Dirichlet series
| L(s) = 1 | + 4·4-s + 4·5-s + 4·9-s + 6·16-s − 24·19-s + 16·20-s + 6·25-s + 32·29-s + 32·31-s + 16·36-s − 48·41-s + 16·45-s − 40·59-s + 24·61-s − 80·71-s − 96·76-s − 16·79-s + 24·80-s + 6·81-s − 88·89-s − 96·95-s + 24·100-s + 48·101-s + 24·109-s + 128·116-s + 52·121-s + 128·124-s + ⋯ |
| L(s) = 1 | + 2·4-s + 1.78·5-s + 4/3·9-s + 3/2·16-s − 5.50·19-s + 3.57·20-s + 6/5·25-s + 5.94·29-s + 5.74·31-s + 8/3·36-s − 7.49·41-s + 2.38·45-s − 5.20·59-s + 3.07·61-s − 9.49·71-s − 11.0·76-s − 1.80·79-s + 2.68·80-s + 2/3·81-s − 9.32·89-s − 9.84·95-s + 12/5·100-s + 4.77·101-s + 2.29·109-s + 11.8·116-s + 4.72·121-s + 11.4·124-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.29870\times 10^{17}\) |
| Root analytic conductor: | \(3.42607\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(5.014348144\) |
| \(L(\frac12)\) | \(\approx\) | \(5.014348144\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 3 | \( ( 1 - T^{2} + T^{4} )^{4} \) | |
| 5 | \( 1 - 4 T + 2 p T^{2} - 16 T^{3} + 18 T^{4} - 28 T^{5} + 224 T^{6} - 868 T^{7} + 2399 T^{8} - 868 p T^{9} + 224 p^{2} T^{10} - 28 p^{3} T^{11} + 18 p^{4} T^{12} - 16 p^{5} T^{13} + 2 p^{7} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 \) | |
| good | 11 | \( ( 1 - 26 T^{2} - 32 T^{3} + 314 T^{4} + 592 T^{5} - 2864 T^{6} - 3360 T^{7} + 31791 T^{8} - 3360 p T^{9} - 2864 p^{2} T^{10} + 592 p^{3} T^{11} + 314 p^{4} T^{12} - 32 p^{5} T^{13} - 26 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 - 4 p T^{2} + 1320 T^{4} - 23228 T^{6} + 329390 T^{8} - 23228 p^{2} T^{10} + 1320 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 + 56 T^{2} + 1868 T^{4} + 48656 T^{6} + 936298 T^{8} + 12285992 T^{10} + 97930288 T^{12} - 398164152 T^{14} - 23883204141 T^{16} - 398164152 p^{2} T^{18} + 97930288 p^{4} T^{20} + 12285992 p^{6} T^{22} + 936298 p^{8} T^{24} + 48656 p^{10} T^{26} + 1868 p^{12} T^{28} + 56 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 + 12 T + 56 T^{2} + 136 T^{3} + 50 T^{4} - 2956 T^{5} - 21056 T^{6} - 74676 T^{7} - 244637 T^{8} - 74676 p T^{9} - 21056 p^{2} T^{10} - 2956 p^{3} T^{11} + 50 p^{4} T^{12} + 136 p^{5} T^{13} + 56 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 + 96 T^{2} + 4996 T^{4} + 171840 T^{6} + 3917514 T^{8} + 36781728 T^{10} - 1323040240 T^{12} - 79025433696 T^{14} - 2275569187117 T^{16} - 79025433696 p^{2} T^{18} - 1323040240 p^{4} T^{20} + 36781728 p^{6} T^{22} + 3917514 p^{8} T^{24} + 171840 p^{10} T^{26} + 4996 p^{12} T^{28} + 96 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 - 8 T + 50 T^{2} - 360 T^{3} + 2786 T^{4} - 360 p T^{5} + 50 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 31 | \( ( 1 - 16 T + 54 T^{2} + 4282 T^{4} - 1136 p T^{5} + 43632 T^{6} - 580080 T^{7} + 7925247 T^{8} - 580080 p T^{9} + 43632 p^{2} T^{10} - 1136 p^{4} T^{11} + 4282 p^{4} T^{12} + 54 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 + 4 p T^{2} + 12344 T^{4} + 673640 T^{6} + 24006274 T^{8} + 339978444 T^{10} - 21738927680 T^{12} - 2023710378468 T^{14} - 93303766201293 T^{16} - 2023710378468 p^{2} T^{18} - 21738927680 p^{4} T^{20} + 339978444 p^{6} T^{22} + 24006274 p^{8} T^{24} + 673640 p^{10} T^{26} + 12344 p^{12} T^{28} + 4 p^{15} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 + 12 T + 190 T^{2} + 1356 T^{3} + 11842 T^{4} + 1356 p T^{5} + 190 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 43 | \( ( 1 - 108 T^{2} + 7400 T^{4} - 450340 T^{6} + 22241646 T^{8} - 450340 p^{2} T^{10} + 7400 p^{4} T^{12} - 108 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 + 172 T^{2} + 14120 T^{4} + 763480 T^{6} + 28239298 T^{8} + 309646612 T^{10} - 49680791168 T^{12} - 4612061316732 T^{14} - 252887442101229 T^{16} - 4612061316732 p^{2} T^{18} - 49680791168 p^{4} T^{20} + 309646612 p^{6} T^{22} + 28239298 p^{8} T^{24} + 763480 p^{10} T^{26} + 14120 p^{12} T^{28} + 172 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 + 240 T^{2} + 32068 T^{4} + 2696480 T^{6} + 146040106 T^{8} + 3454338960 T^{10} - 212919588592 T^{12} - 30977671099760 T^{14} - 2065672492914381 T^{16} - 30977671099760 p^{2} T^{18} - 212919588592 p^{4} T^{20} + 3454338960 p^{6} T^{22} + 146040106 p^{8} T^{24} + 2696480 p^{10} T^{26} + 32068 p^{12} T^{28} + 240 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 + 20 T + 112 T^{2} + 152 T^{3} + 2850 T^{4} + 12428 T^{5} - 357280 T^{6} - 3797740 T^{7} - 23752573 T^{8} - 3797740 p T^{9} - 357280 p^{2} T^{10} + 12428 p^{3} T^{11} + 2850 p^{4} T^{12} + 152 p^{5} T^{13} + 112 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 12 T + 26 T^{2} - 48 T^{3} - 590 T^{4} + 28908 T^{5} + 101216 T^{6} - 2459148 T^{7} + 10939903 T^{8} - 2459148 p T^{9} + 101216 p^{2} T^{10} + 28908 p^{3} T^{11} - 590 p^{4} T^{12} - 48 p^{5} T^{13} + 26 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 + 492 T^{2} + 133672 T^{4} + 25363224 T^{6} + 3713074146 T^{8} + 440689891668 T^{10} + 43682134886528 T^{12} + 3678846760423332 T^{14} + 265757294006694131 T^{16} + 3678846760423332 p^{2} T^{18} + 43682134886528 p^{4} T^{20} + 440689891668 p^{6} T^{22} + 3713074146 p^{8} T^{24} + 25363224 p^{10} T^{26} + 133672 p^{12} T^{28} + 492 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 + 20 T + 248 T^{2} + 2788 T^{3} + 28078 T^{4} + 2788 p T^{5} + 248 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 73 | \( 1 + 268 T^{2} + 41256 T^{4} + 4088344 T^{6} + 258915074 T^{8} + 6354240500 T^{10} - 822261942144 T^{12} - 137649648237724 T^{14} - 12273291779723693 T^{16} - 137649648237724 p^{2} T^{18} - 822261942144 p^{4} T^{20} + 6354240500 p^{6} T^{22} + 258915074 p^{8} T^{24} + 4088344 p^{10} T^{26} + 41256 p^{12} T^{28} + 268 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 + 8 T - 76 T^{2} + 912 T^{3} + 12714 T^{4} - 74088 T^{5} + 845008 T^{6} + 9263368 T^{7} - 62368237 T^{8} + 9263368 p T^{9} + 845008 p^{2} T^{10} - 74088 p^{3} T^{11} + 12714 p^{4} T^{12} + 912 p^{5} T^{13} - 76 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 - 424 T^{2} + 80348 T^{4} - 9509080 T^{6} + 861447654 T^{8} - 9509080 p^{2} T^{10} + 80348 p^{4} T^{12} - 424 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 + 44 T + 922 T^{2} + 13744 T^{3} + 179314 T^{4} + 2092484 T^{5} + 21812736 T^{6} + 217805788 T^{7} + 2113929871 T^{8} + 217805788 p T^{9} + 21812736 p^{2} T^{10} + 2092484 p^{3} T^{11} + 179314 p^{4} T^{12} + 13744 p^{5} T^{13} + 922 p^{6} T^{14} + 44 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 348 T^{2} + 81160 T^{4} - 12144532 T^{6} + 1394593550 T^{8} - 12144532 p^{2} T^{10} + 81160 p^{4} T^{12} - 348 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
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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
−2.38804599730167418181307837195, −2.33321683588260741173676782229, −2.20800197219674756583421243195, −2.19493338618892084401510407904, −2.15766764127010248160785540371, −1.97771949890996707363490889494, −1.94548116903751743208714049860, −1.83499227751362700669276366318, −1.78108590731354167439151311919, −1.75934571695070241681623037282, −1.75033838316297213184306141878, −1.46932070033836511264978468800, −1.46481558855433681448129164952, −1.37827889510769342516008551347, −1.36552089786552584886465207871, −1.29057593373605695783770817577, −1.12830978644674134145053498852, −1.09905491782361360417116727032, −0.995285181066580920919806921774, −0.929519870467632866084681210310, −0.803751471023662167833660458673, −0.42463110739304108093614077970, −0.21670373252245223976623442108, −0.20488375368545718493750525164, −0.15091842481130948664423961251, 0.15091842481130948664423961251, 0.20488375368545718493750525164, 0.21670373252245223976623442108, 0.42463110739304108093614077970, 0.803751471023662167833660458673, 0.929519870467632866084681210310, 0.995285181066580920919806921774, 1.09905491782361360417116727032, 1.12830978644674134145053498852, 1.29057593373605695783770817577, 1.36552089786552584886465207871, 1.37827889510769342516008551347, 1.46481558855433681448129164952, 1.46932070033836511264978468800, 1.75033838316297213184306141878, 1.75934571695070241681623037282, 1.78108590731354167439151311919, 1.83499227751362700669276366318, 1.94548116903751743208714049860, 1.97771949890996707363490889494, 2.15766764127010248160785540371, 2.19493338618892084401510407904, 2.20800197219674756583421243195, 2.33321683588260741173676782229, 2.38804599730167418181307837195
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.