Dirichlet series
| L(s) = 1 | + 12·4-s − 10·7-s − 18·9-s − 16·11-s + 84·16-s − 120·28-s + 48·29-s − 216·36-s + 64·37-s − 108·43-s − 192·44-s + 109·49-s + 176·53-s + 180·63-s + 448·64-s − 4·67-s + 248·71-s + 160·77-s − 208·79-s + 189·81-s + 288·99-s + 576·107-s − 212·109-s − 840·112-s + 56·113-s + 576·116-s − 336·121-s + ⋯ |
| L(s) = 1 | + 3·4-s − 1.42·7-s − 2·9-s − 1.45·11-s + 21/4·16-s − 4.28·28-s + 1.65·29-s − 6·36-s + 1.72·37-s − 2.51·43-s − 4.36·44-s + 2.22·49-s + 3.32·53-s + 20/7·63-s + 7·64-s − 0.0597·67-s + 3.49·71-s + 2.07·77-s − 2.63·79-s + 7/3·81-s + 2.90·99-s + 5.38·107-s − 1.94·109-s − 7.5·112-s + 0.495·113-s + 4.96·116-s − 2.77·121-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{12} \cdot 5^{24} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.00809\times 10^{17}\) |
| Root analytic conductor: | \(5.34887\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{12} \cdot 5^{24} \cdot 7^{12} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(6.527144132\) |
| \(L(\frac12)\) | \(\approx\) | \(6.527144132\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T^{2} )^{6} \) |
| 3 | \( ( 1 + p T^{2} )^{6} \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + 10 T - 9 T^{2} - 950 T^{3} - 5685 T^{4} + 3860 p T^{5} + 9306 p^{2} T^{6} + 3860 p^{3} T^{7} - 5685 p^{4} T^{8} - 950 p^{6} T^{9} - 9 p^{8} T^{10} + 10 p^{10} T^{11} + p^{12} T^{12} \) | |
| good | 11 | \( ( 1 + 8 T + 24 p T^{2} + 60 T^{3} + 23224 T^{4} - 140920 T^{5} + 2574190 T^{6} - 140920 p^{2} T^{7} + 23224 p^{4} T^{8} + 60 p^{6} T^{9} + 24 p^{9} T^{10} + 8 p^{10} T^{11} + p^{12} T^{12} )^{2} \) |
| 13 | \( 1 - 942 T^{2} + 472295 T^{4} - 167763302 T^{6} + 46197204579 T^{8} - 10315226989484 T^{10} + 1909797010892858 T^{12} - 10315226989484 p^{4} T^{14} + 46197204579 p^{8} T^{16} - 167763302 p^{12} T^{18} + 472295 p^{16} T^{20} - 942 p^{20} T^{22} + p^{24} T^{24} \) | |
| 17 | \( 1 - 2156 T^{2} + 2241434 T^{4} - 1512989724 T^{6} + 751159530847 T^{8} - 293252645911096 T^{10} + 322909971725836 p^{2} T^{12} - 293252645911096 p^{4} T^{14} + 751159530847 p^{8} T^{16} - 1512989724 p^{12} T^{18} + 2241434 p^{16} T^{20} - 2156 p^{20} T^{22} + p^{24} T^{24} \) | |
| 19 | \( 1 - 2574 T^{2} + 3125495 T^{4} - 2377155734 T^{6} + 1290078672771 T^{8} - 554626723727324 T^{10} + 208841754502278746 T^{12} - 554626723727324 p^{4} T^{14} + 1290078672771 p^{8} T^{16} - 2377155734 p^{12} T^{18} + 3125495 p^{16} T^{20} - 2574 p^{20} T^{22} + p^{24} T^{24} \) | |
| 23 | \( ( 1 + 1880 T^{2} + 7740 T^{3} + 77400 p T^{4} + 10341120 T^{5} + 1112392990 T^{6} + 10341120 p^{2} T^{7} + 77400 p^{5} T^{8} + 7740 p^{6} T^{9} + 1880 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 29 | \( ( 1 - 24 T + 2468 T^{2} - 46500 T^{3} + 3264648 T^{4} - 44867280 T^{5} + 3003323062 T^{6} - 44867280 p^{2} T^{7} + 3264648 p^{4} T^{8} - 46500 p^{6} T^{9} + 2468 p^{8} T^{10} - 24 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 31 | \( 1 - 4230 T^{2} + 9088079 T^{4} - 14633333366 T^{6} + 19981620474843 T^{8} - 753901615698236 p T^{10} + 23842846561092500522 T^{12} - 753901615698236 p^{5} T^{14} + 19981620474843 p^{8} T^{16} - 14633333366 p^{12} T^{18} + 9088079 p^{16} T^{20} - 4230 p^{20} T^{22} + p^{24} T^{24} \) | |
| 37 | \( ( 1 - 32 T + 5376 T^{2} - 127652 T^{3} + 14018592 T^{4} - 278870752 T^{5} + 23711833206 T^{6} - 278870752 p^{2} T^{7} + 14018592 p^{4} T^{8} - 127652 p^{6} T^{9} + 5376 p^{8} T^{10} - 32 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 41 | \( 1 - 9756 T^{2} + 45272162 T^{4} - 134799694124 T^{6} + 295285070608431 T^{8} - 532039008457912376 T^{10} + \)\(89\!\cdots\!48\)\( T^{12} - 532039008457912376 p^{4} T^{14} + 295285070608431 p^{8} T^{16} - 134799694124 p^{12} T^{18} + 45272162 p^{16} T^{20} - 9756 p^{20} T^{22} + p^{24} T^{24} \) | |
| 43 | \( ( 1 + 54 T + 5873 T^{2} + 78578 T^{3} + 5791254 T^{4} - 440546410 T^{5} - 5275967779 T^{6} - 440546410 p^{2} T^{7} + 5791254 p^{4} T^{8} + 78578 p^{6} T^{9} + 5873 p^{8} T^{10} + 54 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 47 | \( 1 - 12108 T^{2} + 66491354 T^{4} - 201576695612 T^{6} + 297824218172127 T^{8} + 93650953492723912 T^{10} - \)\(10\!\cdots\!36\)\( T^{12} + 93650953492723912 p^{4} T^{14} + 297824218172127 p^{8} T^{16} - 201576695612 p^{12} T^{18} + 66491354 p^{16} T^{20} - 12108 p^{20} T^{22} + p^{24} T^{24} \) | |
| 53 | \( ( 1 - 88 T + 12710 T^{2} - 646936 T^{3} + 56915823 T^{4} - 2001475024 T^{5} + 166001738548 T^{6} - 2001475024 p^{2} T^{7} + 56915823 p^{4} T^{8} - 646936 p^{6} T^{9} + 12710 p^{8} T^{10} - 88 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 59 | \( 1 - 29036 T^{2} + 418144058 T^{4} - 3912168300060 T^{6} + 26352049808161663 T^{8} - \)\(13\!\cdots\!00\)\( T^{10} + \)\(52\!\cdots\!12\)\( T^{12} - \)\(13\!\cdots\!00\)\( p^{4} T^{14} + 26352049808161663 p^{8} T^{16} - 3912168300060 p^{12} T^{18} + 418144058 p^{16} T^{20} - 29036 p^{20} T^{22} + p^{24} T^{24} \) | |
| 61 | \( 1 - 14550 T^{2} + 147830015 T^{4} - 1079303851430 T^{6} + 6377617535191227 T^{8} - 30778007090052182564 T^{10} + \)\(12\!\cdots\!18\)\( T^{12} - 30778007090052182564 p^{4} T^{14} + 6377617535191227 p^{8} T^{16} - 1079303851430 p^{12} T^{18} + 147830015 p^{16} T^{20} - 14550 p^{20} T^{22} + p^{24} T^{24} \) | |
| 67 | \( ( 1 + 2 T + 15625 T^{2} + 222934 T^{3} + 107068198 T^{4} + 2819337506 T^{5} + 512079807173 T^{6} + 2819337506 p^{2} T^{7} + 107068198 p^{4} T^{8} + 222934 p^{6} T^{9} + 15625 p^{8} T^{10} + 2 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 71 | \( ( 1 - 124 T + 22052 T^{2} - 2363176 T^{3} + 248446416 T^{4} - 20951950444 T^{5} + 22570671038 p T^{6} - 20951950444 p^{2} T^{7} + 248446416 p^{4} T^{8} - 2363176 p^{6} T^{9} + 22052 p^{8} T^{10} - 124 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 73 | \( 1 - 16420 T^{2} + 197735674 T^{4} - 1719348663284 T^{6} + 12390862178969503 T^{8} - 78297228167810338984 T^{10} + \)\(43\!\cdots\!52\)\( T^{12} - 78297228167810338984 p^{4} T^{14} + 12390862178969503 p^{8} T^{16} - 1719348663284 p^{12} T^{18} + 197735674 p^{16} T^{20} - 16420 p^{20} T^{22} + p^{24} T^{24} \) | |
| 79 | \( ( 1 + 104 T + 38652 T^{2} + 3101876 T^{3} + 609372096 T^{4} + 37770388048 T^{5} + 5081231518734 T^{6} + 37770388048 p^{2} T^{7} + 609372096 p^{4} T^{8} + 3101876 p^{6} T^{9} + 38652 p^{8} T^{10} + 104 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 83 | \( 1 - 31764 T^{2} + 595556354 T^{4} - 8113078567172 T^{6} + 88346115609234255 T^{8} - \)\(79\!\cdots\!84\)\( T^{10} + \)\(59\!\cdots\!96\)\( T^{12} - \)\(79\!\cdots\!84\)\( p^{4} T^{14} + 88346115609234255 p^{8} T^{16} - 8113078567172 p^{12} T^{18} + 595556354 p^{16} T^{20} - 31764 p^{20} T^{22} + p^{24} T^{24} \) | |
| 89 | \( 1 - 51924 T^{2} + 1298536346 T^{4} - 20815697619236 T^{6} + 243465083989872927 T^{8} - \)\(22\!\cdots\!40\)\( T^{10} + \)\(18\!\cdots\!88\)\( T^{12} - \)\(22\!\cdots\!40\)\( p^{4} T^{14} + 243465083989872927 p^{8} T^{16} - 20815697619236 p^{12} T^{18} + 1298536346 p^{16} T^{20} - 51924 p^{20} T^{22} + p^{24} T^{24} \) | |
| 97 | \( 1 - 2614 T^{2} + 293186815 T^{4} - 2480462083430 T^{6} + 33960899176336411 T^{8} - \)\(54\!\cdots\!20\)\( T^{10} + \)\(28\!\cdots\!34\)\( T^{12} - \)\(54\!\cdots\!20\)\( p^{4} T^{14} + 33960899176336411 p^{8} T^{16} - 2480462083430 p^{12} T^{18} + 293186815 p^{16} T^{20} - 2614 p^{20} T^{22} + p^{24} T^{24} \) | |
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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
−2.92450965134597822661830499951, −2.79968868672033984442757993566, −2.76167404062520683977149531509, −2.64469332724236390578048716460, −2.48871138579965087064377021140, −2.47192566438455107447146909126, −2.40112827603990107223317278624, −2.37296884584890061149721840786, −2.12181624648727018894897611351, −1.96788630345339568416487465339, −1.93135813650626187030593967487, −1.86387511487208812669097777917, −1.82596085724235917585483297040, −1.79156884898234180892119019358, −1.56607514538672291683405740663, −1.19645182395886082381967336403, −1.16450614876658358035143593066, −1.08234957674452935686043440528, −0.988984998638015660030774674793, −0.847547816738058128469007261673, −0.61418027144234117955949671007, −0.49673226279298219007116623888, −0.48567105583137830527059729474, −0.35044554725278513772008193102, −0.07647626731233447889420335894, 0.07647626731233447889420335894, 0.35044554725278513772008193102, 0.48567105583137830527059729474, 0.49673226279298219007116623888, 0.61418027144234117955949671007, 0.847547816738058128469007261673, 0.988984998638015660030774674793, 1.08234957674452935686043440528, 1.16450614876658358035143593066, 1.19645182395886082381967336403, 1.56607514538672291683405740663, 1.79156884898234180892119019358, 1.82596085724235917585483297040, 1.86387511487208812669097777917, 1.93135813650626187030593967487, 1.96788630345339568416487465339, 2.12181624648727018894897611351, 2.37296884584890061149721840786, 2.40112827603990107223317278624, 2.47192566438455107447146909126, 2.48871138579965087064377021140, 2.64469332724236390578048716460, 2.76167404062520683977149531509, 2.79968868672033984442757993566, 2.92450965134597822661830499951
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.