Dirichlet series
| L(s) = 1 | − 272·7-s − 1.49e4·25-s + 4.66e4·37-s + 3.55e4·43-s + 6.68e4·49-s − 2.10e5·67-s + 3.18e5·79-s + 4.85e5·109-s + 1.19e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.72e6·169-s + 173-s + 4.05e6·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2.09·7-s − 4.77·25-s + 5.59·37-s + 2.93·43-s + 3.97·49-s − 5.72·67-s + 5.73·79-s + 3.91·109-s + 7.41·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 4.64·169-s + 2.54e−6·173-s + 10.0·175-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 3^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.17739\times 10^{35}\) |
| Root analytic conductor: | \(12.7148\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 3^{32} \cdot 7^{16} ,\ ( \ : [5/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(7.953784922\) |
| \(L(\frac12)\) | \(\approx\) | \(7.953784922\) |
| \(L(\frac{7}{2})\) | 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 | \( 1 \) | |
| 7 | \( ( 1 + 136 T - 116 p^{2} T^{2} - 15272 p T^{3} + 811834 p^{3} T^{4} - 15272 p^{6} T^{5} - 116 p^{12} T^{6} + 136 p^{15} T^{7} + p^{20} T^{8} )^{2} \) | |
| good | 5 | \( ( 1 + 7456 T^{2} + 20026876 T^{4} + 18914913376 T^{6} + 2123009304166 T^{8} + 18914913376 p^{10} T^{10} + 20026876 p^{20} T^{12} + 7456 p^{30} T^{14} + p^{40} T^{16} )^{2} \) |
| 11 | \( ( 1 - 596776 T^{2} + 205526941396 T^{4} - 48955618804668856 T^{6} + \)\(89\!\cdots\!34\)\( T^{8} - 48955618804668856 p^{10} T^{10} + 205526941396 p^{20} T^{12} - 596776 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 13 | \( ( 1 - 862424 T^{2} + 471478194652 T^{4} - 126376554687929768 T^{6} + \)\(42\!\cdots\!50\)\( T^{8} - 126376554687929768 p^{10} T^{10} + 471478194652 p^{20} T^{12} - 862424 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 17 | \( ( 1 + 158528 p T^{2} + 1852607975068 T^{4} + 4995283249546697536 T^{6} + \)\(14\!\cdots\!58\)\( T^{8} + 4995283249546697536 p^{10} T^{10} + 1852607975068 p^{20} T^{12} + 158528 p^{31} T^{14} + p^{40} T^{16} )^{2} \) | |
| 19 | \( ( 1 + 211576 T^{2} + 11585613310876 T^{4} - 18068878253003016440 T^{6} + \)\(57\!\cdots\!22\)\( T^{8} - 18068878253003016440 p^{10} T^{10} + 11585613310876 p^{20} T^{12} + 211576 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 23 | \( ( 1 - 21865480 T^{2} + 309168260144500 T^{4} - \)\(30\!\cdots\!92\)\( T^{6} + \)\(22\!\cdots\!34\)\( T^{8} - \)\(30\!\cdots\!92\)\( p^{10} T^{10} + 309168260144500 p^{20} T^{12} - 21865480 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 29 | \( ( 1 - 104628928 T^{2} + 5519905557260452 T^{4} - \)\(18\!\cdots\!24\)\( T^{6} + \)\(45\!\cdots\!54\)\( T^{8} - \)\(18\!\cdots\!24\)\( p^{10} T^{10} + 5519905557260452 p^{20} T^{12} - 104628928 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 57370232 T^{2} + 1256540400528412 T^{4} + \)\(26\!\cdots\!96\)\( T^{6} - \)\(17\!\cdots\!70\)\( T^{8} + \)\(26\!\cdots\!96\)\( p^{10} T^{10} + 1256540400528412 p^{20} T^{12} - 57370232 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 11656 T + 254643364 T^{2} - 2154609442168 T^{3} + 26141528187032182 T^{4} - 2154609442168 p^{5} T^{5} + 254643364 p^{10} T^{6} - 11656 p^{15} T^{7} + p^{20} T^{8} )^{4} \) | |
| 41 | \( ( 1 + 13082368 T^{2} + 37218575083702108 T^{4} + \)\(69\!\cdots\!12\)\( p T^{6} + \)\(68\!\cdots\!46\)\( T^{8} + \)\(69\!\cdots\!12\)\( p^{11} T^{10} + 37218575083702108 p^{20} T^{12} + 13082368 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 8888 T + 433685596 T^{2} - 4347964099064 T^{3} + 84261591352052902 T^{4} - 4347964099064 p^{5} T^{5} + 433685596 p^{10} T^{6} - 8888 p^{15} T^{7} + p^{20} T^{8} )^{4} \) | |
| 47 | \( ( 1 + 1402742872 T^{2} + 939203032637090140 T^{4} + \)\(38\!\cdots\!56\)\( T^{6} + \)\(10\!\cdots\!10\)\( T^{8} + \)\(38\!\cdots\!56\)\( p^{10} T^{10} + 939203032637090140 p^{20} T^{12} + 1402742872 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 53 | \( ( 1 - 1719882784 T^{2} + 1537379272307815012 T^{4} - \)\(95\!\cdots\!60\)\( T^{6} + \)\(45\!\cdots\!62\)\( T^{8} - \)\(95\!\cdots\!60\)\( p^{10} T^{10} + 1537379272307815012 p^{20} T^{12} - 1719882784 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 59 | \( ( 1 + 1896153304 T^{2} + 3081511519737058876 T^{4} + \)\(29\!\cdots\!44\)\( T^{6} + \)\(25\!\cdots\!54\)\( T^{8} + \)\(29\!\cdots\!44\)\( p^{10} T^{10} + 3081511519737058876 p^{20} T^{12} + 1896153304 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 3536457896 T^{2} + 6913251186941453308 T^{4} - \)\(91\!\cdots\!52\)\( T^{6} + \)\(89\!\cdots\!02\)\( T^{8} - \)\(91\!\cdots\!52\)\( p^{10} T^{10} + 6913251186941453308 p^{20} T^{12} - 3536457896 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 67 | \( ( 1 + 52616 T + 4275206908 T^{2} + 120884458301768 T^{3} + 6779706135838027798 T^{4} + 120884458301768 p^{5} T^{5} + 4275206908 p^{10} T^{6} + 52616 p^{15} T^{7} + p^{20} T^{8} )^{4} \) | |
| 71 | \( ( 1 - 8709083560 T^{2} + 38850247452739804660 T^{4} - \)\(11\!\cdots\!52\)\( T^{6} + \)\(24\!\cdots\!86\)\( T^{8} - \)\(11\!\cdots\!52\)\( p^{10} T^{10} + 38850247452739804660 p^{20} T^{12} - 8709083560 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 8609840312 T^{2} + 32180604550065582652 T^{4} - \)\(74\!\cdots\!96\)\( T^{6} + \)\(15\!\cdots\!70\)\( T^{8} - \)\(74\!\cdots\!96\)\( p^{10} T^{10} + 32180604550065582652 p^{20} T^{12} - 8609840312 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 79 | \( ( 1 - 79576 T + 7013349868 T^{2} - 120512211334648 T^{3} + 10611814857165843430 T^{4} - 120512211334648 p^{5} T^{5} + 7013349868 p^{10} T^{6} - 79576 p^{15} T^{7} + p^{20} T^{8} )^{4} \) | |
| 83 | \( ( 1 + 14248484344 T^{2} + 55723281380625861052 T^{4} - \)\(14\!\cdots\!04\)\( T^{6} - \)\(15\!\cdots\!82\)\( T^{8} - \)\(14\!\cdots\!04\)\( p^{10} T^{10} + 55723281380625861052 p^{20} T^{12} + 14248484344 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 89 | \( ( 1 + 19734258880 T^{2} + \)\(24\!\cdots\!68\)\( T^{4} + \)\(20\!\cdots\!88\)\( T^{6} + \)\(13\!\cdots\!30\)\( T^{8} + \)\(20\!\cdots\!88\)\( p^{10} T^{10} + \)\(24\!\cdots\!68\)\( p^{20} T^{12} + 19734258880 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 40450347128 T^{2} + \)\(86\!\cdots\!80\)\( T^{4} - \)\(12\!\cdots\!36\)\( T^{6} + \)\(12\!\cdots\!14\)\( T^{8} - \)\(12\!\cdots\!36\)\( p^{10} T^{10} + \)\(86\!\cdots\!80\)\( p^{20} T^{12} - 40450347128 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.89494045163157295237855490174, −1.85728938027747818960092754083, −1.65339449057054872903366720532, −1.62831072403352745414168283924, −1.55594875577153263743876780379, −1.55061931080080482227743962689, −1.38936775825090491291739720204, −1.34257061742312565090616134713, −1.23441245107641587774952075572, −1.18469908822361215651065455870, −0.905079489708170862744729954122, −0.901919609246820688314771773721, −0.899255407379283209471970752038, −0.850194343875267379975934931708, −0.796491657590863610460183682627, −0.71975547172885547467893993021, −0.69768014834937019822537595398, −0.62217209539517914255610457735, −0.43017278693099237147985484115, −0.40886905852888036742411528635, −0.37930548562344117256521215926, −0.34390617573998728549714383360, −0.25994300329511416199749768407, −0.17221488437554683335778739941, −0.03970262918052901320774886922, 0.03970262918052901320774886922, 0.17221488437554683335778739941, 0.25994300329511416199749768407, 0.34390617573998728549714383360, 0.37930548562344117256521215926, 0.40886905852888036742411528635, 0.43017278693099237147985484115, 0.62217209539517914255610457735, 0.69768014834937019822537595398, 0.71975547172885547467893993021, 0.796491657590863610460183682627, 0.850194343875267379975934931708, 0.899255407379283209471970752038, 0.901919609246820688314771773721, 0.905079489708170862744729954122, 1.18469908822361215651065455870, 1.23441245107641587774952075572, 1.34257061742312565090616134713, 1.38936775825090491291739720204, 1.55061931080080482227743962689, 1.55594875577153263743876780379, 1.62831072403352745414168283924, 1.65339449057054872903366720532, 1.85728938027747818960092754083, 1.89494045163157295237855490174
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.