Dirichlet series
| L(s) = 1 | + 10·13-s + 8·19-s − 40·25-s − 8·31-s − 4·37-s − 10·43-s − 10·49-s + 28·61-s + 18·67-s − 20·79-s + 42·97-s − 76·103-s + 24·109-s − 82·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 89·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2.77·13-s + 1.83·19-s − 8·25-s − 1.43·31-s − 0.657·37-s − 1.52·43-s − 1.42·49-s + 3.58·61-s + 2.19·67-s − 2.25·79-s + 4.26·97-s − 7.48·103-s + 2.29·109-s − 7.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{64} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{64} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 3^{64} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.33882\times 10^{20}\) |
| Root analytic conductor: | \(4.25559\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | induced by $\chi_{2268} (1, \cdot )$ |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 3^{64} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.05147150592\) |
| \(L(\frac12)\) | \(\approx\) | \(0.05147150592\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| 7 | \( ( 1 + 5 T^{2} + 57 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} )^{2} \) | |
| good | 5 | \( ( 1 + 4 p T^{2} + 194 T^{4} + 1293 T^{6} + 6989 T^{8} + 1293 p^{2} T^{10} + 194 p^{4} T^{12} + 4 p^{7} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 + 41 T^{2} + 896 T^{4} + 14643 T^{6} + 185615 T^{8} + 14643 p^{2} T^{10} + 896 p^{4} T^{12} + 41 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 13 | \( ( 1 - 5 T - 7 T^{2} - 22 T^{3} + 332 T^{4} + 365 T^{5} - 12 T^{6} - 4560 T^{7} - 33251 T^{8} - 4560 p T^{9} - 12 p^{2} T^{10} + 365 p^{3} T^{11} + 332 p^{4} T^{12} - 22 p^{5} T^{13} - 7 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 - 74 T^{2} + 2453 T^{4} - 57022 T^{6} + 1324580 T^{8} - 32297308 T^{10} + 689971179 T^{12} - 12093908148 T^{14} + 198768259759 T^{16} - 12093908148 p^{2} T^{18} + 689971179 p^{4} T^{20} - 32297308 p^{6} T^{22} + 1324580 p^{8} T^{24} - 57022 p^{10} T^{26} + 2453 p^{12} T^{28} - 74 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 - 4 T - 36 T^{2} + 262 T^{3} + 503 T^{4} - 6597 T^{5} + 9337 T^{6} + 65501 T^{7} - 346131 T^{8} + 65501 p T^{9} + 9337 p^{2} T^{10} - 6597 p^{3} T^{11} + 503 p^{4} T^{12} + 262 p^{5} T^{13} - 36 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( ( 1 + 140 T^{2} + 9086 T^{4} + 363489 T^{6} + 9954809 T^{8} + 363489 p^{2} T^{10} + 9086 p^{4} T^{12} + 140 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 - 20 T^{2} - 1810 T^{4} + 69272 T^{6} + 1450073 T^{8} - 81129256 T^{10} + 309387726 T^{12} + 37288425348 T^{14} - 899163086348 T^{16} + 37288425348 p^{2} T^{18} + 309387726 p^{4} T^{20} - 81129256 p^{6} T^{22} + 1450073 p^{8} T^{24} + 69272 p^{10} T^{26} - 1810 p^{12} T^{28} - 20 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( ( 1 + 4 T - 91 T^{2} - 232 T^{3} + 5354 T^{4} + 7340 T^{5} - 237291 T^{6} - 82998 T^{7} + 8452027 T^{8} - 82998 p T^{9} - 237291 p^{2} T^{10} + 7340 p^{3} T^{11} + 5354 p^{4} T^{12} - 232 p^{5} T^{13} - 91 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 2 T - 84 T^{2} + 106 T^{3} + 3899 T^{4} - 10725 T^{5} - 92621 T^{6} + 246455 T^{7} + 2134827 T^{8} + 246455 p T^{9} - 92621 p^{2} T^{10} - 10725 p^{3} T^{11} + 3899 p^{4} T^{12} + 106 p^{5} T^{13} - 84 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( 1 - 217 T^{2} + 23601 T^{4} - 1836506 T^{6} + 118865078 T^{8} - 6714540021 T^{10} + 338454221758 T^{12} - 15599817316678 T^{14} + 664987011104169 T^{16} - 15599817316678 p^{2} T^{18} + 338454221758 p^{4} T^{20} - 6714540021 p^{6} T^{22} + 118865078 p^{8} T^{24} - 1836506 p^{10} T^{26} + 23601 p^{12} T^{28} - 217 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( ( 1 + 5 T - 33 T^{2} - 68 T^{3} - 610 T^{4} - 12711 T^{5} + 29458 T^{6} + 592034 T^{7} + 2438991 T^{8} + 592034 p T^{9} + 29458 p^{2} T^{10} - 12711 p^{3} T^{11} - 610 p^{4} T^{12} - 68 p^{5} T^{13} - 33 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 + 53 T^{2} - 606 T^{4} - 149165 T^{6} - 8301346 T^{8} - 260309166 T^{10} + 2955616105 T^{12} + 815118488912 T^{14} + 45401270346297 T^{16} + 815118488912 p^{2} T^{18} + 2955616105 p^{4} T^{20} - 260309166 p^{6} T^{22} - 8301346 p^{8} T^{24} - 149165 p^{10} T^{26} - 606 p^{12} T^{28} + 53 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 - 232 T^{2} + 24618 T^{4} - 1815014 T^{6} + 124253255 T^{8} - 8284516923 T^{10} + 519890455663 T^{12} - 31837101584005 T^{14} + 1808174634489315 T^{16} - 31837101584005 p^{2} T^{18} + 519890455663 p^{4} T^{20} - 8284516923 p^{6} T^{22} + 124253255 p^{8} T^{24} - 1815014 p^{10} T^{26} + 24618 p^{12} T^{28} - 232 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 - 272 T^{2} + 33374 T^{4} - 3120238 T^{6} + 295626995 T^{8} - 24770742499 T^{10} + 1741894126659 T^{12} - 117285517486725 T^{14} + 7429142089346515 T^{16} - 117285517486725 p^{2} T^{18} + 1741894126659 p^{4} T^{20} - 24770742499 p^{6} T^{22} + 295626995 p^{8} T^{24} - 3120238 p^{10} T^{26} + 33374 p^{12} T^{28} - 272 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 - 14 T + 47 T^{2} - 154 T^{3} + 590 T^{4} + 23576 T^{5} + 12087 T^{6} - 1191834 T^{7} + 997843 T^{8} - 1191834 p T^{9} + 12087 p^{2} T^{10} + 23576 p^{3} T^{11} + 590 p^{4} T^{12} - 154 p^{5} T^{13} + 47 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 9 T - 125 T^{2} + 420 T^{3} + 14494 T^{4} + 8361 T^{5} - 1191428 T^{6} + 221142 T^{7} + 65070439 T^{8} + 221142 p T^{9} - 1191428 p^{2} T^{10} + 8361 p^{3} T^{11} + 14494 p^{4} T^{12} + 420 p^{5} T^{13} - 125 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 + 263 T^{2} + 33875 T^{4} + 2995944 T^{6} + 221839643 T^{8} + 2995944 p^{2} T^{10} + 33875 p^{4} T^{12} + 263 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 164 T^{2} - 210 T^{3} + 10879 T^{4} + 24885 T^{5} - 867851 T^{6} - 832335 T^{7} + 84012439 T^{8} - 832335 p T^{9} - 867851 p^{2} T^{10} + 24885 p^{3} T^{11} + 10879 p^{4} T^{12} - 210 p^{5} T^{13} - 164 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 10 T - 166 T^{2} - 742 T^{3} + 25697 T^{4} + 25865 T^{5} - 2591211 T^{6} - 2246829 T^{7} + 176237245 T^{8} - 2246829 p T^{9} - 2591211 p^{2} T^{10} + 25865 p^{3} T^{11} + 25697 p^{4} T^{12} - 742 p^{5} T^{13} - 166 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 + 67 T^{2} - 14083 T^{4} - 979054 T^{6} + 1275034 p T^{8} + 6247214963 T^{10} - 571573673790 T^{12} - 20512900578186 T^{14} + 2895764855646409 T^{16} - 20512900578186 p^{2} T^{18} - 571573673790 p^{4} T^{20} + 6247214963 p^{6} T^{22} + 1275034 p^{9} T^{24} - 979054 p^{10} T^{26} - 14083 p^{12} T^{28} + 67 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( 1 - 412 T^{2} + 82722 T^{4} - 11456822 T^{6} + 1296977555 T^{8} - 127940738655 T^{10} + 11108997498859 T^{12} - 888294241775269 T^{14} + 74110599938321811 T^{16} - 888294241775269 p^{2} T^{18} + 11108997498859 p^{4} T^{20} - 127940738655 p^{6} T^{22} + 1296977555 p^{8} T^{24} - 11456822 p^{10} T^{26} + 82722 p^{12} T^{28} - 412 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( ( 1 - 21 T + 61 T^{2} + 1722 T^{3} - 15308 T^{4} + 115227 T^{5} - 1176506 T^{6} - 17201520 T^{7} + 420283777 T^{8} - 17201520 p T^{9} - 1176506 p^{2} T^{10} + 115227 p^{3} T^{11} - 15308 p^{4} T^{12} + 1722 p^{5} T^{13} + 61 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
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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
−2.22926191744209476519025877888, −2.13461242841696202437570008036, −2.06907201367401830813800157370, −1.92962805230050361218593135276, −1.90109390078348733104043078930, −1.85064183248297682996548616088, −1.83754835463933192945742667879, −1.75299045016264478971754591125, −1.63631004082488045661576793476, −1.61787794508871829875415896190, −1.46107567666792661649050290385, −1.41440040089737748040723810057, −1.23869712912680887713431248331, −1.21171569968898655456573410304, −1.12945739400494823974472594182, −1.10844480889146587706395569054, −1.05729006709867879411382822688, −1.01517578541608682711139045942, −1.00338045956189045835978193877, −0.56004983034709982358861311727, −0.46121279619479019697439877650, −0.35886743517351276645601851471, −0.32720666367031902302941417175, −0.05408454524469829932598124193, −0.05261109509803606120208239509, 0.05261109509803606120208239509, 0.05408454524469829932598124193, 0.32720666367031902302941417175, 0.35886743517351276645601851471, 0.46121279619479019697439877650, 0.56004983034709982358861311727, 1.00338045956189045835978193877, 1.01517578541608682711139045942, 1.05729006709867879411382822688, 1.10844480889146587706395569054, 1.12945739400494823974472594182, 1.21171569968898655456573410304, 1.23869712912680887713431248331, 1.41440040089737748040723810057, 1.46107567666792661649050290385, 1.61787794508871829875415896190, 1.63631004082488045661576793476, 1.75299045016264478971754591125, 1.83754835463933192945742667879, 1.85064183248297682996548616088, 1.90109390078348733104043078930, 1.92962805230050361218593135276, 2.06907201367401830813800157370, 2.13461242841696202437570008036, 2.22926191744209476519025877888
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.