Dirichlet series
| L(s) = 1 | − 2·2-s + 2·4-s − 16·7-s − 2·8-s − 8·9-s + 32·14-s + 3·16-s + 16·18-s − 8·25-s − 32·28-s − 4·32-s − 16·36-s + 32·47-s + 136·49-s + 16·50-s + 32·56-s + 128·63-s − 4·64-s − 64·71-s + 16·72-s + 48·79-s + 36·81-s − 64·94-s − 272·98-s − 16·100-s − 48·103-s − 48·112-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 6.04·7-s − 0.707·8-s − 8/3·9-s + 8.55·14-s + 3/4·16-s + 3.77·18-s − 8/5·25-s − 6.04·28-s − 0.707·32-s − 8/3·36-s + 4.66·47-s + 19.4·49-s + 2.26·50-s + 4.27·56-s + 16.1·63-s − 1/2·64-s − 7.59·71-s + 1.88·72-s + 5.40·79-s + 4·81-s − 6.60·94-s − 27.4·98-s − 8/5·100-s − 4.72·103-s − 4.53·112-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{48} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.67842\times 10^{13}\) |
| Root analytic conductor: | \(2.58987\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{48} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.3095982243\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3095982243\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + p T + p T^{2} + p T^{3} + T^{4} + p^{3} T^{6} + 3 p^{3} T^{7} + p^{5} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} + p^{4} T^{12} + p^{6} T^{13} + p^{7} T^{14} + p^{8} T^{15} + p^{8} T^{16} \) |
| 3 | \( ( 1 + T^{2} )^{8} \) | |
| 5 | \( ( 1 + T^{2} )^{8} \) | |
| 7 | \( ( 1 + T )^{16} \) | |
| good | 11 | \( 1 - 72 T^{2} + 2824 T^{4} - 78552 T^{6} + 1718716 T^{8} - 31168776 T^{10} + 482235704 T^{12} - 6471175256 T^{14} + 75991506886 T^{16} - 6471175256 p^{2} T^{18} + 482235704 p^{4} T^{20} - 31168776 p^{6} T^{22} + 1718716 p^{8} T^{24} - 78552 p^{10} T^{26} + 2824 p^{12} T^{28} - 72 p^{14} T^{30} + p^{16} T^{32} \) |
| 13 | \( 1 - 48 T^{2} + 136 p T^{4} - 47568 T^{6} + 1087356 T^{8} - 21295088 T^{10} + 366244952 T^{12} - 433269072 p T^{14} + 77016950214 T^{16} - 433269072 p^{3} T^{18} + 366244952 p^{4} T^{20} - 21295088 p^{6} T^{22} + 1087356 p^{8} T^{24} - 47568 p^{10} T^{26} + 136 p^{13} T^{28} - 48 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( ( 1 + 52 T^{2} + 80 T^{3} + 1444 T^{4} + 3344 T^{5} + 34764 T^{6} + 65696 T^{7} + 688182 T^{8} + 65696 p T^{9} + 34764 p^{2} T^{10} + 3344 p^{3} T^{11} + 1444 p^{4} T^{12} + 80 p^{5} T^{13} + 52 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 - 104 T^{2} + 5576 T^{4} - 226232 T^{6} + 7807676 T^{8} - 228982568 T^{10} + 305488296 p T^{12} - 131378881592 T^{14} + 2655541261638 T^{16} - 131378881592 p^{2} T^{18} + 305488296 p^{5} T^{20} - 228982568 p^{6} T^{22} + 7807676 p^{8} T^{24} - 226232 p^{10} T^{26} + 5576 p^{12} T^{28} - 104 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( ( 1 + 76 T^{2} - 160 T^{3} + 2820 T^{4} - 9312 T^{5} + 89844 T^{6} - 250048 T^{7} + 2414390 T^{8} - 250048 p T^{9} + 89844 p^{2} T^{10} - 9312 p^{3} T^{11} + 2820 p^{4} T^{12} - 160 p^{5} T^{13} + 76 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 - 152 T^{2} + 12376 T^{4} - 700936 T^{6} + 30056860 T^{8} - 1018356376 T^{10} + 28518902248 T^{12} - 718944065352 T^{14} + 19173038412422 T^{16} - 718944065352 p^{2} T^{18} + 28518902248 p^{4} T^{20} - 1018356376 p^{6} T^{22} + 30056860 p^{8} T^{24} - 700936 p^{10} T^{26} + 12376 p^{12} T^{28} - 152 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( ( 1 + 120 T^{2} - 176 T^{3} + 7332 T^{4} - 16048 T^{5} + 325480 T^{6} - 712832 T^{7} + 11400694 T^{8} - 712832 p T^{9} + 325480 p^{2} T^{10} - 16048 p^{3} T^{11} + 7332 p^{4} T^{12} - 176 p^{5} T^{13} + 120 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 - 424 T^{2} + 86104 T^{4} - 11188280 T^{6} + 1049124380 T^{8} - 75901344424 T^{10} + 4417616980712 T^{12} - 212390270335480 T^{14} + 8561568190519814 T^{16} - 212390270335480 p^{2} T^{18} + 4417616980712 p^{4} T^{20} - 75901344424 p^{6} T^{22} + 1049124380 p^{8} T^{24} - 11188280 p^{10} T^{26} + 86104 p^{12} T^{28} - 424 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 + 112 T^{2} - 224 T^{3} + 9052 T^{4} - 17376 T^{5} + 558224 T^{6} - 1062464 T^{7} + 25205126 T^{8} - 1062464 p T^{9} + 558224 p^{2} T^{10} - 17376 p^{3} T^{11} + 9052 p^{4} T^{12} - 224 p^{5} T^{13} + 112 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 - 216 T^{2} + 20664 T^{4} - 1275592 T^{6} + 67740892 T^{8} - 83614984 p T^{10} + 186200874120 T^{12} - 9181766409096 T^{14} + 417451835547526 T^{16} - 9181766409096 p^{2} T^{18} + 186200874120 p^{4} T^{20} - 83614984 p^{7} T^{22} + 67740892 p^{8} T^{24} - 1275592 p^{10} T^{26} + 20664 p^{12} T^{28} - 216 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( ( 1 - 16 T + 380 T^{2} - 4320 T^{3} + 59700 T^{4} - 527040 T^{5} + 5354116 T^{6} - 38210288 T^{7} + 308643094 T^{8} - 38210288 p T^{9} + 5354116 p^{2} T^{10} - 527040 p^{3} T^{11} + 59700 p^{4} T^{12} - 4320 p^{5} T^{13} + 380 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 - 360 T^{2} + 61352 T^{4} - 6619192 T^{6} + 520109052 T^{8} - 33338877736 T^{10} + 1936667618456 T^{12} - 108570604943800 T^{14} + 5885365267878726 T^{16} - 108570604943800 p^{2} T^{18} + 1936667618456 p^{4} T^{20} - 33338877736 p^{6} T^{22} + 520109052 p^{8} T^{24} - 6619192 p^{10} T^{26} + 61352 p^{12} T^{28} - 360 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 - 416 T^{2} + 91000 T^{4} - 13795040 T^{6} + 1616387804 T^{8} - 155188221600 T^{10} + 12666244788680 T^{12} - 899735530617568 T^{14} + 56387594275420806 T^{16} - 899735530617568 p^{2} T^{18} + 12666244788680 p^{4} T^{20} - 155188221600 p^{6} T^{22} + 1616387804 p^{8} T^{24} - 13795040 p^{10} T^{26} + 91000 p^{12} T^{28} - 416 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( 1 - 440 T^{2} + 100760 T^{4} - 15672168 T^{6} + 1842816028 T^{8} - 174536393912 T^{10} + 13958212904104 T^{12} - 981168033645736 T^{14} + 62501078738509702 T^{16} - 981168033645736 p^{2} T^{18} + 13958212904104 p^{4} T^{20} - 174536393912 p^{6} T^{22} + 1842816028 p^{8} T^{24} - 15672168 p^{10} T^{26} + 100760 p^{12} T^{28} - 440 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( 1 - 504 T^{2} + 121592 T^{4} - 19138728 T^{6} + 2287412700 T^{8} - 228724422008 T^{10} + 20163231832904 T^{12} - 1592528304782952 T^{14} + 112747680608235526 T^{16} - 1592528304782952 p^{2} T^{18} + 20163231832904 p^{4} T^{20} - 228724422008 p^{6} T^{22} + 2287412700 p^{8} T^{24} - 19138728 p^{10} T^{26} + 121592 p^{12} T^{28} - 504 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 + 32 T + 664 T^{2} + 9952 T^{3} + 124196 T^{4} + 1313600 T^{5} + 12671688 T^{6} + 112512000 T^{7} + 968740278 T^{8} + 112512000 p T^{9} + 12671688 p^{2} T^{10} + 1313600 p^{3} T^{11} + 124196 p^{4} T^{12} + 9952 p^{5} T^{13} + 664 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 + 280 T^{2} + 384 T^{3} + 43764 T^{4} + 90784 T^{5} + 4762184 T^{6} + 11265504 T^{7} + 392302230 T^{8} + 11265504 p T^{9} + 4762184 p^{2} T^{10} + 90784 p^{3} T^{11} + 43764 p^{4} T^{12} + 384 p^{5} T^{13} + 280 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 - 24 T + 656 T^{2} - 10136 T^{3} + 165212 T^{4} - 1941176 T^{5} + 23842160 T^{6} - 227897912 T^{7} + 2274147014 T^{8} - 227897912 p T^{9} + 23842160 p^{2} T^{10} - 1941176 p^{3} T^{11} + 165212 p^{4} T^{12} - 10136 p^{5} T^{13} + 656 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 432 T^{2} + 108728 T^{4} - 20115344 T^{6} + 3008716764 T^{8} - 382671494576 T^{10} + 42578063242632 T^{12} - 4199503519970832 T^{14} + 368916256295379718 T^{16} - 4199503519970832 p^{2} T^{18} + 42578063242632 p^{4} T^{20} - 382671494576 p^{6} T^{22} + 3008716764 p^{8} T^{24} - 20115344 p^{10} T^{26} + 108728 p^{12} T^{28} - 432 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 352 T^{2} + 160 T^{3} + 60860 T^{4} + 6304 T^{5} + 7473568 T^{6} - 4616768 T^{7} + 737540166 T^{8} - 4616768 p T^{9} + 7473568 p^{2} T^{10} + 6304 p^{3} T^{11} + 60860 p^{4} T^{12} + 160 p^{5} T^{13} + 352 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 + 440 T^{2} + 32 T^{3} + 78132 T^{4} + 24768 T^{5} + 7608616 T^{6} + 5674016 T^{7} + 631877846 T^{8} + 5674016 p T^{9} + 7608616 p^{2} T^{10} + 24768 p^{3} T^{11} + 78132 p^{4} T^{12} + 32 p^{5} T^{13} + 440 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
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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
−2.72132182159216383010066003251, −2.64728538461673053716542535172, −2.50302996109418489356944167003, −2.46491112873632075843835892862, −2.46024082383777845584782794774, −2.37964396021329607176702850962, −2.21236932372098739778431409476, −2.18478608541467775946335813821, −1.95648226608659195829809005558, −1.94430296340110374354680120554, −1.88235415207616651772605461718, −1.75817301203454028652427158608, −1.75443845657724113318048743773, −1.64991538557305856274510980708, −1.42337791159200064211173788892, −1.23694800591260032056278000595, −1.01911840956166058357647264509, −0.947803893868049867318126047917, −0.910398927042086078712782438559, −0.828411997456586016282155148234, −0.64608033735916847206474051402, −0.37608620825975123004793881673, −0.37069758145400501676277992610, −0.25296189599720252530224011904, −0.21893450936345267252537407537, 0.21893450936345267252537407537, 0.25296189599720252530224011904, 0.37069758145400501676277992610, 0.37608620825975123004793881673, 0.64608033735916847206474051402, 0.828411997456586016282155148234, 0.910398927042086078712782438559, 0.947803893868049867318126047917, 1.01911840956166058357647264509, 1.23694800591260032056278000595, 1.42337791159200064211173788892, 1.64991538557305856274510980708, 1.75443845657724113318048743773, 1.75817301203454028652427158608, 1.88235415207616651772605461718, 1.94430296340110374354680120554, 1.95648226608659195829809005558, 2.18478608541467775946335813821, 2.21236932372098739778431409476, 2.37964396021329607176702850962, 2.46024082383777845584782794774, 2.46491112873632075843835892862, 2.50302996109418489356944167003, 2.64728538461673053716542535172, 2.72132182159216383010066003251
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.