3^{14} \cdot 5^{7} \cdot 23^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{21} \cdot 3^{14} \cdot 5^{7} \cdot 23^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.52270\times 10^{12}\) |
| Root analytic conductor: | \(8.13118\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{21} \cdot 3^{14} \cdot 5^{7} \cdot 23^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.5460895366\) |
| \(L(\frac12)\) | \(\approx\) | \(0.5460895366\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| 5 | \( ( 1 + T )^{7} \) | |
| 23 | \( ( 1 + T )^{7} \) | |
| good | 7 | \( 1 + 6 T + 33 T^{2} + 118 T^{3} + 8 p^{2} T^{4} + 1174 T^{5} + 3280 T^{6} + 9244 T^{7} + 3280 p T^{8} + 1174 p^{2} T^{9} + 8 p^{5} T^{10} + 118 p^{4} T^{11} + 33 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) |
| 11 | \( 1 - 2 T + 27 T^{2} - 4 T^{3} + 347 T^{4} + 74 T^{5} + 6161 T^{6} - 3112 T^{7} + 6161 p T^{8} + 74 p^{2} T^{9} + 347 p^{3} T^{10} - 4 p^{4} T^{11} + 27 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 6 T + 33 T^{2} + 220 T^{3} + 1191 T^{4} + 4386 T^{5} + 19575 T^{6} + 80376 T^{7} + 19575 p T^{8} + 4386 p^{2} T^{9} + 1191 p^{3} T^{10} + 220 p^{4} T^{11} + 33 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 4 T + 31 T^{2} - 44 T^{3} + 188 T^{4} - 428 T^{5} + 5748 T^{6} - 25736 T^{7} + 5748 p T^{8} - 428 p^{2} T^{9} + 188 p^{3} T^{10} - 44 p^{4} T^{11} + 31 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 8 T + 59 T^{2} + 396 T^{3} + 2091 T^{4} + 10760 T^{5} + 53985 T^{6} + 229640 T^{7} + 53985 p T^{8} + 10760 p^{2} T^{9} + 2091 p^{3} T^{10} + 396 p^{4} T^{11} + 59 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 2 T + 105 T^{2} + 246 T^{3} + 5960 T^{4} + 13694 T^{5} + 243530 T^{6} + 477700 T^{7} + 243530 p T^{8} + 13694 p^{2} T^{9} + 5960 p^{3} T^{10} + 246 p^{4} T^{11} + 105 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 8 T + 55 T^{2} + 264 T^{3} + 2080 T^{4} + 8824 T^{5} + 51350 T^{6} + 222192 T^{7} + 51350 p T^{8} + 8824 p^{2} T^{9} + 2080 p^{3} T^{10} + 264 p^{4} T^{11} + 55 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 16 T + 219 T^{2} + 1918 T^{3} + 14316 T^{4} + 90054 T^{5} + 13656 p T^{6} + 3098944 T^{7} + 13656 p^{2} T^{8} + 90054 p^{2} T^{9} + 14316 p^{3} T^{10} + 1918 p^{4} T^{11} + 219 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 2 T + 97 T^{2} + 490 T^{3} + 7340 T^{4} + 35154 T^{5} + 389214 T^{6} + 1901076 T^{7} + 389214 p T^{8} + 35154 p^{2} T^{9} + 7340 p^{3} T^{10} + 490 p^{4} T^{11} + 97 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 10 T + 137 T^{2} + 1404 T^{3} + 12929 T^{4} + 101862 T^{5} + 740097 T^{6} + 5407112 T^{7} + 740097 p T^{8} + 101862 p^{2} T^{9} + 12929 p^{3} T^{10} + 1404 p^{4} T^{11} + 137 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 8 T + 167 T^{2} - 764 T^{3} + 8075 T^{4} + 4920 T^{5} + 8277 T^{6} + 2172568 T^{7} + 8277 p T^{8} + 4920 p^{2} T^{9} + 8075 p^{3} T^{10} - 764 p^{4} T^{11} + 167 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 10 T + 227 T^{2} - 1440 T^{3} + 420 p T^{4} - 103448 T^{5} + 27076 p T^{6} - 5542636 T^{7} + 27076 p^{2} T^{8} - 103448 p^{2} T^{9} + 420 p^{4} T^{10} - 1440 p^{4} T^{11} + 227 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 24 T + 439 T^{2} - 5646 T^{3} + 62028 T^{4} - 562974 T^{5} + 4793650 T^{6} - 37001200 T^{7} + 4793650 p T^{8} - 562974 p^{2} T^{9} + 62028 p^{3} T^{10} - 5646 p^{4} T^{11} + 439 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 8 T + 149 T^{2} - 940 T^{3} + 15171 T^{4} - 92440 T^{5} + 1152647 T^{6} - 6156392 T^{7} + 1152647 p T^{8} - 92440 p^{2} T^{9} + 15171 p^{3} T^{10} - 940 p^{4} T^{11} + 149 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 20 T + 293 T^{2} + 3462 T^{3} + 36580 T^{4} + 343950 T^{5} + 3091200 T^{6} + 26654016 T^{7} + 3091200 p T^{8} + 343950 p^{2} T^{9} + 36580 p^{3} T^{10} + 3462 p^{4} T^{11} + 293 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 8 T + 207 T^{2} - 982 T^{3} + 272 p T^{4} - 33686 T^{5} + 15266 p T^{6} + 537032 T^{7} + 15266 p^{2} T^{8} - 33686 p^{2} T^{9} + 272 p^{4} T^{10} - 982 p^{4} T^{11} + 207 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 2 T + 185 T^{2} - 68 T^{3} + 25315 T^{4} - 14086 T^{5} + 2389155 T^{6} - 335144 T^{7} + 2389155 p T^{8} - 14086 p^{2} T^{9} + 25315 p^{3} T^{10} - 68 p^{4} T^{11} + 185 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 2 T + 281 T^{2} + 692 T^{3} + 42565 T^{4} + 120670 T^{5} + 4537101 T^{6} + 12474200 T^{7} + 4537101 p T^{8} + 120670 p^{2} T^{9} + 42565 p^{3} T^{10} + 692 p^{4} T^{11} + 281 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 22 T + 553 T^{2} - 8176 T^{3} + 127120 T^{4} - 1444144 T^{5} + 16806392 T^{6} - 151937092 T^{7} + 16806392 p T^{8} - 1444144 p^{2} T^{9} + 127120 p^{3} T^{10} - 8176 p^{4} T^{11} + 553 p^{5} T^{12} - 22 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 16 T + 347 T^{2} - 3840 T^{3} + 58753 T^{4} - 571120 T^{5} + 7185331 T^{6} - 59722240 T^{7} + 7185331 p T^{8} - 571120 p^{2} T^{9} + 58753 p^{3} T^{10} - 3840 p^{4} T^{11} + 347 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 4 T + 439 T^{2} - 1848 T^{3} + 97605 T^{4} - 391100 T^{5} + 13827779 T^{6} - 48684048 T^{7} + 13827779 p T^{8} - 391100 p^{2} T^{9} + 97605 p^{3} T^{10} - 1848 p^{4} T^{11} + 439 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.62697526250809974222098268465, −3.58703329016221771538373609873, −3.44480098219122757321893823108, −3.15795390529430113532167720822, −2.92485064099297848048926575309, −2.89315226302397280941841447118, −2.84913517630237701829382019422, −2.70753988461150631540574935338, −2.68313812389255078473513518096, −2.61337326127592544255698879410, −2.52526045505017375436786463193, −1.89312989709893550645436675065, −1.85521080649305126421985330562, −1.85165958755323143916787719184, −1.83640526174772949237104591922, −1.80304550695603007254987775038, −1.58850041425631154559592112524, −1.58667384438298413357274807031, −0.852421312517057076272356270917, −0.73284740694536883638330308932, −0.72027251833946690790423496568, −0.55705924827714916080852002623, −0.43575992711945093142150898099, −0.41353656776546325059939378381, −0.10897282743110876172984305429, 0.10897282743110876172984305429, 0.41353656776546325059939378381, 0.43575992711945093142150898099, 0.55705924827714916080852002623, 0.72027251833946690790423496568, 0.73284740694536883638330308932, 0.852421312517057076272356270917, 1.58667384438298413357274807031, 1.58850041425631154559592112524, 1.80304550695603007254987775038, 1.83640526174772949237104591922, 1.85165958755323143916787719184, 1.85521080649305126421985330562, 1.89312989709893550645436675065, 2.52526045505017375436786463193, 2.61337326127592544255698879410, 2.68313812389255078473513518096, 2.70753988461150631540574935338, 2.84913517630237701829382019422, 2.89315226302397280941841447118, 2.92485064099297848048926575309, 3.15795390529430113532167720822, 3.44480098219122757321893823108, 3.58703329016221771538373609873, 3.62697526250809974222098268465
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.