Dirichlet series
| L(s) = 1 | − 7·9-s + 18·11-s + 20·23-s − 23·25-s − 18·29-s − 18·37-s + 10·43-s − 38·53-s + 42·67-s + 56·71-s + 56·79-s + 21·81-s − 126·99-s + 94·107-s + 30·109-s + 40·113-s + 127·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 27·169-s + ⋯ |
| L(s) = 1 | − 7/3·9-s + 5.42·11-s + 4.17·23-s − 4.59·25-s − 3.34·29-s − 2.95·37-s + 1.52·43-s − 5.21·53-s + 5.13·67-s + 6.64·71-s + 6.30·79-s + 7/3·81-s − 12.6·99-s + 9.08·107-s + 2.87·109-s + 3.76·113-s + 11.5·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.07·169-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{36} \cdot 7^{36}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.22443\times 10^{16}\) |
| Root analytic conductor: | \(4.68091\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{36} \cdot 7^{36} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(18.40866262\) |
| \(L(\frac12)\) | \(\approx\) | \(18.40866262\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| good | 3 | \( 1 + 7 T^{2} + 28 T^{4} + 98 T^{6} + p^{5} T^{8} + 161 p T^{10} + 1232 T^{12} + 161 p^{3} T^{14} + p^{9} T^{16} + 98 p^{6} T^{18} + 28 p^{8} T^{20} + 7 p^{10} T^{22} + p^{12} T^{24} \) |
| 5 | \( 1 + 23 T^{2} + 274 T^{4} + 2322 T^{6} + 16479 T^{8} + 827 p^{3} T^{10} + 561652 T^{12} + 827 p^{5} T^{14} + 16479 p^{4} T^{16} + 2322 p^{6} T^{18} + 274 p^{8} T^{20} + 23 p^{10} T^{22} + p^{12} T^{24} \) | |
| 11 | \( ( 1 - 9 T + 58 T^{2} - 270 T^{3} + 1127 T^{4} - 4053 T^{5} + 14148 T^{6} - 4053 p T^{7} + 1127 p^{2} T^{8} - 270 p^{3} T^{9} + 58 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 13 | \( 1 + 27 T^{2} + 876 T^{4} + 18506 T^{6} + 341033 T^{8} + 5681023 T^{10} + 74834948 T^{12} + 5681023 p^{2} T^{14} + 341033 p^{4} T^{16} + 18506 p^{6} T^{18} + 876 p^{8} T^{20} + 27 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( 1 + 96 T^{2} + 4286 T^{4} + 122920 T^{6} + 2738767 T^{8} + 3167736 p T^{10} + 964237924 T^{12} + 3167736 p^{3} T^{14} + 2738767 p^{4} T^{16} + 122920 p^{6} T^{18} + 4286 p^{8} T^{20} + 96 p^{10} T^{22} + p^{12} T^{24} \) | |
| 19 | \( 1 + 66 T^{2} + 3094 T^{4} + 103362 T^{6} + 2896063 T^{8} + 67258124 T^{10} + 1382642292 T^{12} + 67258124 p^{2} T^{14} + 2896063 p^{4} T^{16} + 103362 p^{6} T^{18} + 3094 p^{8} T^{20} + 66 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( ( 1 - 10 T + 127 T^{2} - 850 T^{3} + 6391 T^{4} - 33320 T^{5} + 186593 T^{6} - 33320 p T^{7} + 6391 p^{2} T^{8} - 850 p^{3} T^{9} + 127 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( ( 1 + 9 T + 80 T^{2} + 410 T^{3} + 3477 T^{4} + 18297 T^{5} + 125188 T^{6} + 18297 p T^{7} + 3477 p^{2} T^{8} + 410 p^{3} T^{9} + 80 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( 1 + 18 T^{2} + 718 T^{4} - 19518 T^{6} + 614735 T^{8} + 12745260 T^{10} + 2152064964 T^{12} + 12745260 p^{2} T^{14} + 614735 p^{4} T^{16} - 19518 p^{6} T^{18} + 718 p^{8} T^{20} + 18 p^{10} T^{22} + p^{12} T^{24} \) | |
| 37 | \( ( 1 + 9 T + 76 T^{2} + 12 p T^{3} + 4621 T^{4} + 28703 T^{5} + 206268 T^{6} + 28703 p T^{7} + 4621 p^{2} T^{8} + 12 p^{4} T^{9} + 76 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 + 216 T^{2} + 18494 T^{4} + 844768 T^{6} + 34398895 T^{8} + 2166596040 T^{10} + 114486723940 T^{12} + 2166596040 p^{2} T^{14} + 34398895 p^{4} T^{16} + 844768 p^{6} T^{18} + 18494 p^{8} T^{20} + 216 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( ( 1 - 5 T + 168 T^{2} - 840 T^{3} + 14561 T^{4} - 61655 T^{5} + 784612 T^{6} - 61655 p T^{7} + 14561 p^{2} T^{8} - 840 p^{3} T^{9} + 168 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 + 490 T^{2} + 112910 T^{4} + 16177882 T^{6} + 1605187023 T^{8} + 116185339004 T^{10} + 6290248283844 T^{12} + 116185339004 p^{2} T^{14} + 1605187023 p^{4} T^{16} + 16177882 p^{6} T^{18} + 112910 p^{8} T^{20} + 490 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( ( 1 + 19 T + 368 T^{2} + 4310 T^{3} + 49861 T^{4} + 425347 T^{5} + 3540004 T^{6} + 425347 p T^{7} + 49861 p^{2} T^{8} + 4310 p^{3} T^{9} + 368 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 + 205 T^{2} + 25790 T^{4} + 2329096 T^{6} + 187530003 T^{8} + 13413949163 T^{10} + 860540336676 T^{12} + 13413949163 p^{2} T^{14} + 187530003 p^{4} T^{16} + 2329096 p^{6} T^{18} + 25790 p^{8} T^{20} + 205 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( 1 + 539 T^{2} + 138992 T^{4} + 22826594 T^{6} + 2675879373 T^{8} + 237079047751 T^{10} + 16330671219828 T^{12} + 237079047751 p^{2} T^{14} + 2675879373 p^{4} T^{16} + 22826594 p^{6} T^{18} + 138992 p^{8} T^{20} + 539 p^{10} T^{22} + p^{12} T^{24} \) | |
| 67 | \( ( 1 - 21 T + 516 T^{2} - 6874 T^{3} + 95597 T^{4} - 908201 T^{5} + 8784020 T^{6} - 908201 p T^{7} + 95597 p^{2} T^{8} - 6874 p^{3} T^{9} + 516 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( ( 1 - 28 T + 503 T^{2} - 6902 T^{3} + 81509 T^{4} - 836612 T^{5} + 7542597 T^{6} - 836612 p T^{7} + 81509 p^{2} T^{8} - 6902 p^{3} T^{9} + 503 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 73 | \( 1 + 428 T^{2} + 105230 T^{4} + 17756036 T^{6} + 2276537199 T^{8} + 229397964304 T^{10} + 18635823769668 T^{12} + 229397964304 p^{2} T^{14} + 2276537199 p^{4} T^{16} + 17756036 p^{6} T^{18} + 105230 p^{8} T^{20} + 428 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( ( 1 - 28 T + 597 T^{2} - 8806 T^{3} + 112347 T^{4} - 1184442 T^{5} + 11327107 T^{6} - 1184442 p T^{7} + 112347 p^{2} T^{8} - 8806 p^{3} T^{9} + 597 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 + 617 T^{2} + 193860 T^{4} + 40455784 T^{6} + 6202349269 T^{8} + 732821907011 T^{10} + 68268057989292 T^{12} + 732821907011 p^{2} T^{14} + 6202349269 p^{4} T^{16} + 40455784 p^{6} T^{18} + 193860 p^{8} T^{20} + 617 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( 1 + 264 T^{2} + 37398 T^{4} + 4347264 T^{6} + 444634127 T^{8} + 37768150776 T^{10} + 3115620312692 T^{12} + 37768150776 p^{2} T^{14} + 444634127 p^{4} T^{16} + 4347264 p^{6} T^{18} + 37398 p^{8} T^{20} + 264 p^{10} T^{22} + p^{12} T^{24} \) | |
| 97 | \( 1 + 608 T^{2} + 197694 T^{4} + 44137192 T^{6} + 7446216847 T^{8} + 993198084536 T^{10} + 106896809192804 T^{12} + 993198084536 p^{2} T^{14} + 7446216847 p^{4} T^{16} + 44137192 p^{6} T^{18} + 197694 p^{8} T^{20} + 608 p^{10} T^{22} + p^{12} T^{24} \) | |
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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
−2.75869186612610958812693783096, −2.38644551908933902972919501131, −2.27034282958770242087546560371, −2.26100842028666206049990689574, −2.23522870262809361059667794865, −2.20331046204367791887043135212, −2.19375270968792008309938663582, −2.01807283193730835904367408422, −2.01762963110735418968216151202, −1.99848768306900604103485949228, −1.81992134178139990239372202194, −1.75746039889895137420307509657, −1.63833149400073566057207362459, −1.54306051158839122304017505042, −1.50736078884967932577035021912, −1.26267779713096543809348528095, −1.03169328266977772030721577941, −0.960922098337462033098053333821, −0.891957148338429571416558511566, −0.853040210381991800686258327992, −0.66704831222827499262441417097, −0.64686744592065137547737595743, −0.50038207203557490214824503136, −0.39907006518405449687150978115, −0.13884148271989064950534601263, 0.13884148271989064950534601263, 0.39907006518405449687150978115, 0.50038207203557490214824503136, 0.64686744592065137547737595743, 0.66704831222827499262441417097, 0.853040210381991800686258327992, 0.891957148338429571416558511566, 0.960922098337462033098053333821, 1.03169328266977772030721577941, 1.26267779713096543809348528095, 1.50736078884967932577035021912, 1.54306051158839122304017505042, 1.63833149400073566057207362459, 1.75746039889895137420307509657, 1.81992134178139990239372202194, 1.99848768306900604103485949228, 2.01762963110735418968216151202, 2.01807283193730835904367408422, 2.19375270968792008309938663582, 2.20331046204367791887043135212, 2.23522870262809361059667794865, 2.26100842028666206049990689574, 2.27034282958770242087546560371, 2.38644551908933902972919501131, 2.75869186612610958812693783096
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.