Dirichlet series
| L(s) = 1 | + 2·5-s + 4·9-s + 24·19-s + 24·29-s − 16·31-s + 16·41-s + 8·45-s − 20·49-s − 4·59-s + 16·61-s + 56·71-s + 16·79-s + 6·81-s + 16·89-s + 48·95-s − 52·101-s − 16·109-s + 52·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + 149-s + 151-s − 32·155-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 4/3·9-s + 5.50·19-s + 4.45·29-s − 2.87·31-s + 2.49·41-s + 1.19·45-s − 2.85·49-s − 0.520·59-s + 2.04·61-s + 6.64·71-s + 1.80·79-s + 2/3·81-s + 1.69·89-s + 4.92·95-s − 5.17·101-s − 1.53·109-s + 4.72·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + 0.0819·149-s + 0.0813·151-s − 2.57·155-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.09997\times 10^{18}\) |
| Root analytic conductor: | \(3.66263\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(80.71999281\) |
| \(L(\frac12)\) | \(\approx\) | \(80.71999281\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 - T^{2} + T^{4} )^{4} \) | |
| 5 | \( 1 - 2 T + 4 T^{2} - 4 p T^{3} + 62 T^{4} - 118 T^{5} + 8 p^{2} T^{6} - 134 p T^{7} + 1639 T^{8} - 134 p^{2} T^{9} + 8 p^{4} T^{10} - 118 p^{3} T^{11} + 62 p^{4} T^{12} - 4 p^{6} T^{13} + 4 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 + 20 T^{2} + 202 T^{4} + 1244 T^{6} + 7267 T^{8} + 1244 p^{2} T^{10} + 202 p^{4} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} \) | |
| good | 11 | \( ( 1 - 26 T^{2} + 28 T^{3} + 316 T^{4} - 518 T^{5} - 2872 T^{6} + 2968 T^{7} + 31007 T^{8} + 2968 p T^{9} - 2872 p^{2} T^{10} - 518 p^{3} T^{11} + 316 p^{4} T^{12} + 28 p^{5} T^{13} - 26 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 - 44 T^{2} + 1234 T^{4} - 23988 T^{6} + 358939 T^{8} - 23988 p^{2} T^{10} + 1234 p^{4} T^{12} - 44 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 + 76 T^{2} + 2664 T^{4} + 67392 T^{6} + 1583990 T^{8} + 33141472 T^{10} + 35499008 p T^{12} + 11039990476 T^{14} + 198790801859 T^{16} + 11039990476 p^{2} T^{18} + 35499008 p^{5} T^{20} + 33141472 p^{6} T^{22} + 1583990 p^{8} T^{24} + 67392 p^{10} T^{26} + 2664 p^{12} T^{28} + 76 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 - 12 T + 30 T^{2} + 80 T^{3} + 689 T^{4} - 9436 T^{5} + 28846 T^{6} - 59944 T^{7} + 291332 T^{8} - 59944 p T^{9} + 28846 p^{2} T^{10} - 9436 p^{3} T^{11} + 689 p^{4} T^{12} + 80 p^{5} T^{13} + 30 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 + 20 T^{2} - 1512 T^{4} - 18448 T^{6} + 1612902 T^{8} + 10149224 T^{10} - 54279328 p T^{12} - 1840459676 T^{14} + 770565238915 T^{16} - 1840459676 p^{2} T^{18} - 54279328 p^{5} T^{20} + 10149224 p^{6} T^{22} + 1612902 p^{8} T^{24} - 18448 p^{10} T^{26} - 1512 p^{12} T^{28} + 20 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 - 6 T + 78 T^{2} - 332 T^{3} + 2820 T^{4} - 332 p T^{5} + 78 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 31 | \( ( 1 + 8 T - 54 T^{2} - 352 T^{3} + 3285 T^{4} + 9272 T^{5} - 157694 T^{6} - 60416 T^{7} + 6248764 T^{8} - 60416 p T^{9} - 157694 p^{2} T^{10} + 9272 p^{3} T^{11} + 3285 p^{4} T^{12} - 352 p^{5} T^{13} - 54 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 + 128 T^{2} + 6574 T^{4} + 187656 T^{6} + 5183961 T^{8} + 158877780 T^{10} + 5690568750 T^{12} + 459641012708 T^{14} + 25029766954564 T^{16} + 459641012708 p^{2} T^{18} + 5690568750 p^{4} T^{20} + 158877780 p^{6} T^{22} + 5183961 p^{8} T^{24} + 187656 p^{10} T^{26} + 6574 p^{12} T^{28} + 128 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 4 T + 114 T^{2} - 346 T^{3} + 5976 T^{4} - 346 p T^{5} + 114 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 43 | \( ( 1 - 216 T^{2} + 22138 T^{4} - 1464812 T^{6} + 71619331 T^{8} - 1464812 p^{2} T^{10} + 22138 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 + 308 T^{2} + 50892 T^{4} + 5950136 T^{6} + 547353834 T^{8} + 41768130380 T^{10} + 2726163659344 T^{12} + 155042751818668 T^{14} + 7757667166613347 T^{16} + 155042751818668 p^{2} T^{18} + 2726163659344 p^{4} T^{20} + 41768130380 p^{6} T^{22} + 547353834 p^{8} T^{24} + 5950136 p^{10} T^{26} + 50892 p^{12} T^{28} + 308 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 + 264 T^{2} + 35652 T^{4} + 3177712 T^{6} + 212585578 T^{8} + 12279652216 T^{10} + 710817442960 T^{12} + 42611108403128 T^{14} + 2398437218467955 T^{16} + 42611108403128 p^{2} T^{18} + 710817442960 p^{4} T^{20} + 12279652216 p^{6} T^{22} + 212585578 p^{8} T^{24} + 3177712 p^{10} T^{26} + 35652 p^{12} T^{28} + 264 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 + 2 T - 226 T^{2} - 4 p T^{3} + 32016 T^{4} + 19820 T^{5} - 2961940 T^{6} - 407278 T^{7} + 205790979 T^{8} - 407278 p T^{9} - 2961940 p^{2} T^{10} + 19820 p^{3} T^{11} + 32016 p^{4} T^{12} - 4 p^{6} T^{13} - 226 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 8 T - 80 T^{2} + 376 T^{3} + 3998 T^{4} + 13396 T^{5} - 247632 T^{6} - 798752 T^{7} + 15355491 T^{8} - 798752 p T^{9} - 247632 p^{2} T^{10} + 13396 p^{3} T^{11} + 3998 p^{4} T^{12} + 376 p^{5} T^{13} - 80 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 + 356 T^{2} + 63502 T^{4} + 8090208 T^{6} + 858425049 T^{8} + 80231005476 T^{10} + 6774300927054 T^{12} + 525083738078264 T^{14} + 37066264561921156 T^{16} + 525083738078264 p^{2} T^{18} + 6774300927054 p^{4} T^{20} + 80231005476 p^{6} T^{22} + 858425049 p^{8} T^{24} + 8090208 p^{10} T^{26} + 63502 p^{12} T^{28} + 356 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 - 14 T + 194 T^{2} - 1648 T^{3} + 14264 T^{4} - 1648 p T^{5} + 194 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 73 | \( 1 + 312 T^{2} + 46894 T^{4} + 4662008 T^{6} + 358067385 T^{8} + 22613492772 T^{10} + 1186535858958 T^{12} + 57781939528716 T^{14} + 3463839236542756 T^{16} + 57781939528716 p^{2} T^{18} + 1186535858958 p^{4} T^{20} + 22613492772 p^{6} T^{22} + 358067385 p^{8} T^{24} + 4662008 p^{10} T^{26} + 46894 p^{12} T^{28} + 312 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 - 8 T - 98 T^{2} + 1128 T^{3} - 3647 T^{4} + 31028 T^{5} - 546546 T^{6} - 5486636 T^{7} + 152161556 T^{8} - 5486636 p T^{9} - 546546 p^{2} T^{10} + 31028 p^{3} T^{11} - 3647 p^{4} T^{12} + 1128 p^{5} T^{13} - 98 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 - 516 T^{2} + 125904 T^{4} - 18859016 T^{6} + 1891189930 T^{8} - 18859016 p^{2} T^{10} + 125904 p^{4} T^{12} - 516 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 8 T - 98 T^{2} + 1004 T^{3} + 4584 T^{4} - 89290 T^{5} + 1051584 T^{6} + 32888 p T^{7} - 1604897 p T^{8} + 32888 p^{2} T^{9} + 1051584 p^{2} T^{10} - 89290 p^{3} T^{11} + 4584 p^{4} T^{12} + 1004 p^{5} T^{13} - 98 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 756 T^{2} + 251932 T^{4} - 48333356 T^{6} + 5838717718 T^{8} - 48333356 p^{2} T^{10} + 251932 p^{4} T^{12} - 756 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.38434441098616219941927776967, −2.36402257014520284464758268459, −2.19402434558170309101938815260, −2.18399289950082754027548638553, −2.11194492846324396010755219409, −2.02877504797785102237549518046, −1.82737055833696659935869948791, −1.79747632206549500894687027418, −1.64397178127814168090516166126, −1.64075539298886924499442289563, −1.55697394187308836910280493898, −1.47770104318603635285553438167, −1.37739611719529637500157554516, −1.22331923829352667649465670471, −1.21063428784305887483752987514, −1.17605381639675393090990346317, −1.10347858237847259015688205272, −0.965638438997812942556293038325, −0.911339463709197491646545475813, −0.829114927459818394211839250307, −0.60026667008355336884904350996, −0.57456055326585053514124437150, −0.49904705240309182715518923890, −0.36692422029192091202512423853, −0.17659922311406160633181998618, 0.17659922311406160633181998618, 0.36692422029192091202512423853, 0.49904705240309182715518923890, 0.57456055326585053514124437150, 0.60026667008355336884904350996, 0.829114927459818394211839250307, 0.911339463709197491646545475813, 0.965638438997812942556293038325, 1.10347858237847259015688205272, 1.17605381639675393090990346317, 1.21063428784305887483752987514, 1.22331923829352667649465670471, 1.37739611719529637500157554516, 1.47770104318603635285553438167, 1.55697394187308836910280493898, 1.64075539298886924499442289563, 1.64397178127814168090516166126, 1.79747632206549500894687027418, 1.82737055833696659935869948791, 2.02877504797785102237549518046, 2.11194492846324396010755219409, 2.18399289950082754027548638553, 2.19402434558170309101938815260, 2.36402257014520284464758268459, 2.38434441098616219941927776967
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.