Dirichlet series
| L(s) = 1 | − 8·3-s − 5·4-s + 28·9-s + 40·12-s + 4·13-s + 13·16-s − 8·17-s − 8·23-s − 23·25-s − 48·27-s − 36·29-s − 140·36-s − 32·39-s + 32·43-s − 104·48-s − 23·49-s + 64·51-s − 20·52-s + 36·53-s + 12·61-s − 26·64-s + 40·68-s + 64·69-s + 184·75-s + 8·79-s + 6·81-s + 288·87-s + ⋯ |
| L(s) = 1 | − 4.61·3-s − 5/2·4-s + 28/3·9-s + 11.5·12-s + 1.10·13-s + 13/4·16-s − 1.94·17-s − 1.66·23-s − 4.59·25-s − 9.23·27-s − 6.68·29-s − 23.3·36-s − 5.12·39-s + 4.87·43-s − 15.0·48-s − 3.28·49-s + 8.96·51-s − 2.77·52-s + 4.94·53-s + 1.53·61-s − 3.25·64-s + 4.85·68-s + 7.70·69-s + 21.2·75-s + 0.900·79-s + 2/3·81-s + 30.8·87-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{16} \cdot 7^{16} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(260037.\) |
| Root analytic conductor: | \(1.47645\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{16} \cdot 7^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.004777489820\) |
| \(L(\frac12)\) | \(\approx\) | \(0.004777489820\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T + T^{2} )^{8} \) |
| 7 | \( 1 + 23 T^{2} + 283 T^{4} + 2459 T^{6} + 17749 T^{8} + 2459 p^{2} T^{10} + 283 p^{4} T^{12} + 23 p^{6} T^{14} + p^{8} T^{16} \) | |
| 13 | \( ( 1 - 2 T - 16 T^{2} - 14 T^{3} + 382 T^{4} - 14 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| good | 2 | \( 1 + 5 T^{2} + 3 p^{2} T^{4} + 21 T^{6} + 13 p T^{8} + 11 p^{2} T^{10} + 31 T^{12} - 101 p T^{14} - 607 T^{16} - 101 p^{3} T^{18} + 31 p^{4} T^{20} + 11 p^{8} T^{22} + 13 p^{9} T^{24} + 21 p^{10} T^{26} + 3 p^{14} T^{28} + 5 p^{14} T^{30} + p^{16} T^{32} \) |
| 5 | \( 1 + 23 T^{2} + 243 T^{4} + 1862 T^{6} + 13746 T^{8} + 97802 T^{10} + 610438 T^{12} + 3329353 T^{14} + 16895881 T^{16} + 3329353 p^{2} T^{18} + 610438 p^{4} T^{20} + 97802 p^{6} T^{22} + 13746 p^{8} T^{24} + 1862 p^{10} T^{26} + 243 p^{12} T^{28} + 23 p^{14} T^{30} + p^{16} T^{32} \) | |
| 11 | \( 1 + 32 T^{2} + 447 T^{4} + 1442 T^{6} - 46182 T^{8} - 891898 T^{10} - 474700 p T^{12} + 47222251 T^{14} + 1171976998 T^{16} + 47222251 p^{2} T^{18} - 474700 p^{5} T^{20} - 891898 p^{6} T^{22} - 46182 p^{8} T^{24} + 1442 p^{10} T^{26} + 447 p^{12} T^{28} + 32 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( ( 1 + 4 T - 3 T^{2} + 158 T^{3} + 626 T^{4} - 668 T^{5} + 17370 T^{6} + 72243 T^{7} - 49010 T^{8} + 72243 p T^{9} + 17370 p^{2} T^{10} - 668 p^{3} T^{11} + 626 p^{4} T^{12} + 158 p^{5} T^{13} - 3 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 + 120 T^{2} + 7598 T^{4} + 345248 T^{6} + 12583401 T^{8} + 385029928 T^{10} + 10133593214 T^{12} + 232884450304 T^{14} + 4710725449588 T^{16} + 232884450304 p^{2} T^{18} + 10133593214 p^{4} T^{20} + 385029928 p^{6} T^{22} + 12583401 p^{8} T^{24} + 345248 p^{10} T^{26} + 7598 p^{12} T^{28} + 120 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( ( 1 + 4 T - 27 T^{2} - 266 T^{3} - 441 T^{4} + 3142 T^{5} + 12505 T^{6} + 16140 T^{7} + 27675 T^{8} + 16140 p T^{9} + 12505 p^{2} T^{10} + 3142 p^{3} T^{11} - 441 p^{4} T^{12} - 266 p^{5} T^{13} - 27 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 + 9 T + 129 T^{2} + 751 T^{3} + 5811 T^{4} + 751 p T^{5} + 129 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 31 | \( 1 + 140 T^{2} + 10935 T^{4} + 550538 T^{6} + 590961 p T^{8} + 281644586 T^{10} - 283494689 p T^{12} - 832901236676 T^{14} - 33061081267151 T^{16} - 832901236676 p^{2} T^{18} - 283494689 p^{5} T^{20} + 281644586 p^{6} T^{22} + 590961 p^{9} T^{24} + 550538 p^{10} T^{26} + 10935 p^{12} T^{28} + 140 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( 1 + 193 T^{2} + 18655 T^{4} + 1272346 T^{6} + 71596640 T^{8} + 3581702572 T^{10} + 163816449648 T^{12} + 6881841830779 T^{14} + 265558186178013 T^{16} + 6881841830779 p^{2} T^{18} + 163816449648 p^{4} T^{20} + 3581702572 p^{6} T^{22} + 71596640 p^{8} T^{24} + 1272346 p^{10} T^{26} + 18655 p^{12} T^{28} + 193 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 228 T^{2} + 25557 T^{4} - 1814009 T^{6} + 88609954 T^{8} - 1814009 p^{2} T^{10} + 25557 p^{4} T^{12} - 228 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 8 T + 57 T^{2} + 331 T^{3} - 2348 T^{4} + 331 p T^{5} + 57 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 47 | \( 1 + 121 T^{2} + 5967 T^{4} + 1116 p T^{6} - 10195477 T^{8} - 650667311 T^{10} - 17847569918 T^{12} - 161406103067 T^{14} + 6099112596176 T^{16} - 161406103067 p^{2} T^{18} - 17847569918 p^{4} T^{20} - 650667311 p^{6} T^{22} - 10195477 p^{8} T^{24} + 1116 p^{11} T^{26} + 5967 p^{12} T^{28} + 121 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( ( 1 - 18 T + 85 T^{2} + 72 T^{3} + 388 T^{4} - 13392 T^{5} - 111440 T^{6} + 2659725 T^{7} - 23755742 T^{8} + 2659725 p T^{9} - 111440 p^{2} T^{10} - 13392 p^{3} T^{11} + 388 p^{4} T^{12} + 72 p^{5} T^{13} + 85 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 + 164 T^{2} + 63 p T^{4} - 41910 T^{6} + 83457062 T^{8} + 5749873004 T^{10} - 56646715538 T^{12} + 9337743825017 T^{14} + 1984621430912030 T^{16} + 9337743825017 p^{2} T^{18} - 56646715538 p^{4} T^{20} + 5749873004 p^{6} T^{22} + 83457062 p^{8} T^{24} - 41910 p^{10} T^{26} + 63 p^{13} T^{28} + 164 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 - 6 T - 143 T^{2} - 8 T^{3} + 15528 T^{4} + 31860 T^{5} - 962256 T^{6} - 815481 T^{7} + 43232026 T^{8} - 815481 p T^{9} - 962256 p^{2} T^{10} + 31860 p^{3} T^{11} + 15528 p^{4} T^{12} - 8 p^{5} T^{13} - 143 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 + 356 T^{2} + 67843 T^{4} + 8595486 T^{6} + 792495150 T^{8} + 54574953306 T^{10} + 2833225724484 T^{12} + 115825705517951 T^{14} + 5529327173624518 T^{16} + 115825705517951 p^{2} T^{18} + 2833225724484 p^{4} T^{20} + 54574953306 p^{6} T^{22} + 792495150 p^{8} T^{24} + 8595486 p^{10} T^{26} + 67843 p^{12} T^{28} + 356 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 - 236 T^{2} + 35567 T^{4} - 3837077 T^{6} + 308393071 T^{8} - 3837077 p^{2} T^{10} + 35567 p^{4} T^{12} - 236 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 + 300 T^{2} + 47483 T^{4} + 4872010 T^{6} + 324665403 T^{8} + 8247330470 T^{10} - 1129936012843 T^{12} - 193777476125200 T^{14} - 17351669002215959 T^{16} - 193777476125200 p^{2} T^{18} - 1129936012843 p^{4} T^{20} + 8247330470 p^{6} T^{22} + 324665403 p^{8} T^{24} + 4872010 p^{10} T^{26} + 47483 p^{12} T^{28} + 300 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 - 4 T - 15 T^{2} - 378 T^{3} - 5071 T^{4} + 66290 T^{5} + 240463 T^{6} - 2537506 T^{7} + 12326975 T^{8} - 2537506 p T^{9} + 240463 p^{2} T^{10} + 66290 p^{3} T^{11} - 5071 p^{4} T^{12} - 378 p^{5} T^{13} - 15 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 - 412 T^{2} + 83535 T^{4} - 11280169 T^{6} + 1097824914 T^{8} - 11280169 p^{2} T^{10} + 83535 p^{4} T^{12} - 412 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 + 659 T^{2} + 239933 T^{4} + 61108080 T^{6} + 12020546462 T^{8} + 1916015703050 T^{10} + 254607554635214 T^{12} + 28655261995263903 T^{14} + 2755629347265377805 T^{16} + 28655261995263903 p^{2} T^{18} + 254607554635214 p^{4} T^{20} + 1916015703050 p^{6} T^{22} + 12020546462 p^{8} T^{24} + 61108080 p^{10} T^{26} + 239933 p^{12} T^{28} + 659 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( ( 1 - 713 T^{2} + 227494 T^{4} - 42490164 T^{6} + 5086526224 T^{8} - 42490164 p^{2} T^{10} + 227494 p^{4} T^{12} - 713 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
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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
−3.61629215006135726756476537531, −3.50427109701050013676640354159, −3.40592628066501234519694669158, −3.16392909525715900287270507496, −3.10275770865853027999663625325, −3.08404091836720946810223505640, −2.80024317378229077749733152029, −2.61622183842162789022455019420, −2.47634895917672510493592677971, −2.43113505741050634599693966198, −2.40372759951019108954633772242, −2.31054175877019514651991694868, −2.19293243324152481574953170648, −1.95794151819401967779420111767, −1.87352524215023467071224723268, −1.83576891528729060117202252886, −1.77611838853621486981409085334, −1.62165775846887326118015183632, −1.33753213832628815272820034084, −1.11902489759947870406925967392, −0.930193756395866512996321885256, −0.805875758913945465097792374660, −0.49911874940107730086682852944, −0.22163788301526042071666887973, −0.10738818357161833465737719828, 0.10738818357161833465737719828, 0.22163788301526042071666887973, 0.49911874940107730086682852944, 0.805875758913945465097792374660, 0.930193756395866512996321885256, 1.11902489759947870406925967392, 1.33753213832628815272820034084, 1.62165775846887326118015183632, 1.77611838853621486981409085334, 1.83576891528729060117202252886, 1.87352524215023467071224723268, 1.95794151819401967779420111767, 2.19293243324152481574953170648, 2.31054175877019514651991694868, 2.40372759951019108954633772242, 2.43113505741050634599693966198, 2.47634895917672510493592677971, 2.61622183842162789022455019420, 2.80024317378229077749733152029, 3.08404091836720946810223505640, 3.10275770865853027999663625325, 3.16392909525715900287270507496, 3.40592628066501234519694669158, 3.50427109701050013676640354159, 3.61629215006135726756476537531
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.