Dirichlet series
| L(s) = 1 | − 42·7-s + 186·19-s + 252·25-s + 738·31-s − 336·37-s + 756·43-s + 273·49-s − 426·61-s − 1.44e3·67-s + 2.73e3·73-s + 1.99e3·79-s + 744·103-s − 4.55e3·109-s − 2.26e3·121-s + 127-s + 131-s − 7.81e3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.85e4·169-s + 173-s − 1.05e4·175-s + ⋯ |
| L(s) = 1 | − 2.26·7-s + 2.24·19-s + 2.01·25-s + 4.27·31-s − 1.49·37-s + 2.68·43-s + 0.795·49-s − 0.894·61-s − 2.62·67-s + 4.37·73-s + 2.83·79-s + 0.711·103-s − 4.00·109-s − 1.69·121-s + 0.000698·127-s + 0.000666·131-s − 5.09·133-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 + 8.42·169-s + 0.000439·173-s − 4.57·175-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{36} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{36} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 3^{36} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.20375\times 10^{19}\) |
| Root analytic conductor: | \(6.67873\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 3^{36} \cdot 7^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.7556787179\) |
| \(L(\frac12)\) | \(\approx\) | \(0.7556787179\) |
| \(L(\frac{5}{2})\) | 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 | \( 1 \) | |
| 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 | \( 1 - 252 T^{2} - 2007 T^{4} + 96044 T^{6} + 235174986 p T^{8} - 89523362988 T^{10} - 6002206885911 T^{12} - 89523362988 p^{6} T^{14} + 235174986 p^{13} T^{16} + 96044 p^{18} T^{18} - 2007 p^{24} T^{20} - 252 p^{30} T^{22} + p^{36} T^{24} \) |
| 11 | \( 1 + 2262 T^{2} + 1152381 T^{4} - 1993184654 T^{6} - 3703705681734 T^{8} - 2947777032065490 T^{10} - 2746837752853299771 T^{12} - 2947777032065490 p^{6} T^{14} - 3703705681734 p^{12} T^{16} - 1993184654 p^{18} T^{18} + 1152381 p^{24} T^{20} + 2262 p^{30} T^{22} + p^{36} T^{24} \) | |
| 13 | \( ( 1 - 9252 T^{2} + 40000104 T^{4} - 107401034198 T^{6} + 40000104 p^{6} T^{8} - 9252 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 17 | \( 1 - 9540 T^{2} + 21877281 T^{4} + 70007208692 T^{6} - 369127210547790 T^{8} + 1062812883569839596 T^{10} - \)\(72\!\cdots\!15\)\( T^{12} + 1062812883569839596 p^{6} T^{14} - 369127210547790 p^{12} T^{16} + 70007208692 p^{18} T^{18} + 21877281 p^{24} T^{20} - 9540 p^{30} T^{22} + p^{36} T^{24} \) | |
| 19 | \( ( 1 - 93 T + 20364 T^{2} - 1625733 T^{3} + 225498900 T^{4} - 18076944885 T^{5} + 1864876936246 T^{6} - 18076944885 p^{3} T^{7} + 225498900 p^{6} T^{8} - 1625733 p^{9} T^{9} + 20364 p^{12} T^{10} - 93 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 23 | \( 1 + 65208 T^{2} + 2392610457 T^{4} + 62190399053704 T^{6} + 1256709579409953018 T^{8} + \)\(20\!\cdots\!32\)\( T^{10} + \)\(27\!\cdots\!09\)\( T^{12} + \)\(20\!\cdots\!32\)\( p^{6} T^{14} + 1256709579409953018 p^{12} T^{16} + 62190399053704 p^{18} T^{18} + 2392610457 p^{24} T^{20} + 65208 p^{30} T^{22} + p^{36} T^{24} \) | |
| 29 | \( ( 1 - 55128 T^{2} + 1152955527 T^{4} - 16578040471792 T^{6} + 1152955527 p^{6} T^{8} - 55128 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 31 | \( ( 1 - 369 T + 121377 T^{2} - 28040310 T^{3} + 6206799255 T^{4} - 1338197883561 T^{5} + 239026592646286 T^{6} - 1338197883561 p^{3} T^{7} + 6206799255 p^{6} T^{8} - 28040310 p^{9} T^{9} + 121377 p^{12} T^{10} - 369 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 37 | \( ( 1 + 168 T - 22134 T^{2} + 12842392 T^{3} + 353253222 T^{4} - 536361933744 T^{5} + 63857072514822 T^{6} - 536361933744 p^{3} T^{7} + 353253222 p^{6} T^{8} + 12842392 p^{9} T^{9} - 22134 p^{12} T^{10} + 168 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 41 | \( ( 1 + 198036 T^{2} + 24786408159 T^{4} + 2010074092026136 T^{6} + 24786408159 p^{6} T^{8} + 198036 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 43 | \( ( 1 - 189 T + 215136 T^{2} - 28810969 T^{3} + 215136 p^{3} T^{4} - 189 p^{6} T^{5} + p^{9} T^{6} )^{4} \) | |
| 47 | \( 1 - 344376 T^{2} + 60551620065 T^{4} - 6419291338225768 T^{6} + \)\(40\!\cdots\!18\)\( T^{8} - \)\(28\!\cdots\!52\)\( T^{10} - \)\(15\!\cdots\!91\)\( T^{12} - \)\(28\!\cdots\!52\)\( p^{6} T^{14} + \)\(40\!\cdots\!18\)\( p^{12} T^{16} - 6419291338225768 p^{18} T^{18} + 60551620065 p^{24} T^{20} - 344376 p^{30} T^{22} + p^{36} T^{24} \) | |
| 53 | \( 1 + 841980 T^{2} + 406567853265 T^{4} + 135973155202150036 T^{6} + \)\(34\!\cdots\!50\)\( T^{8} + \)\(70\!\cdots\!00\)\( T^{10} + \)\(11\!\cdots\!77\)\( T^{12} + \)\(70\!\cdots\!00\)\( p^{6} T^{14} + \)\(34\!\cdots\!50\)\( p^{12} T^{16} + 135973155202150036 p^{18} T^{18} + 406567853265 p^{24} T^{20} + 841980 p^{30} T^{22} + p^{36} T^{24} \) | |
| 59 | \( 1 + 33930 T^{2} + 38826258813 T^{4} + 5616950881151534 T^{6} + \)\(31\!\cdots\!62\)\( T^{8} + \)\(29\!\cdots\!66\)\( T^{10} + \)\(10\!\cdots\!57\)\( T^{12} + \)\(29\!\cdots\!66\)\( p^{6} T^{14} + \)\(31\!\cdots\!62\)\( p^{12} T^{16} + 5616950881151534 p^{18} T^{18} + 38826258813 p^{24} T^{20} + 33930 p^{30} T^{22} + p^{36} T^{24} \) | |
| 61 | \( ( 1 + 213 T + 565635 T^{2} + 117259056 T^{3} + 186573295329 T^{4} + 54665779799811 T^{5} + 49361109958114342 T^{6} + 54665779799811 p^{3} T^{7} + 186573295329 p^{6} T^{8} + 117259056 p^{9} T^{9} + 565635 p^{12} T^{10} + 213 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 67 | \( ( 1 + 720 T - 395802 T^{2} - 126377356 T^{3} + 278350154118 T^{4} + 27833041794636 T^{5} - 89947447613726754 T^{6} + 27833041794636 p^{3} T^{7} + 278350154118 p^{6} T^{8} - 126377356 p^{9} T^{9} - 395802 p^{12} T^{10} + 720 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 71 | \( ( 1 - 966630 T^{2} + 694440115647 T^{4} - 280667473593553492 T^{6} + 694440115647 p^{6} T^{8} - 966630 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 73 | \( ( 1 - 1365 T + 17934 p T^{2} - 939266055 T^{3} + 452440890300 T^{4} - 166420534299033 T^{5} + 70404227186208424 T^{6} - 166420534299033 p^{3} T^{7} + 452440890300 p^{6} T^{8} - 939266055 p^{9} T^{9} + 17934 p^{13} T^{10} - 1365 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 79 | \( ( 1 - 996 T - 744744 T^{2} + 272772248 T^{3} + 1162757450376 T^{4} - 275389544651076 T^{5} - 486027513279189282 T^{6} - 275389544651076 p^{3} T^{7} + 1162757450376 p^{6} T^{8} + 272772248 p^{9} T^{9} - 744744 p^{12} T^{10} - 996 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 83 | \( ( 1 + 387450 T^{2} + 223986727431 T^{4} + 196949419681664428 T^{6} + 223986727431 p^{6} T^{8} + 387450 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 89 | \( 1 - 2900820 T^{2} + 4204844963409 T^{4} - 4797069783497410684 T^{6} + \)\(48\!\cdots\!70\)\( T^{8} - \)\(41\!\cdots\!68\)\( T^{10} + \)\(31\!\cdots\!45\)\( T^{12} - \)\(41\!\cdots\!68\)\( p^{6} T^{14} + \)\(48\!\cdots\!70\)\( p^{12} T^{16} - 4797069783497410684 p^{18} T^{18} + 4204844963409 p^{24} T^{20} - 2900820 p^{30} T^{22} + p^{36} T^{24} \) | |
| 97 | \( ( 1 - 4297605 T^{2} + 8591021261430 T^{4} - 10000867429565198417 T^{6} + 8591021261430 p^{6} T^{8} - 4297605 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.81692940164406852915031461384, −2.75080373007396616856289703838, −2.66579071355549317393076311147, −2.59147020630782306920936577374, −2.58462676684205134379619391065, −2.55882731826056423465880249974, −2.30028571933249977563891683886, −2.28168344070002638234122909250, −2.11681046849539970798280743709, −1.88260850012461048774750058184, −1.82287429097849120517522064615, −1.77351709545356571697950552677, −1.45685099193859315041131106376, −1.44152882885183632520027543238, −1.38271176782227245668620498326, −1.13796344091312212568622214545, −1.07897875627408515459387226148, −0.884236406624797257753993546156, −0.802534698701013831433879894440, −0.802364738590720348404173934885, −0.74324863002087904343055989322, −0.61363081667639528766752435429, −0.29921776641251474261957497415, −0.13538895343395155514021467339, −0.06181434714995350480738663833, 0.06181434714995350480738663833, 0.13538895343395155514021467339, 0.29921776641251474261957497415, 0.61363081667639528766752435429, 0.74324863002087904343055989322, 0.802364738590720348404173934885, 0.802534698701013831433879894440, 0.884236406624797257753993546156, 1.07897875627408515459387226148, 1.13796344091312212568622214545, 1.38271176782227245668620498326, 1.44152882885183632520027543238, 1.45685099193859315041131106376, 1.77351709545356571697950552677, 1.82287429097849120517522064615, 1.88260850012461048774750058184, 2.11681046849539970798280743709, 2.28168344070002638234122909250, 2.30028571933249977563891683886, 2.55882731826056423465880249974, 2.58462676684205134379619391065, 2.59147020630782306920936577374, 2.66579071355549317393076311147, 2.75080373007396616856289703838, 2.81692940164406852915031461384
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.