Dirichlet series
| L(s) = 1 | + 12·7-s − 12·9-s + 8·17-s + 16·23-s − 20·25-s − 8·31-s + 32·41-s − 16·47-s + 78·49-s − 144·63-s + 8·71-s − 24·79-s + 74·81-s + 24·89-s + 48·97-s − 32·103-s + 32·113-s + 96·119-s − 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 96·153-s + 157-s + ⋯ |
| L(s) = 1 | + 4.53·7-s − 4·9-s + 1.94·17-s + 3.33·23-s − 4·25-s − 1.43·31-s + 4.99·41-s − 2.33·47-s + 78/7·49-s − 18.1·63-s + 0.949·71-s − 2.70·79-s + 74/9·81-s + 2.54·89-s + 4.87·97-s − 3.15·103-s + 3.01·113-s + 8.80·119-s − 5.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 7.76·153-s + 0.0798·157-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{120} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{120} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{120} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.23623\times 10^{21}\) |
| Root analytic conductor: | \(7.56549\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{120} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(41.77929815\) |
| \(L(\frac12)\) | \(\approx\) | \(41.77929815\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 - T )^{12} \) | |
| good | 3 | \( 1 + 4 p T^{2} + 70 T^{4} + 292 T^{6} + 1123 T^{8} + 1376 p T^{10} + 13420 T^{12} + 1376 p^{3} T^{14} + 1123 p^{4} T^{16} + 292 p^{6} T^{18} + 70 p^{8} T^{20} + 4 p^{11} T^{22} + p^{12} T^{24} \) |
| 5 | \( 1 + 4 p T^{2} + 46 p T^{4} + 2012 T^{6} + 14627 T^{8} + 89664 T^{10} + 476844 T^{12} + 89664 p^{2} T^{14} + 14627 p^{4} T^{16} + 2012 p^{6} T^{18} + 46 p^{9} T^{20} + 4 p^{11} T^{22} + p^{12} T^{24} \) | |
| 11 | \( 1 + 56 T^{2} + 1774 T^{4} + 39288 T^{6} + 61773 p T^{8} + 9623696 T^{10} + 114956260 T^{12} + 9623696 p^{2} T^{14} + 61773 p^{5} T^{16} + 39288 p^{6} T^{18} + 1774 p^{8} T^{20} + 56 p^{10} T^{22} + p^{12} T^{24} \) | |
| 13 | \( 1 + 44 T^{2} + 1142 T^{4} + 21716 T^{6} + 362403 T^{8} + 5344880 T^{10} + 72892876 T^{12} + 5344880 p^{2} T^{14} + 362403 p^{4} T^{16} + 21716 p^{6} T^{18} + 1142 p^{8} T^{20} + 44 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( ( 1 - 4 T + 58 T^{2} - 140 T^{3} + 1383 T^{4} - 1648 T^{5} + 23228 T^{6} - 1648 p T^{7} + 1383 p^{2} T^{8} - 140 p^{3} T^{9} + 58 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 + 100 T^{2} + 5254 T^{4} + 191324 T^{6} + 5450531 T^{8} + 129419344 T^{10} + 2641798636 T^{12} + 129419344 p^{2} T^{14} + 5450531 p^{4} T^{16} + 191324 p^{6} T^{18} + 5254 p^{8} T^{20} + 100 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( ( 1 - 8 T + 102 T^{2} - 600 T^{3} + 4963 T^{4} - 22896 T^{5} + 141532 T^{6} - 22896 p T^{7} + 4963 p^{2} T^{8} - 600 p^{3} T^{9} + 102 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( 1 + 192 T^{2} + 18454 T^{4} + 1174464 T^{6} + 55711935 T^{8} + 2110668160 T^{10} + 66590211124 T^{12} + 2110668160 p^{2} T^{14} + 55711935 p^{4} T^{16} + 1174464 p^{6} T^{18} + 18454 p^{8} T^{20} + 192 p^{10} T^{22} + p^{12} T^{24} \) | |
| 31 | \( ( 1 + 4 T + 170 T^{2} + 548 T^{3} + 12407 T^{4} + 31840 T^{5} + 502140 T^{6} + 31840 p T^{7} + 12407 p^{2} T^{8} + 548 p^{3} T^{9} + 170 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( 1 + 272 T^{2} + 36694 T^{4} + 3260624 T^{6} + 213580575 T^{8} + 10918611104 T^{10} + 448570168116 T^{12} + 10918611104 p^{2} T^{14} + 213580575 p^{4} T^{16} + 3260624 p^{6} T^{18} + 36694 p^{8} T^{20} + 272 p^{10} T^{22} + p^{12} T^{24} \) | |
| 41 | \( ( 1 - 16 T + 314 T^{2} - 3240 T^{3} + 35767 T^{4} - 263592 T^{5} + 2015228 T^{6} - 263592 p T^{7} + 35767 p^{2} T^{8} - 3240 p^{3} T^{9} + 314 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 43 | \( 1 + 296 T^{2} + 43950 T^{4} + 4359272 T^{6} + 323325647 T^{8} + 18949666608 T^{10} + 901095067748 T^{12} + 18949666608 p^{2} T^{14} + 323325647 p^{4} T^{16} + 4359272 p^{6} T^{18} + 43950 p^{8} T^{20} + 296 p^{10} T^{22} + p^{12} T^{24} \) | |
| 47 | \( ( 1 + 8 T + 178 T^{2} + 848 T^{3} + 12375 T^{4} + 40136 T^{5} + 604268 T^{6} + 40136 p T^{7} + 12375 p^{2} T^{8} + 848 p^{3} T^{9} + 178 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 53 | \( 1 + 288 T^{2} + 41110 T^{4} + 4098208 T^{6} + 329446367 T^{8} + 22387365696 T^{10} + 1290285511092 T^{12} + 22387365696 p^{2} T^{14} + 329446367 p^{4} T^{16} + 4098208 p^{6} T^{18} + 41110 p^{8} T^{20} + 288 p^{10} T^{22} + p^{12} T^{24} \) | |
| 59 | \( 1 + 468 T^{2} + 108742 T^{4} + 16525420 T^{6} + 1825859267 T^{8} + 154400486544 T^{10} + 10245169196652 T^{12} + 154400486544 p^{2} T^{14} + 1825859267 p^{4} T^{16} + 16525420 p^{6} T^{18} + 108742 p^{8} T^{20} + 468 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( 1 + 468 T^{2} + 100646 T^{4} + 13234908 T^{6} + 19905087 p T^{8} + 86963788992 T^{10} + 5482018820524 T^{12} + 86963788992 p^{2} T^{14} + 19905087 p^{5} T^{16} + 13234908 p^{6} T^{18} + 100646 p^{8} T^{20} + 468 p^{10} T^{22} + p^{12} T^{24} \) | |
| 67 | \( 1 + 568 T^{2} + 149134 T^{4} + 24190392 T^{6} + 2762415151 T^{8} + 243129788368 T^{10} + 17671972639012 T^{12} + 243129788368 p^{2} T^{14} + 2762415151 p^{4} T^{16} + 24190392 p^{6} T^{18} + 149134 p^{8} T^{20} + 568 p^{10} T^{22} + p^{12} T^{24} \) | |
| 71 | \( ( 1 - 4 T + 218 T^{2} - 540 T^{3} + 21871 T^{4} - 7416 T^{5} + 1626188 T^{6} - 7416 p T^{7} + 21871 p^{2} T^{8} - 540 p^{3} T^{9} + 218 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 73 | \( ( 1 + 130 T^{2} - 384 T^{3} + 9759 T^{4} - 20096 T^{5} + 762076 T^{6} - 20096 p T^{7} + 9759 p^{2} T^{8} - 384 p^{3} T^{9} + 130 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 79 | \( ( 1 + 12 T + 430 T^{2} + 4164 T^{3} + 79599 T^{4} + 615800 T^{5} + 8193220 T^{6} + 615800 p T^{7} + 79599 p^{2} T^{8} + 4164 p^{3} T^{9} + 430 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 + 428 T^{2} + 112134 T^{4} + 20399492 T^{6} + 2879450147 T^{8} + 323040242592 T^{10} + 29674593548780 T^{12} + 323040242592 p^{2} T^{14} + 2879450147 p^{4} T^{16} + 20399492 p^{6} T^{18} + 112134 p^{8} T^{20} + 428 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( ( 1 - 12 T + 330 T^{2} - 3772 T^{3} + 57407 T^{4} - 575064 T^{5} + 6212172 T^{6} - 575064 p T^{7} + 57407 p^{2} T^{8} - 3772 p^{3} T^{9} + 330 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( ( 1 - 24 T + 522 T^{2} - 9072 T^{3} + 126023 T^{4} - 1521976 T^{5} + 16411804 T^{6} - 1521976 p T^{7} + 126023 p^{2} T^{8} - 9072 p^{3} T^{9} + 522 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
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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
−2.47986913804666874153672912882, −2.05991930001089169657434823219, −2.02966437252992766293724716102, −2.02809558348514907112078222568, −1.96887649902592818995295566441, −1.94772220840727728796996861648, −1.92835853295009250402454485688, −1.73684947806213516633822837480, −1.72509218443276304869626887101, −1.67746025936760133556056533750, −1.62379929252141829101917459893, −1.40352909403991245133007333277, −1.31418954024752428194182321565, −1.29863647657838211136972972191, −1.22658366007809374580090547931, −1.01106440179191916117526916237, −0.928373451516354017919492544876, −0.844456814316674024736163979703, −0.835622272634963551750588312308, −0.61645648354570318517794993640, −0.59129891825306483413272534566, −0.50629316471580010727049771084, −0.43054529417049644873820841204, −0.21752885009332813984612112300, −0.16669212510909786777841530682, 0.16669212510909786777841530682, 0.21752885009332813984612112300, 0.43054529417049644873820841204, 0.50629316471580010727049771084, 0.59129891825306483413272534566, 0.61645648354570318517794993640, 0.835622272634963551750588312308, 0.844456814316674024736163979703, 0.928373451516354017919492544876, 1.01106440179191916117526916237, 1.22658366007809374580090547931, 1.29863647657838211136972972191, 1.31418954024752428194182321565, 1.40352909403991245133007333277, 1.62379929252141829101917459893, 1.67746025936760133556056533750, 1.72509218443276304869626887101, 1.73684947806213516633822837480, 1.92835853295009250402454485688, 1.94772220840727728796996861648, 1.96887649902592818995295566441, 2.02809558348514907112078222568, 2.02966437252992766293724716102, 2.05991930001089169657434823219, 2.47986913804666874153672912882
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.