Dirichlet series
| L(s) = 1 | + 6·3-s − 8·4-s + 9-s − 48·12-s + 32·16-s + 34·17-s + 6·23-s − 35·25-s − 68·27-s + 2·29-s − 8·36-s + 22·43-s + 192·48-s + 204·51-s + 16·53-s − 10·61-s − 91·64-s − 272·68-s + 36·69-s − 210·75-s + 70·79-s − 128·81-s + 12·87-s − 48·92-s + 280·100-s + 38·101-s − 30·103-s + ⋯ |
| L(s) = 1 | + 3.46·3-s − 4·4-s + 1/3·9-s − 13.8·12-s + 8·16-s + 8.24·17-s + 1.25·23-s − 7·25-s − 13.0·27-s + 0.371·29-s − 4/3·36-s + 3.35·43-s + 27.7·48-s + 28.5·51-s + 2.19·53-s − 1.28·61-s − 11.3·64-s − 32.9·68-s + 4.33·69-s − 24.2·75-s + 7.87·79-s − 14.2·81-s + 1.28·87-s − 5.00·92-s + 28·100-s + 3.78·101-s − 2.95·103-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(7^{24} \cdot 13^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.98744\times 10^{21}\) |
| Root analytic conductor: | \(8.13167\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 7^{24} \cdot 13^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(68.35380685\) |
| \(L(\frac12)\) | \(\approx\) | \(68.35380685\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + p^{3} T^{2} + p^{5} T^{4} + 91 T^{6} + 7 p^{5} T^{8} + 33 p^{4} T^{10} + 1137 T^{12} + 33 p^{6} T^{14} + 7 p^{9} T^{16} + 91 p^{6} T^{18} + p^{13} T^{20} + p^{13} T^{22} + p^{12} T^{24} \) |
| 3 | \( ( 1 - p T + 13 T^{2} - 29 T^{3} + 79 T^{4} - 145 T^{5} + 301 T^{6} - 145 p T^{7} + 79 p^{2} T^{8} - 29 p^{3} T^{9} + 13 p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} )^{2} \) | |
| 5 | \( 1 + 7 p T^{2} + 601 T^{4} + 6779 T^{6} + 56873 T^{8} + 379271 T^{10} + 2080181 T^{12} + 379271 p^{2} T^{14} + 56873 p^{4} T^{16} + 6779 p^{6} T^{18} + 601 p^{8} T^{20} + 7 p^{11} T^{22} + p^{12} T^{24} \) | |
| 11 | \( 1 + 70 T^{2} + 2521 T^{4} + 61186 T^{6} + 1119878 T^{8} + 16383787 T^{10} + 197579915 T^{12} + 16383787 p^{2} T^{14} + 1119878 p^{4} T^{16} + 61186 p^{6} T^{18} + 2521 p^{8} T^{20} + 70 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( ( 1 - p T + 198 T^{2} - 1643 T^{3} + 10919 T^{4} - 59082 T^{5} + 266647 T^{6} - 59082 p T^{7} + 10919 p^{2} T^{8} - 1643 p^{3} T^{9} + 198 p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 + 149 T^{2} + 10800 T^{4} + 511360 T^{6} + 17748002 T^{8} + 476368077 T^{10} + 10131618103 T^{12} + 476368077 p^{2} T^{14} + 17748002 p^{4} T^{16} + 511360 p^{6} T^{18} + 10800 p^{8} T^{20} + 149 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( ( 1 - 3 T + 88 T^{2} - 86 T^{3} + 3150 T^{4} + 1355 T^{5} + 76923 T^{6} + 1355 p T^{7} + 3150 p^{2} T^{8} - 86 p^{3} T^{9} + 88 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( ( 1 - T + 88 T^{2} + 45 T^{3} + 4077 T^{4} + 3080 T^{5} + 141237 T^{6} + 3080 p T^{7} + 4077 p^{2} T^{8} + 45 p^{3} T^{9} + 88 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( 1 + 140 T^{2} + 9930 T^{4} + 450715 T^{6} + 15153149 T^{8} + 429707637 T^{10} + 12513324193 T^{12} + 429707637 p^{2} T^{14} + 15153149 p^{4} T^{16} + 450715 p^{6} T^{18} + 9930 p^{8} T^{20} + 140 p^{10} T^{22} + p^{12} T^{24} \) | |
| 37 | \( 1 + 297 T^{2} + 43128 T^{4} + 4084528 T^{6} + 281694330 T^{8} + 14898657933 T^{10} + 619410154695 T^{12} + 14898657933 p^{2} T^{14} + 281694330 p^{4} T^{16} + 4084528 p^{6} T^{18} + 43128 p^{8} T^{20} + 297 p^{10} T^{22} + p^{12} T^{24} \) | |
| 41 | \( 1 + 138 T^{2} + 10369 T^{4} + 551194 T^{6} + 24197606 T^{8} + 984355043 T^{10} + 39607623959 T^{12} + 984355043 p^{2} T^{14} + 24197606 p^{4} T^{16} + 551194 p^{6} T^{18} + 10369 p^{8} T^{20} + 138 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( ( 1 - 11 T + 209 T^{2} - 1549 T^{3} + 17930 T^{4} - 105870 T^{5} + 948579 T^{6} - 105870 p T^{7} + 17930 p^{2} T^{8} - 1549 p^{3} T^{9} + 209 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 + 523 T^{2} + 127033 T^{4} + 18954871 T^{6} + 1935563261 T^{8} + 142521847195 T^{10} + 7760213706737 T^{12} + 142521847195 p^{2} T^{14} + 1935563261 p^{4} T^{16} + 18954871 p^{6} T^{18} + 127033 p^{8} T^{20} + 523 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( ( 1 - 8 T + 280 T^{2} - 1716 T^{3} + 33468 T^{4} - 160748 T^{5} + 2272305 T^{6} - 160748 p T^{7} + 33468 p^{2} T^{8} - 1716 p^{3} T^{9} + 280 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 + 155 T^{2} + 24317 T^{4} + 2572015 T^{6} + 230267981 T^{8} + 17468522907 T^{10} + 1098167895897 T^{12} + 17468522907 p^{2} T^{14} + 230267981 p^{4} T^{16} + 2572015 p^{6} T^{18} + 24317 p^{8} T^{20} + 155 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( ( 1 + 5 T + 291 T^{2} + 1171 T^{3} + 38615 T^{4} + 126093 T^{5} + 3001147 T^{6} + 126093 p T^{7} + 38615 p^{2} T^{8} + 1171 p^{3} T^{9} + 291 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( 1 + 365 T^{2} + 70430 T^{4} + 9408718 T^{6} + 14623229 p T^{8} + 84195136134 T^{10} + 6117380213691 T^{12} + 84195136134 p^{2} T^{14} + 14623229 p^{5} T^{16} + 9408718 p^{6} T^{18} + 70430 p^{8} T^{20} + 365 p^{10} T^{22} + p^{12} T^{24} \) | |
| 71 | \( 1 + 406 T^{2} + 82657 T^{4} + 11557366 T^{6} + 1259471246 T^{8} + 113239527451 T^{10} + 8663606903147 T^{12} + 113239527451 p^{2} T^{14} + 1259471246 p^{4} T^{16} + 11557366 p^{6} T^{18} + 82657 p^{8} T^{20} + 406 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 + 476 T^{2} + 113546 T^{4} + 17894407 T^{6} + 2099846477 T^{8} + 197467415277 T^{10} + 15594902689281 T^{12} + 197467415277 p^{2} T^{14} + 2099846477 p^{4} T^{16} + 17894407 p^{6} T^{18} + 113546 p^{8} T^{20} + 476 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( ( 1 - 35 T + 818 T^{2} - 13321 T^{3} + 181646 T^{4} - 2028165 T^{5} + 19730991 T^{6} - 2028165 p T^{7} + 181646 p^{2} T^{8} - 13321 p^{3} T^{9} + 818 p^{4} T^{10} - 35 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 + 533 T^{2} + 130220 T^{4} + 19632136 T^{6} + 2123453318 T^{8} + 188332543809 T^{10} + 15642556571895 T^{12} + 188332543809 p^{2} T^{14} + 2123453318 p^{4} T^{16} + 19632136 p^{6} T^{18} + 130220 p^{8} T^{20} + 533 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( 1 + 638 T^{2} + 182947 T^{4} + 30717524 T^{6} + 3345408557 T^{8} + 264875421410 T^{10} + 20634099784895 T^{12} + 264875421410 p^{2} T^{14} + 3345408557 p^{4} T^{16} + 30717524 p^{6} T^{18} + 182947 p^{8} T^{20} + 638 p^{10} T^{22} + p^{12} T^{24} \) | |
| 97 | \( 1 + 803 T^{2} + 300803 T^{4} + 70985683 T^{6} + 12056931086 T^{8} + 1591425970020 T^{10} + 170288319348129 T^{12} + 1591425970020 p^{2} T^{14} + 12056931086 p^{4} T^{16} + 70985683 p^{6} T^{18} + 300803 p^{8} T^{20} + 803 p^{10} T^{22} + p^{12} T^{24} \) | |
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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
−2.23237893311081306028734741547, −2.22331691827370922675276364758, −2.18627192837306150339072580661, −2.11703605235426695684043166395, −2.09510634200149993100592086143, −2.09090763024783243909580073592, −1.86021985350655270774144879825, −1.77523766906501670294428154673, −1.60690712512778993744994400324, −1.51976802435098711799672306096, −1.42661681007584098568429719016, −1.40802647404667995050420777135, −1.25888486424146368621801481685, −1.22919010345442936445694117404, −1.11983146406576827570707547033, −1.10959023511523463277169636860, −0.74853910896985506728218575008, −0.73821112401080174142257832791, −0.60049377029348509788900463433, −0.58273335385365322963524140050, −0.57628664883330566700845846464, −0.46398386607818375891804417352, −0.35423015336388461633855987576, −0.31155193370487468896046402997, −0.25744442300102052837029991019, 0.25744442300102052837029991019, 0.31155193370487468896046402997, 0.35423015336388461633855987576, 0.46398386607818375891804417352, 0.57628664883330566700845846464, 0.58273335385365322963524140050, 0.60049377029348509788900463433, 0.73821112401080174142257832791, 0.74853910896985506728218575008, 1.10959023511523463277169636860, 1.11983146406576827570707547033, 1.22919010345442936445694117404, 1.25888486424146368621801481685, 1.40802647404667995050420777135, 1.42661681007584098568429719016, 1.51976802435098711799672306096, 1.60690712512778993744994400324, 1.77523766906501670294428154673, 1.86021985350655270774144879825, 2.09090763024783243909580073592, 2.09510634200149993100592086143, 2.11703605235426695684043166395, 2.18627192837306150339072580661, 2.22331691827370922675276364758, 2.23237893311081306028734741547
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.