Dirichlet series
| L(s) = 1 | + 2·2-s − 2·3-s + 5·4-s + 7·5-s − 4·6-s + 2·7-s + 8·8-s + 10·9-s + 14·10-s − 10·11-s − 10·12-s + 6·13-s + 4·14-s − 14·15-s + 13·16-s − 17-s + 20·18-s + 6·19-s + 35·20-s − 4·21-s − 20·22-s + 12·23-s − 16·24-s + 21·25-s + 12·26-s − 14·27-s + 10·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 5/2·4-s + 3.13·5-s − 1.63·6-s + 0.755·7-s + 2.82·8-s + 10/3·9-s + 4.42·10-s − 3.01·11-s − 2.88·12-s + 1.66·13-s + 1.06·14-s − 3.61·15-s + 13/4·16-s − 0.242·17-s + 4.71·18-s + 1.37·19-s + 7.82·20-s − 0.872·21-s − 4.26·22-s + 2.50·23-s − 3.26·24-s + 21/5·25-s + 2.35·26-s − 2.69·27-s + 1.88·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(5^{14} \cdot 37^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(235.658\) |
| Root analytic conductor: | \(1.21541\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 5^{14} \cdot 37^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(16.83812331\) |
| \(L(\frac12)\) | \(\approx\) | \(16.83812331\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 - T + T^{2} )^{7} \) |
| 37 | \( 1 - 12 T + 77 T^{2} - 530 T^{3} + 2235 T^{4} - 8080 T^{5} - 5310 T^{6} + 434817 T^{7} - 5310 p T^{8} - 8080 p^{2} T^{9} + 2235 p^{3} T^{10} - 530 p^{4} T^{11} + 77 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| good | 2 | \( 1 - p T - T^{2} + p^{2} T^{3} - 3 p T^{5} + 3 p T^{6} - p T^{7} + T^{8} - 25 T^{10} + 57 T^{11} - 5 p^{2} T^{12} - 37 T^{13} + 43 T^{14} - 37 p T^{15} - 5 p^{4} T^{16} + 57 p^{3} T^{17} - 25 p^{4} T^{18} + p^{6} T^{20} - p^{8} T^{21} + 3 p^{9} T^{22} - 3 p^{10} T^{23} + p^{13} T^{25} - p^{12} T^{26} - p^{14} T^{27} + p^{14} T^{28} \) |
| 3 | \( 1 + 2 T - 2 p T^{2} - 2 p^{2} T^{3} - T^{4} + 35 T^{5} + 47 T^{6} + 122 T^{7} + 77 p T^{8} - 245 T^{9} - 172 p^{2} T^{10} - 665 p T^{11} + 341 T^{12} + 1657 p T^{13} + 10699 T^{14} + 1657 p^{2} T^{15} + 341 p^{2} T^{16} - 665 p^{4} T^{17} - 172 p^{6} T^{18} - 245 p^{5} T^{19} + 77 p^{7} T^{20} + 122 p^{7} T^{21} + 47 p^{8} T^{22} + 35 p^{9} T^{23} - p^{10} T^{24} - 2 p^{13} T^{25} - 2 p^{13} T^{26} + 2 p^{13} T^{27} + p^{14} T^{28} \) | |
| 7 | \( 1 - 2 T - 19 T^{2} - 6 T^{3} + 235 T^{4} + 446 T^{5} - 1262 T^{6} - 5119 T^{7} - 888 T^{8} + 23963 T^{9} + 49818 T^{10} - 2630 p T^{11} - 259614 T^{12} - 60800 T^{13} + 540023 T^{14} - 60800 p T^{15} - 259614 p^{2} T^{16} - 2630 p^{4} T^{17} + 49818 p^{4} T^{18} + 23963 p^{5} T^{19} - 888 p^{6} T^{20} - 5119 p^{7} T^{21} - 1262 p^{8} T^{22} + 446 p^{9} T^{23} + 235 p^{10} T^{24} - 6 p^{11} T^{25} - 19 p^{12} T^{26} - 2 p^{13} T^{27} + p^{14} T^{28} \) | |
| 11 | \( ( 1 + 5 T + 48 T^{2} + 156 T^{3} + 914 T^{4} + 2058 T^{5} + 10829 T^{6} + 20501 T^{7} + 10829 p T^{8} + 2058 p^{2} T^{9} + 914 p^{3} T^{10} + 156 p^{4} T^{11} + 48 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 13 | \( 1 - 6 T - 33 T^{2} + 266 T^{3} + 35 p T^{4} - 5269 T^{5} - 4950 T^{6} + 56715 T^{7} + 138842 T^{8} - 364275 T^{9} - 3770595 T^{10} + 2045614 T^{11} + 67384158 T^{12} - 9339605 T^{13} - 942811663 T^{14} - 9339605 p T^{15} + 67384158 p^{2} T^{16} + 2045614 p^{3} T^{17} - 3770595 p^{4} T^{18} - 364275 p^{5} T^{19} + 138842 p^{6} T^{20} + 56715 p^{7} T^{21} - 4950 p^{8} T^{22} - 5269 p^{9} T^{23} + 35 p^{11} T^{24} + 266 p^{11} T^{25} - 33 p^{12} T^{26} - 6 p^{13} T^{27} + p^{14} T^{28} \) | |
| 17 | \( 1 + T - 61 T^{2} - 8 T^{3} + 1594 T^{4} - 4 p^{2} T^{5} - 31213 T^{6} + 13754 T^{7} + 720997 T^{8} + 852 p^{2} T^{9} - 16357766 T^{10} + 1821109 T^{11} + 311698769 T^{12} - 85365254 T^{13} - 5410277981 T^{14} - 85365254 p T^{15} + 311698769 p^{2} T^{16} + 1821109 p^{3} T^{17} - 16357766 p^{4} T^{18} + 852 p^{7} T^{19} + 720997 p^{6} T^{20} + 13754 p^{7} T^{21} - 31213 p^{8} T^{22} - 4 p^{11} T^{23} + 1594 p^{10} T^{24} - 8 p^{11} T^{25} - 61 p^{12} T^{26} + p^{13} T^{27} + p^{14} T^{28} \) | |
| 19 | \( 1 - 6 T - 71 T^{2} + 312 T^{3} + 3658 T^{4} - 9145 T^{5} - 130964 T^{6} + 130696 T^{7} + 190539 p T^{8} - 240373 T^{9} - 78603255 T^{10} - 31347982 T^{11} + 1495132055 T^{12} + 350340880 T^{13} - 27618196471 T^{14} + 350340880 p T^{15} + 1495132055 p^{2} T^{16} - 31347982 p^{3} T^{17} - 78603255 p^{4} T^{18} - 240373 p^{5} T^{19} + 190539 p^{7} T^{20} + 130696 p^{7} T^{21} - 130964 p^{8} T^{22} - 9145 p^{9} T^{23} + 3658 p^{10} T^{24} + 312 p^{11} T^{25} - 71 p^{12} T^{26} - 6 p^{13} T^{27} + p^{14} T^{28} \) | |
| 23 | \( ( 1 - 6 T + 80 T^{2} - 258 T^{3} + 1412 T^{4} + 4361 T^{5} - 28804 T^{6} + 327641 T^{7} - 28804 p T^{8} + 4361 p^{2} T^{9} + 1412 p^{3} T^{10} - 258 p^{4} T^{11} + 80 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 29 | \( ( 1 + 6 T + 135 T^{2} + 834 T^{3} + 9375 T^{4} + 51830 T^{5} + 412788 T^{6} + 1893371 T^{7} + 412788 p T^{8} + 51830 p^{2} T^{9} + 9375 p^{3} T^{10} + 834 p^{4} T^{11} + 135 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 31 | \( ( 1 + 4 T + 118 T^{2} + 270 T^{3} + 7041 T^{4} + 11844 T^{5} + 297899 T^{6} + 413135 T^{7} + 297899 p T^{8} + 11844 p^{2} T^{9} + 7041 p^{3} T^{10} + 270 p^{4} T^{11} + 118 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 41 | \( 1 + 3 T - 132 T^{2} + 219 T^{3} + 9501 T^{4} - 45631 T^{5} - 258957 T^{6} + 2740826 T^{7} - 2786166 T^{8} - 28793740 T^{9} + 150222736 T^{10} - 3258693180 T^{11} + 28014711812 T^{12} + 100963739871 T^{13} - 2053507332125 T^{14} + 100963739871 p T^{15} + 28014711812 p^{2} T^{16} - 3258693180 p^{3} T^{17} + 150222736 p^{4} T^{18} - 28793740 p^{5} T^{19} - 2786166 p^{6} T^{20} + 2740826 p^{7} T^{21} - 258957 p^{8} T^{22} - 45631 p^{9} T^{23} + 9501 p^{10} T^{24} + 219 p^{11} T^{25} - 132 p^{12} T^{26} + 3 p^{13} T^{27} + p^{14} T^{28} \) | |
| 43 | \( ( 1 + 19 T + 238 T^{2} + 2019 T^{3} + 16130 T^{4} + 125883 T^{5} + 1029703 T^{6} + 7244919 T^{7} + 1029703 p T^{8} + 125883 p^{2} T^{9} + 16130 p^{3} T^{10} + 2019 p^{4} T^{11} + 238 p^{5} T^{12} + 19 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 47 | \( ( 1 - 2 T + 320 T^{2} - 547 T^{3} + 44291 T^{4} - 63105 T^{5} + 3437391 T^{6} - 3930513 T^{7} + 3437391 p T^{8} - 63105 p^{2} T^{9} + 44291 p^{3} T^{10} - 547 p^{4} T^{11} + 320 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 53 | \( 1 + 2 T - 127 T^{2} + 1268 T^{3} + 8755 T^{4} - 2892 p T^{5} + 831891 T^{6} + 8097483 T^{7} - 101615229 T^{8} + 513019129 T^{9} + 4221002934 T^{10} - 57306112812 T^{11} + 285397794925 T^{12} + 2051405818738 T^{13} - 25941522997123 T^{14} + 2051405818738 p T^{15} + 285397794925 p^{2} T^{16} - 57306112812 p^{3} T^{17} + 4221002934 p^{4} T^{18} + 513019129 p^{5} T^{19} - 101615229 p^{6} T^{20} + 8097483 p^{7} T^{21} + 831891 p^{8} T^{22} - 2892 p^{10} T^{23} + 8755 p^{10} T^{24} + 1268 p^{11} T^{25} - 127 p^{12} T^{26} + 2 p^{13} T^{27} + p^{14} T^{28} \) | |
| 59 | \( 1 + 18 T + 53 T^{2} + 504 T^{3} + 22362 T^{4} + 148299 T^{5} - 8344 T^{6} + 9002010 T^{7} + 136278493 T^{8} + 349871229 T^{9} + 2385685435 T^{10} + 61208352606 T^{11} + 396044441735 T^{12} + 2332537905570 T^{13} + 22687543993929 T^{14} + 2332537905570 p T^{15} + 396044441735 p^{2} T^{16} + 61208352606 p^{3} T^{17} + 2385685435 p^{4} T^{18} + 349871229 p^{5} T^{19} + 136278493 p^{6} T^{20} + 9002010 p^{7} T^{21} - 8344 p^{8} T^{22} + 148299 p^{9} T^{23} + 22362 p^{10} T^{24} + 504 p^{11} T^{25} + 53 p^{12} T^{26} + 18 p^{13} T^{27} + p^{14} T^{28} \) | |
| 61 | \( 1 + 20 T - 21 T^{2} - 1310 T^{3} + 20850 T^{4} + 171199 T^{5} - 1748120 T^{6} + 1955540 T^{7} + 200730705 T^{8} - 382422771 T^{9} - 5513294787 T^{10} + 76097253270 T^{11} + 175164995925 T^{12} - 721613454826 T^{13} + 19385114629999 T^{14} - 721613454826 p T^{15} + 175164995925 p^{2} T^{16} + 76097253270 p^{3} T^{17} - 5513294787 p^{4} T^{18} - 382422771 p^{5} T^{19} + 200730705 p^{6} T^{20} + 1955540 p^{7} T^{21} - 1748120 p^{8} T^{22} + 171199 p^{9} T^{23} + 20850 p^{10} T^{24} - 1310 p^{11} T^{25} - 21 p^{12} T^{26} + 20 p^{13} T^{27} + p^{14} T^{28} \) | |
| 67 | \( 1 + 20 T - 128 T^{2} - 4372 T^{3} + 17181 T^{4} + 644764 T^{5} - 1773494 T^{6} - 61081543 T^{7} + 261506048 T^{8} + 66607188 p T^{9} - 31850143600 T^{10} - 219732662866 T^{11} + 3259466467250 T^{12} + 5745933232287 T^{13} - 246025846105395 T^{14} + 5745933232287 p T^{15} + 3259466467250 p^{2} T^{16} - 219732662866 p^{3} T^{17} - 31850143600 p^{4} T^{18} + 66607188 p^{6} T^{19} + 261506048 p^{6} T^{20} - 61081543 p^{7} T^{21} - 1773494 p^{8} T^{22} + 644764 p^{9} T^{23} + 17181 p^{10} T^{24} - 4372 p^{11} T^{25} - 128 p^{12} T^{26} + 20 p^{13} T^{27} + p^{14} T^{28} \) | |
| 71 | \( 1 + 11 T - 208 T^{2} - 917 T^{3} + 39971 T^{4} - 13769 T^{5} - 4074953 T^{6} + 23038258 T^{7} + 281978978 T^{8} - 2908272514 T^{9} - 5103419146 T^{10} + 221582973432 T^{11} - 743536639962 T^{12} - 6068539464651 T^{13} + 97884259065037 T^{14} - 6068539464651 p T^{15} - 743536639962 p^{2} T^{16} + 221582973432 p^{3} T^{17} - 5103419146 p^{4} T^{18} - 2908272514 p^{5} T^{19} + 281978978 p^{6} T^{20} + 23038258 p^{7} T^{21} - 4074953 p^{8} T^{22} - 13769 p^{9} T^{23} + 39971 p^{10} T^{24} - 917 p^{11} T^{25} - 208 p^{12} T^{26} + 11 p^{13} T^{27} + p^{14} T^{28} \) | |
| 73 | \( ( 1 + 18 T + 211 T^{2} + 26 p T^{3} + 30890 T^{4} + 348146 T^{5} + 3039018 T^{6} + 20816303 T^{7} + 3039018 p T^{8} + 348146 p^{2} T^{9} + 30890 p^{3} T^{10} + 26 p^{5} T^{11} + 211 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 79 | \( 1 - 23 T - 59 T^{2} + 5770 T^{3} - 20821 T^{4} - 787185 T^{5} + 6058896 T^{6} + 58075960 T^{7} - 709761400 T^{8} - 2085600577 T^{9} + 49141993563 T^{10} - 55015484785 T^{11} - 1456104699135 T^{12} + 4070054318398 T^{13} + 21724350399443 T^{14} + 4070054318398 p T^{15} - 1456104699135 p^{2} T^{16} - 55015484785 p^{3} T^{17} + 49141993563 p^{4} T^{18} - 2085600577 p^{5} T^{19} - 709761400 p^{6} T^{20} + 58075960 p^{7} T^{21} + 6058896 p^{8} T^{22} - 787185 p^{9} T^{23} - 20821 p^{10} T^{24} + 5770 p^{11} T^{25} - 59 p^{12} T^{26} - 23 p^{13} T^{27} + p^{14} T^{28} \) | |
| 83 | \( 1 + 9 T - 357 T^{2} - 38 p T^{3} + 67672 T^{4} + 538774 T^{5} - 10030984 T^{6} - 63401162 T^{7} + 1303818451 T^{8} + 5636932063 T^{9} - 149195088274 T^{10} - 347973530538 T^{11} + 15130057053447 T^{12} + 10253989605968 T^{13} - 1348093825435307 T^{14} + 10253989605968 p T^{15} + 15130057053447 p^{2} T^{16} - 347973530538 p^{3} T^{17} - 149195088274 p^{4} T^{18} + 5636932063 p^{5} T^{19} + 1303818451 p^{6} T^{20} - 63401162 p^{7} T^{21} - 10030984 p^{8} T^{22} + 538774 p^{9} T^{23} + 67672 p^{10} T^{24} - 38 p^{12} T^{25} - 357 p^{12} T^{26} + 9 p^{13} T^{27} + p^{14} T^{28} \) | |
| 89 | \( 1 + 16 T + 15 T^{2} + 608 T^{3} + 19888 T^{4} + 110026 T^{5} + 1762379 T^{6} + 18603401 T^{7} + 11834709 T^{8} + 1168516275 T^{9} + 14048369817 T^{10} + 8327520636 T^{11} + 1078488527133 T^{12} + 7744542785120 T^{13} - 49730327540927 T^{14} + 7744542785120 p T^{15} + 1078488527133 p^{2} T^{16} + 8327520636 p^{3} T^{17} + 14048369817 p^{4} T^{18} + 1168516275 p^{5} T^{19} + 11834709 p^{6} T^{20} + 18603401 p^{7} T^{21} + 1762379 p^{8} T^{22} + 110026 p^{9} T^{23} + 19888 p^{10} T^{24} + 608 p^{11} T^{25} + 15 p^{12} T^{26} + 16 p^{13} T^{27} + p^{14} T^{28} \) | |
| 97 | \( ( 1 + 31 T + 608 T^{2} + 9736 T^{3} + 133294 T^{4} + 1545074 T^{5} + 16719681 T^{6} + 170470865 T^{7} + 16719681 p T^{8} + 1545074 p^{2} T^{9} + 133294 p^{3} T^{10} + 9736 p^{4} T^{11} + 608 p^{5} T^{12} + 31 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.19645906791795638925069289392, −4.02125461620334444858689425613, −3.93234909917639854572488259697, −3.56264921086112335179284918566, −3.53945762625993817309088525932, −3.50268332517667265698659000127, −3.40110495938444861606819623869, −3.26872563980173923392221758344, −3.06720469135722570673977541632, −2.96482769735574168976703878192, −2.91051973802388526826710928289, −2.85705018877607305505293746622, −2.78970049191084226161669154708, −2.67961802799751842147547409410, −2.40709289836287892493367375454, −2.17667146703163883836235564393, −1.98075743165114652958384504085, −1.83181559817393479854572829900, −1.82700343890040531540529183737, −1.73797614666267442176078988048, −1.72273070065560010113177091014, −1.47485728759692458017305214418, −1.29180912892548905053797050500, −1.18347042204952085329508519468, −0.72077516449810680305565486129, 0.72077516449810680305565486129, 1.18347042204952085329508519468, 1.29180912892548905053797050500, 1.47485728759692458017305214418, 1.72273070065560010113177091014, 1.73797614666267442176078988048, 1.82700343890040531540529183737, 1.83181559817393479854572829900, 1.98075743165114652958384504085, 2.17667146703163883836235564393, 2.40709289836287892493367375454, 2.67961802799751842147547409410, 2.78970049191084226161669154708, 2.85705018877607305505293746622, 2.91051973802388526826710928289, 2.96482769735574168976703878192, 3.06720469135722570673977541632, 3.26872563980173923392221758344, 3.40110495938444861606819623869, 3.50268332517667265698659000127, 3.53945762625993817309088525932, 3.56264921086112335179284918566, 3.93234909917639854572488259697, 4.02125461620334444858689425613, 4.19645906791795638925069289392
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.