Dirichlet series
| L(s) = 1 | + 42·7-s − 42·9-s − 204·19-s + 153·25-s − 1.45e3·31-s + 240·37-s + 273·49-s + 2.14e3·61-s − 1.76e3·63-s − 1.98e3·67-s − 3.08e3·73-s + 438·79-s + 1.13e3·81-s + 1.38e3·103-s + 1.29e3·109-s − 2.01e3·121-s + 127-s + 131-s − 8.56e3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.91e3·169-s + ⋯ |
| L(s) = 1 | + 2.26·7-s − 1.55·9-s − 2.46·19-s + 1.22·25-s − 8.44·31-s + 1.06·37-s + 0.795·49-s + 4.50·61-s − 3.52·63-s − 3.61·67-s − 4.94·73-s + 0.623·79-s + 14/9·81-s + 1.32·103-s + 1.13·109-s − 1.51·121-s + 0.000698·127-s + 0.000666·131-s − 5.58·133-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.32·169-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 3^{12} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.68522\times 10^{15}\) |
| Root analytic conductor: | \(4.45248\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{48} \cdot 3^{12} \cdot 7^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(1.907080887\) |
| \(L(\frac12)\) | \(\approx\) | \(1.907080887\) |
| \(L(\frac{5}{2})\) | 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 | \( 1 + 14 p T^{2} + 70 p^{2} T^{4} - 224 p^{3} T^{5} + 562 p^{3} T^{6} - 224 p^{6} T^{7} + 70 p^{8} T^{8} + 14 p^{13} T^{10} + p^{18} T^{12} \) | |
| 7 | \( ( 1 - 3 p T + 75 p T^{2} - 230 p^{2} T^{3} + 75 p^{4} T^{4} - 3 p^{7} T^{5} + p^{9} T^{6} )^{2} \) | |
| good | 5 | \( 1 - 153 T^{2} + 4023 T^{4} + 4723304 T^{6} - 515325447 T^{8} - 19197188991 T^{10} + 10823738069886 T^{12} - 19197188991 p^{6} T^{14} - 515325447 p^{12} T^{16} + 4723304 p^{18} T^{18} + 4023 p^{24} T^{20} - 153 p^{30} T^{22} + p^{36} T^{24} \) |
| 11 | \( 1 + 2019 T^{2} + 1867287 T^{4} - 2418805472 T^{6} - 5977777080879 T^{8} - 169839608003577 p T^{10} + 12488769899646846 p^{2} T^{12} - 169839608003577 p^{7} T^{14} - 5977777080879 p^{12} T^{16} - 2418805472 p^{18} T^{18} + 1867287 p^{24} T^{20} + 2019 p^{30} T^{22} + p^{36} T^{24} \) | |
| 13 | \( ( 1 - 1458 T^{2} + 12022119 T^{4} - 10536506876 T^{6} + 12022119 p^{6} T^{8} - 1458 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 17 | \( 1 - 9576 T^{2} + 21936096 T^{4} + 245735653964 T^{6} - 1192193818198992 T^{8} - 5203415829879660456 T^{10} + \)\(65\!\cdots\!66\)\( T^{12} - 5203415829879660456 p^{6} T^{14} - 1192193818198992 p^{12} T^{16} + 245735653964 p^{18} T^{18} + 21936096 p^{24} T^{20} - 9576 p^{30} T^{22} + p^{36} T^{24} \) | |
| 19 | \( ( 1 + 102 T + 18078 T^{2} + 1490220 T^{3} + 150002706 T^{4} + 10997564478 T^{5} + 1045197598246 T^{6} + 10997564478 p^{3} T^{7} + 150002706 p^{6} T^{8} + 1490220 p^{9} T^{9} + 18078 p^{12} T^{10} + 102 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 23 | \( 1 + 41664 T^{2} + 1001918568 T^{4} + 12327072546316 T^{6} + 49810596479841960 T^{8} - \)\(14\!\cdots\!68\)\( T^{10} - \)\(28\!\cdots\!10\)\( T^{12} - \)\(14\!\cdots\!68\)\( p^{6} T^{14} + 49810596479841960 p^{12} T^{16} + 12327072546316 p^{18} T^{18} + 1001918568 p^{24} T^{20} + 41664 p^{30} T^{22} + p^{36} T^{24} \) | |
| 29 | \( ( 1 - 90957 T^{2} + 3685482627 T^{4} - 100535121841678 T^{6} + 3685482627 p^{6} T^{8} - 90957 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 31 | \( ( 1 + 729 T + 312465 T^{2} + 98646822 T^{3} + 25465575543 T^{4} + 5582021984433 T^{5} + 1041731236510702 T^{6} + 5582021984433 p^{3} T^{7} + 25465575543 p^{6} T^{8} + 98646822 p^{9} T^{9} + 312465 p^{12} T^{10} + 729 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 37 | \( ( 1 - 120 T - 131448 T^{2} + 5838484 T^{3} + 12483585720 T^{4} - 236917823976 T^{5} - 718303568739474 T^{6} - 236917823976 p^{3} T^{7} + 12483585720 p^{6} T^{8} + 5838484 p^{9} T^{9} - 131448 p^{12} T^{10} - 120 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 41 | \( ( 1 + 284094 T^{2} + 38573492751 T^{4} + 3254645758116292 T^{6} + 38573492751 p^{6} T^{8} + 284094 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 43 | \( ( 1 + 137673 T^{2} - 12209024 T^{3} + 137673 p^{3} T^{4} + p^{9} T^{6} )^{4} \) | |
| 47 | \( 1 - 90936 T^{2} + 9775711944 T^{4} - 2661755644478884 T^{6} + \)\(14\!\cdots\!12\)\( T^{8} - \)\(15\!\cdots\!88\)\( T^{10} + \)\(34\!\cdots\!62\)\( T^{12} - \)\(15\!\cdots\!88\)\( p^{6} T^{14} + \)\(14\!\cdots\!12\)\( p^{12} T^{16} - 2661755644478884 p^{18} T^{18} + 9775711944 p^{24} T^{20} - 90936 p^{30} T^{22} + p^{36} T^{24} \) | |
| 53 | \( 1 + 289047 T^{2} + 67795147191 T^{4} + 7626248096694088 T^{6} + \)\(34\!\cdots\!89\)\( T^{8} - \)\(84\!\cdots\!19\)\( T^{10} - \)\(22\!\cdots\!46\)\( T^{12} - \)\(84\!\cdots\!19\)\( p^{6} T^{14} + \)\(34\!\cdots\!89\)\( p^{12} T^{16} + 7626248096694088 p^{18} T^{18} + 67795147191 p^{24} T^{20} + 289047 p^{30} T^{22} + p^{36} T^{24} \) | |
| 59 | \( 1 - 682605 T^{2} + 202333862871 T^{4} - 47742310131832768 T^{6} + \)\(22\!\cdots\!83\)\( p T^{8} - \)\(34\!\cdots\!19\)\( T^{10} + \)\(75\!\cdots\!62\)\( T^{12} - \)\(34\!\cdots\!19\)\( p^{6} T^{14} + \)\(22\!\cdots\!83\)\( p^{13} T^{16} - 47742310131832768 p^{18} T^{18} + 202333862871 p^{24} T^{20} - 682605 p^{30} T^{22} + p^{36} T^{24} \) | |
| 61 | \( ( 1 - 1074 T + 1045506 T^{2} - 709929036 T^{3} + 445184851902 T^{4} - 236922017752806 T^{5} + 121139495988089182 T^{6} - 236922017752806 p^{3} T^{7} + 445184851902 p^{6} T^{8} - 709929036 p^{9} T^{9} + 1045506 p^{12} T^{10} - 1074 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 67 | \( ( 1 + 990 T + 105192 T^{2} - 111231284 T^{3} + 18515902680 T^{4} - 3900057863994 T^{5} - 25645540430114634 T^{6} - 3900057863994 p^{3} T^{7} + 18515902680 p^{6} T^{8} - 111231284 p^{9} T^{9} + 105192 p^{12} T^{10} + 990 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 71 | \( ( 1 - 1325730 T^{2} + 947795292591 T^{4} - 415349195473127932 T^{6} + 947795292591 p^{6} T^{8} - 1325730 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 73 | \( ( 1 + 1542 T + 2061258 T^{2} + 1956289140 T^{3} + 1700342003046 T^{4} + 1208959068686442 T^{5} + 819269771023534534 T^{6} + 1208959068686442 p^{3} T^{7} + 1700342003046 p^{6} T^{8} + 1956289140 p^{9} T^{9} + 2061258 p^{12} T^{10} + 1542 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 79 | \( ( 1 - 219 T - 1229151 T^{2} + 54253534 T^{3} + 964782490887 T^{4} + 9317496689613 T^{5} - 548230401702137298 T^{6} + 9317496689613 p^{3} T^{7} + 964782490887 p^{6} T^{8} + 54253534 p^{9} T^{9} - 1229151 p^{12} T^{10} - 219 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 83 | \( ( 1 + 2213631 T^{2} + 2402637322995 T^{4} + 1655758504150116154 T^{6} + 2402637322995 p^{6} T^{8} + 2213631 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 89 | \( 1 - 337656 T^{2} - 733502310000 T^{4} + 455056539179522540 T^{6} + \)\(17\!\cdots\!32\)\( T^{8} - \)\(11\!\cdots\!32\)\( T^{10} - \)\(22\!\cdots\!66\)\( T^{12} - \)\(11\!\cdots\!32\)\( p^{6} T^{14} + \)\(17\!\cdots\!32\)\( p^{12} T^{16} + 455056539179522540 p^{18} T^{18} - 733502310000 p^{24} T^{20} - 337656 p^{30} T^{22} + p^{36} T^{24} \) | |
| 97 | \( ( 1 - 2338539 T^{2} + 36704173443 p T^{4} - 3525808839119151986 T^{6} + 36704173443 p^{7} T^{8} - 2338539 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.35317207589350624478412401089, −3.30265957259891093114153877546, −3.27322842652311513757966153395, −3.09352736597239194338829969736, −3.02962311354107885460866945459, −2.73533972315050766053938166798, −2.73116302795668080707118204990, −2.37466706850550428458536471400, −2.37020079348961116425062769820, −2.35248593123528744118637814406, −2.20537378942367203700598505565, −2.05832932290059607892095484374, −1.79585833126338168801890229506, −1.72759779144827027252304579029, −1.70515644110826213685323374163, −1.61632570407636255572883128080, −1.59555367792608793782634159577, −1.40010046073328580279215718376, −1.29848403703844992942965928584, −0.62812023348324703906889868824, −0.60472237426237063462739737041, −0.58226258138946958261864647228, −0.47970838043604534339964665514, −0.43143123580394007792583441018, −0.06777483802828525081241196429, 0.06777483802828525081241196429, 0.43143123580394007792583441018, 0.47970838043604534339964665514, 0.58226258138946958261864647228, 0.60472237426237063462739737041, 0.62812023348324703906889868824, 1.29848403703844992942965928584, 1.40010046073328580279215718376, 1.59555367792608793782634159577, 1.61632570407636255572883128080, 1.70515644110826213685323374163, 1.72759779144827027252304579029, 1.79585833126338168801890229506, 2.05832932290059607892095484374, 2.20537378942367203700598505565, 2.35248593123528744118637814406, 2.37020079348961116425062769820, 2.37466706850550428458536471400, 2.73116302795668080707118204990, 2.73533972315050766053938166798, 3.02962311354107885460866945459, 3.09352736597239194338829969736, 3.27322842652311513757966153395, 3.30265957259891093114153877546, 3.35317207589350624478412401089
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.