Dirichlet series
| L(s) = 1 | − 6·4-s − 4·5-s + 4·11-s + 20·13-s + 21·16-s + 8·17-s − 4·19-s + 24·20-s + 6·25-s + 12·31-s − 16·37-s + 8·41-s + 16·43-s − 24·44-s + 32·47-s − 52·49-s − 120·52-s − 16·53-s − 16·55-s + 20·59-s + 16·61-s − 56·64-s − 80·65-s − 48·68-s − 32·71-s + 24·76-s − 84·80-s + ⋯ |
| L(s) = 1 | − 3·4-s − 1.78·5-s + 1.20·11-s + 5.54·13-s + 21/4·16-s + 1.94·17-s − 0.917·19-s + 5.36·20-s + 6/5·25-s + 2.15·31-s − 2.63·37-s + 1.24·41-s + 2.43·43-s − 3.61·44-s + 4.66·47-s − 7.42·49-s − 16.6·52-s − 2.19·53-s − 2.15·55-s + 2.60·59-s + 2.04·61-s − 7·64-s − 9.92·65-s − 5.82·68-s − 3.79·71-s + 2.75·76-s − 9.39·80-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(831956.\) |
| Root analytic conductor: | \(1.76469\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1235649582\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1235649582\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{2} )^{6} \) |
| 3 | \( ( 1 + T^{4} )^{3} \) | |
| 5 | \( 1 + 4 T + 2 p T^{2} + 44 T^{3} + 111 T^{4} + 256 T^{5} + 716 T^{6} + 256 p T^{7} + 111 p^{2} T^{8} + 44 p^{3} T^{9} + 2 p^{5} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) | |
| 13 | \( 1 - 20 T + 188 T^{2} - 1036 T^{3} + 3467 T^{4} - 6672 T^{5} + 11656 T^{6} - 6672 p T^{7} + 3467 p^{2} T^{8} - 1036 p^{3} T^{9} + 188 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 7 | \( ( 1 + 26 T^{2} + 20 T^{3} + 311 T^{4} + 388 T^{5} + 2488 T^{6} + 388 p T^{7} + 311 p^{2} T^{8} + 20 p^{3} T^{9} + 26 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 11 | \( 1 - 4 T + 8 T^{2} - 20 T^{3} - 134 T^{4} + 404 T^{5} - 344 T^{6} - 1116 T^{7} + 27103 T^{8} - 53640 T^{9} + 55568 T^{10} + 100376 T^{11} - 2178260 T^{12} + 100376 p T^{13} + 55568 p^{2} T^{14} - 53640 p^{3} T^{15} + 27103 p^{4} T^{16} - 1116 p^{5} T^{17} - 344 p^{6} T^{18} + 404 p^{7} T^{19} - 134 p^{8} T^{20} - 20 p^{9} T^{21} + 8 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 8 T + 32 T^{2} - 200 T^{3} + 878 T^{4} - 1256 T^{5} + 1952 T^{6} - 328 T^{7} - 170641 T^{8} + 958992 T^{9} - 3070912 T^{10} + 16843760 T^{11} - 92096812 T^{12} + 16843760 p T^{13} - 3070912 p^{2} T^{14} + 958992 p^{3} T^{15} - 170641 p^{4} T^{16} - 328 p^{5} T^{17} + 1952 p^{6} T^{18} - 1256 p^{7} T^{19} + 878 p^{8} T^{20} - 200 p^{9} T^{21} + 32 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 4 T + 8 T^{2} + 124 T^{3} + 706 T^{4} + 4084 T^{5} + 18376 T^{6} + 80092 T^{7} + 580943 T^{8} + 2111848 T^{9} + 9097328 T^{10} + 50992840 T^{11} + 206057740 T^{12} + 50992840 p T^{13} + 9097328 p^{2} T^{14} + 2111848 p^{3} T^{15} + 580943 p^{4} T^{16} + 80092 p^{5} T^{17} + 18376 p^{6} T^{18} + 4084 p^{7} T^{19} + 706 p^{8} T^{20} + 124 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 104 T^{3} + 1214 T^{4} + 520 T^{5} + 5408 T^{6} + 101712 T^{7} + 917807 T^{8} + 1814384 T^{9} + 4147936 T^{10} + 58434688 T^{11} + 612580212 T^{12} + 58434688 p T^{13} + 4147936 p^{2} T^{14} + 1814384 p^{3} T^{15} + 917807 p^{4} T^{16} + 101712 p^{5} T^{17} + 5408 p^{6} T^{18} + 520 p^{7} T^{19} + 1214 p^{8} T^{20} + 104 p^{9} T^{21} + p^{12} T^{24} \) | |
| 29 | \( 1 - 16 T^{2} + 590 T^{4} - 36488 T^{6} - 10545 T^{8} + 15982936 T^{10} + 434859636 T^{12} + 15982936 p^{2} T^{14} - 10545 p^{4} T^{16} - 36488 p^{6} T^{18} + 590 p^{8} T^{20} - 16 p^{10} T^{22} + p^{12} T^{24} \) | |
| 31 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{6} \) | |
| 37 | \( ( 1 + 8 T + 104 T^{2} + 640 T^{3} + 6067 T^{4} + 37928 T^{5} + 290656 T^{6} + 37928 p T^{7} + 6067 p^{2} T^{8} + 640 p^{3} T^{9} + 104 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 - 8 T + 32 T^{2} - 616 T^{3} + 326 T^{4} + 19768 T^{5} + 21152 T^{6} + 1227928 T^{7} - 7391377 T^{8} - 34675920 T^{9} - 53980864 T^{10} - 343138192 T^{11} + 20532326676 T^{12} - 343138192 p T^{13} - 53980864 p^{2} T^{14} - 34675920 p^{3} T^{15} - 7391377 p^{4} T^{16} + 1227928 p^{5} T^{17} + 21152 p^{6} T^{18} + 19768 p^{7} T^{19} + 326 p^{8} T^{20} - 616 p^{9} T^{21} + 32 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 16 T + 128 T^{2} - 832 T^{3} + 3750 T^{4} - 23456 T^{5} + 241408 T^{6} - 2703888 T^{7} + 28876831 T^{8} - 193007936 T^{9} + 1011339392 T^{10} - 5003229408 T^{11} + 24640786964 T^{12} - 5003229408 p T^{13} + 1011339392 p^{2} T^{14} - 193007936 p^{3} T^{15} + 28876831 p^{4} T^{16} - 2703888 p^{5} T^{17} + 241408 p^{6} T^{18} - 23456 p^{7} T^{19} + 3750 p^{8} T^{20} - 832 p^{9} T^{21} + 128 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( ( 1 - 16 T + 258 T^{2} - 2960 T^{3} + 29759 T^{4} - 246080 T^{5} + 1855900 T^{6} - 246080 p T^{7} + 29759 p^{2} T^{8} - 2960 p^{3} T^{9} + 258 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 53 | \( 1 + 16 T + 128 T^{2} + 1184 T^{3} + 17430 T^{4} + 175264 T^{5} + 1274112 T^{6} + 11104144 T^{7} + 111947871 T^{8} + 870351008 T^{9} + 5894557824 T^{10} + 48596916160 T^{11} + 398492641204 T^{12} + 48596916160 p T^{13} + 5894557824 p^{2} T^{14} + 870351008 p^{3} T^{15} + 111947871 p^{4} T^{16} + 11104144 p^{5} T^{17} + 1274112 p^{6} T^{18} + 175264 p^{7} T^{19} + 17430 p^{8} T^{20} + 1184 p^{9} T^{21} + 128 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 20 T + 200 T^{2} - 1124 T^{3} + 410 T^{4} + 26308 T^{5} + 23528 T^{6} - 2001132 T^{7} + 10980959 T^{8} - 21459848 T^{9} + 818686480 T^{10} - 20962118376 T^{11} + 231452753004 T^{12} - 20962118376 p T^{13} + 818686480 p^{2} T^{14} - 21459848 p^{3} T^{15} + 10980959 p^{4} T^{16} - 2001132 p^{5} T^{17} + 23528 p^{6} T^{18} + 26308 p^{7} T^{19} + 410 p^{8} T^{20} - 1124 p^{9} T^{21} + 200 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( ( 1 - 8 T + 206 T^{2} - 1960 T^{3} + 22279 T^{4} - 214832 T^{5} + 1605892 T^{6} - 214832 p T^{7} + 22279 p^{2} T^{8} - 1960 p^{3} T^{9} + 206 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( 1 - 376 T^{2} + 77430 T^{4} - 11139832 T^{6} + 1238206463 T^{8} - 110880362064 T^{10} + 8171062800884 T^{12} - 110880362064 p^{2} T^{14} + 1238206463 p^{4} T^{16} - 11139832 p^{6} T^{18} + 77430 p^{8} T^{20} - 376 p^{10} T^{22} + p^{12} T^{24} \) | |
| 71 | \( 1 + 32 T + 512 T^{2} + 7152 T^{3} + 85350 T^{4} + 765968 T^{5} + 6387328 T^{6} + 52223392 T^{7} + 382047087 T^{8} + 3386151168 T^{9} + 34445472128 T^{10} + 325170274336 T^{11} + 2867009203028 T^{12} + 325170274336 p T^{13} + 34445472128 p^{2} T^{14} + 3386151168 p^{3} T^{15} + 382047087 p^{4} T^{16} + 52223392 p^{5} T^{17} + 6387328 p^{6} T^{18} + 765968 p^{7} T^{19} + 85350 p^{8} T^{20} + 7152 p^{9} T^{21} + 512 p^{10} T^{22} + 32 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 340 T^{2} + 60258 T^{4} - 7884924 T^{6} + 845631487 T^{8} - 76885694448 T^{10} + 6031931709068 T^{12} - 76885694448 p^{2} T^{14} + 845631487 p^{4} T^{16} - 7884924 p^{6} T^{18} + 60258 p^{8} T^{20} - 340 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( 1 - 580 T^{2} + 162434 T^{4} - 29215828 T^{6} + 3822461807 T^{8} - 393961353960 T^{10} + 33773071899804 T^{12} - 393961353960 p^{2} T^{14} + 3822461807 p^{4} T^{16} - 29215828 p^{6} T^{18} + 162434 p^{8} T^{20} - 580 p^{10} T^{22} + p^{12} T^{24} \) | |
| 83 | \( ( 1 - 16 T + 466 T^{2} - 6120 T^{3} + 94367 T^{4} - 972216 T^{5} + 10386524 T^{6} - 972216 p T^{7} + 94367 p^{2} T^{8} - 6120 p^{3} T^{9} + 466 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 - 16 T + 128 T^{2} - 368 T^{3} - 7898 T^{4} + 29264 T^{5} + 610432 T^{6} - 11480784 T^{7} + 109099759 T^{8} - 337804704 T^{9} + 914417408 T^{10} - 3378243424 T^{11} - 35519685164 T^{12} - 3378243424 p T^{13} + 914417408 p^{2} T^{14} - 337804704 p^{3} T^{15} + 109099759 p^{4} T^{16} - 11480784 p^{5} T^{17} + 610432 p^{6} T^{18} + 29264 p^{7} T^{19} - 7898 p^{8} T^{20} - 368 p^{9} T^{21} + 128 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 324 T^{2} + 84034 T^{4} - 15044460 T^{6} + 2302233567 T^{8} - 281278260944 T^{10} + 30118731888844 T^{12} - 281278260944 p^{2} T^{14} + 2302233567 p^{4} T^{16} - 15044460 p^{6} T^{18} + 84034 p^{8} T^{20} - 324 p^{10} T^{22} + p^{12} T^{24} \) | |
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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
−3.81621917178467426408086381154, −3.78400233380584513543623386497, −3.65711472819625609184836429060, −3.60576156695004547069260553314, −3.48890956583873731586745874032, −3.47311333546592841735033770078, −3.22568761715257913059125161802, −3.16913460468694797374956219007, −3.12681006156015015056474028638, −3.04485175487592781943673151321, −2.80958647077236745465534011252, −2.60718869040013214402876617418, −2.36966186154445499391598212084, −2.31500871691703349190654909656, −2.23369168910741702763953607931, −1.90753167717470840646514486063, −1.71082687026627518456743258674, −1.61834604418741823740797542860, −1.32098887135582128731039259608, −1.26756815002001218473645312769, −1.03515030369852661744986314256, −1.01536066822958718458185143223, −0.937071821559497292547772200639, −0.73256836690906133853819310156, −0.06452049434120595888920203796, 0.06452049434120595888920203796, 0.73256836690906133853819310156, 0.937071821559497292547772200639, 1.01536066822958718458185143223, 1.03515030369852661744986314256, 1.26756815002001218473645312769, 1.32098887135582128731039259608, 1.61834604418741823740797542860, 1.71082687026627518456743258674, 1.90753167717470840646514486063, 2.23369168910741702763953607931, 2.31500871691703349190654909656, 2.36966186154445499391598212084, 2.60718869040013214402876617418, 2.80958647077236745465534011252, 3.04485175487592781943673151321, 3.12681006156015015056474028638, 3.16913460468694797374956219007, 3.22568761715257913059125161802, 3.47311333546592841735033770078, 3.48890956583873731586745874032, 3.60576156695004547069260553314, 3.65711472819625609184836429060, 3.78400233380584513543623386497, 3.81621917178467426408086381154
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.