Dirichlet series
| L(s) = 1 | + 14·3-s + 105·9-s − 4·13-s − 4·17-s + 2·23-s + 45·25-s + 560·27-s − 10·29-s − 56·39-s + 22·43-s + 53·49-s − 56·51-s − 20·53-s − 2·61-s + 28·69-s + 630·75-s − 40·79-s + 2.38e3·81-s − 140·87-s + 56·101-s + 26·103-s + 30·107-s − 34·113-s − 420·117-s − 7·121-s + 127-s + 308·129-s + ⋯ |
| L(s) = 1 | + 8.08·3-s + 35·9-s − 1.10·13-s − 0.970·17-s + 0.417·23-s + 9·25-s + 107.·27-s − 1.85·29-s − 8.96·39-s + 3.35·43-s + 53/7·49-s − 7.84·51-s − 2.74·53-s − 0.256·61-s + 3.37·69-s + 72.7·75-s − 4.50·79-s + 264.·81-s − 15.0·87-s + 5.57·101-s + 2.56·103-s + 2.90·107-s − 3.19·113-s − 38.8·117-s − 0.636·121-s + 0.0887·127-s + 27.1·129-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 3^{14} \cdot 11^{14} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 3^{14} \cdot 11^{14} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{42} \cdot 3^{14} \cdot 11^{14} \cdot 13^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.34753\times 10^{20}\) |
| Root analytic conductor: | \(5.23494\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{42} \cdot 3^{14} \cdot 11^{14} \cdot 13^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(14472.32636\) |
| \(L(\frac12)\) | \(\approx\) | \(14472.32636\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{14} \) | |
| 11 | \( ( 1 + T^{2} )^{7} \) | |
| 13 | \( 1 + 4 T + 47 T^{2} + 216 T^{3} + 1201 T^{4} + 5404 T^{5} + 21607 T^{6} + 85328 T^{7} + 21607 p T^{8} + 5404 p^{2} T^{9} + 1201 p^{3} T^{10} + 216 p^{4} T^{11} + 47 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| good | 5 | \( 1 - 9 p T^{2} + 1011 T^{4} - 3002 p T^{6} + 164149 T^{8} - 1395919 T^{10} + 1901039 p T^{12} - 52599484 T^{14} + 1901039 p^{3} T^{16} - 1395919 p^{4} T^{18} + 164149 p^{6} T^{20} - 3002 p^{9} T^{22} + 1011 p^{10} T^{24} - 9 p^{13} T^{26} + p^{14} T^{28} \) |
| 7 | \( 1 - 53 T^{2} + 1399 T^{4} - 24550 T^{6} + 322645 T^{8} - 3396251 T^{10} + 29871011 T^{12} - 225009140 T^{14} + 29871011 p^{2} T^{16} - 3396251 p^{4} T^{18} + 322645 p^{6} T^{20} - 24550 p^{8} T^{22} + 1399 p^{10} T^{24} - 53 p^{12} T^{26} + p^{14} T^{28} \) | |
| 17 | \( ( 1 + 2 T + 73 T^{2} + 116 T^{3} + 2633 T^{4} + 3508 T^{5} + 62625 T^{6} + 70932 T^{7} + 62625 p T^{8} + 3508 p^{2} T^{9} + 2633 p^{3} T^{10} + 116 p^{4} T^{11} + 73 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 19 | \( 1 - 106 T^{2} + 5071 T^{4} - 147836 T^{6} + 2880149 T^{8} - 33897030 T^{10} + 64908971 T^{12} + 4102892824 T^{14} + 64908971 p^{2} T^{16} - 33897030 p^{4} T^{18} + 2880149 p^{6} T^{20} - 147836 p^{8} T^{22} + 5071 p^{10} T^{24} - 106 p^{12} T^{26} + p^{14} T^{28} \) | |
| 23 | \( ( 1 - T + 85 T^{2} + 118 T^{3} + 3321 T^{4} + 10037 T^{5} + 97361 T^{6} + 309468 T^{7} + 97361 p T^{8} + 10037 p^{2} T^{9} + 3321 p^{3} T^{10} + 118 p^{4} T^{11} + 85 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 29 | \( ( 1 + 5 T + 5 p T^{2} + 726 T^{3} + 10027 T^{4} + 46363 T^{5} + 432181 T^{6} + 1712408 T^{7} + 432181 p T^{8} + 46363 p^{2} T^{9} + 10027 p^{3} T^{10} + 726 p^{4} T^{11} + 5 p^{6} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 31 | \( 1 - 258 T^{2} + 33863 T^{4} - 2976232 T^{6} + 194766237 T^{8} - 10011963386 T^{10} + 416267298683 T^{12} - 14208896408952 T^{14} + 416267298683 p^{2} T^{16} - 10011963386 p^{4} T^{18} + 194766237 p^{6} T^{20} - 2976232 p^{8} T^{22} + 33863 p^{10} T^{24} - 258 p^{12} T^{26} + p^{14} T^{28} \) | |
| 37 | \( 1 - 254 T^{2} + 34835 T^{4} - 3298988 T^{6} + 238914825 T^{8} - 13902121346 T^{10} + 668696020019 T^{12} - 728644295880 p T^{14} + 668696020019 p^{2} T^{16} - 13902121346 p^{4} T^{18} + 238914825 p^{6} T^{20} - 3298988 p^{8} T^{22} + 34835 p^{10} T^{24} - 254 p^{12} T^{26} + p^{14} T^{28} \) | |
| 41 | \( 1 - 297 T^{2} + 45103 T^{4} - 4640646 T^{6} + 363444829 T^{8} - 23004539703 T^{10} + 1213385086755 T^{12} - 54043023389780 T^{14} + 1213385086755 p^{2} T^{16} - 23004539703 p^{4} T^{18} + 363444829 p^{6} T^{20} - 4640646 p^{8} T^{22} + 45103 p^{10} T^{24} - 297 p^{12} T^{26} + p^{14} T^{28} \) | |
| 43 | \( ( 1 - 11 T + 209 T^{2} - 1344 T^{3} + 15251 T^{4} - 58761 T^{5} + 620009 T^{6} - 1785776 T^{7} + 620009 p T^{8} - 58761 p^{2} T^{9} + 15251 p^{3} T^{10} - 1344 p^{4} T^{11} + 209 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 47 | \( 1 - 450 T^{2} + 93227 T^{4} - 11823732 T^{6} + 1036442137 T^{8} - 68245267486 T^{10} + 3680067277947 T^{12} - 178313961692248 T^{14} + 3680067277947 p^{2} T^{16} - 68245267486 p^{4} T^{18} + 1036442137 p^{6} T^{20} - 11823732 p^{8} T^{22} + 93227 p^{10} T^{24} - 450 p^{12} T^{26} + p^{14} T^{28} \) | |
| 53 | \( ( 1 + 10 T + 267 T^{2} + 2012 T^{3} + 33493 T^{4} + 209654 T^{5} + 2649895 T^{6} + 13761832 T^{7} + 2649895 p T^{8} + 209654 p^{2} T^{9} + 33493 p^{3} T^{10} + 2012 p^{4} T^{11} + 267 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 59 | \( 1 - 557 T^{2} + 155827 T^{4} - 28665170 T^{6} + 3849916329 T^{8} - 397816108211 T^{10} + 32539094315891 T^{12} - 2137471362287484 T^{14} + 32539094315891 p^{2} T^{16} - 397816108211 p^{4} T^{18} + 3849916329 p^{6} T^{20} - 28665170 p^{8} T^{22} + 155827 p^{10} T^{24} - 557 p^{12} T^{26} + p^{14} T^{28} \) | |
| 61 | \( ( 1 + T + 303 T^{2} - 178 T^{3} + 41237 T^{4} - 72961 T^{5} + 3492315 T^{6} - 7121180 T^{7} + 3492315 p T^{8} - 72961 p^{2} T^{9} + 41237 p^{3} T^{10} - 178 p^{4} T^{11} + 303 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 67 | \( 1 - 365 T^{2} + 78575 T^{4} - 12134858 T^{6} + 1479102041 T^{8} - 148475396911 T^{10} + 12579829703467 T^{12} - 910290813188380 T^{14} + 12579829703467 p^{2} T^{16} - 148475396911 p^{4} T^{18} + 1479102041 p^{6} T^{20} - 12134858 p^{8} T^{22} + 78575 p^{10} T^{24} - 365 p^{12} T^{26} + p^{14} T^{28} \) | |
| 71 | \( 1 - 462 T^{2} + 106459 T^{4} - 16729612 T^{6} + 2053863033 T^{8} - 210952611090 T^{10} + 18640092119371 T^{12} - 1423985876617896 T^{14} + 18640092119371 p^{2} T^{16} - 210952611090 p^{4} T^{18} + 2053863033 p^{6} T^{20} - 16729612 p^{8} T^{22} + 106459 p^{10} T^{24} - 462 p^{12} T^{26} + p^{14} T^{28} \) | |
| 73 | \( 1 - 345 T^{2} + 60471 T^{4} - 7378206 T^{6} + 742061605 T^{8} - 66133475383 T^{10} + 5340678434323 T^{12} - 400504557152388 T^{14} + 5340678434323 p^{2} T^{16} - 66133475383 p^{4} T^{18} + 742061605 p^{6} T^{20} - 7378206 p^{8} T^{22} + 60471 p^{10} T^{24} - 345 p^{12} T^{26} + p^{14} T^{28} \) | |
| 79 | \( ( 1 + 20 T + 435 T^{2} + 5108 T^{3} + 64049 T^{4} + 550478 T^{5} + 5585359 T^{6} + 43724820 T^{7} + 5585359 p T^{8} + 550478 p^{2} T^{9} + 64049 p^{3} T^{10} + 5108 p^{4} T^{11} + 435 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 83 | \( 1 - 610 T^{2} + 190339 T^{4} - 40418036 T^{6} + 6516816873 T^{8} - 843240808478 T^{10} + 90355961380291 T^{12} - 8151939595462744 T^{14} + 90355961380291 p^{2} T^{16} - 843240808478 p^{4} T^{18} + 6516816873 p^{6} T^{20} - 40418036 p^{8} T^{22} + 190339 p^{10} T^{24} - 610 p^{12} T^{26} + p^{14} T^{28} \) | |
| 89 | \( 1 - 630 T^{2} + 205663 T^{4} - 46401556 T^{6} + 8034728525 T^{8} - 1121527816102 T^{10} + 129625058738867 T^{12} - 12570839789334688 T^{14} + 129625058738867 p^{2} T^{16} - 1121527816102 p^{4} T^{18} + 8034728525 p^{6} T^{20} - 46401556 p^{8} T^{22} + 205663 p^{10} T^{24} - 630 p^{12} T^{26} + p^{14} T^{28} \) | |
| 97 | \( 1 - 338 T^{2} + 89883 T^{4} - 17338452 T^{6} + 2838599017 T^{8} - 387362241166 T^{10} + 46271580225019 T^{12} - 4782560588438936 T^{14} + 46271580225019 p^{2} T^{16} - 387362241166 p^{4} T^{18} + 2838599017 p^{6} T^{20} - 17338452 p^{8} T^{22} + 89883 p^{10} T^{24} - 338 p^{12} T^{26} + p^{14} T^{28} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.37113224124955552061446069038, −2.35376924658873216670847574504, −2.30623872357260491974502061926, −2.19874045077615079836292435149, −2.14082445251169540332775443447, −1.98974693912643723091793791776, −1.95555609297742365259568948829, −1.88595704186782133955462625949, −1.76070510385758001593402671859, −1.70628621449781434295205458531, −1.52797723580380790717111543349, −1.46405715724631064436795840652, −1.46388383503798225325044396265, −1.39185768838827022699272884779, −1.31878872990101505637627397274, −1.26110700353172745081326724602, −1.15390116448273413631959895469, −0.935254529789664300122089944739, −0.819874833532940181698179947472, −0.68729759395112468744307228468, −0.67658066817852354820312858245, −0.66945307523072899445079977163, −0.66136174743554035014245997354, −0.45508488703866759548761626096, −0.31212777877734438597966880129, 0.31212777877734438597966880129, 0.45508488703866759548761626096, 0.66136174743554035014245997354, 0.66945307523072899445079977163, 0.67658066817852354820312858245, 0.68729759395112468744307228468, 0.819874833532940181698179947472, 0.935254529789664300122089944739, 1.15390116448273413631959895469, 1.26110700353172745081326724602, 1.31878872990101505637627397274, 1.39185768838827022699272884779, 1.46388383503798225325044396265, 1.46405715724631064436795840652, 1.52797723580380790717111543349, 1.70628621449781434295205458531, 1.76070510385758001593402671859, 1.88595704186782133955462625949, 1.95555609297742365259568948829, 1.98974693912643723091793791776, 2.14082445251169540332775443447, 2.19874045077615079836292435149, 2.30623872357260491974502061926, 2.35376924658873216670847574504, 2.37113224124955552061446069038
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.