Dirichlet series
| L(s) = 1 | − 4·4-s + 2·5-s + 4·9-s + 12·16-s − 24·19-s − 8·20-s + 24·29-s + 16·31-s − 16·36-s + 16·41-s + 8·45-s − 20·49-s + 4·59-s + 16·61-s − 24·64-s − 56·71-s + 96·76-s − 16·79-s + 24·80-s + 6·81-s + 16·89-s − 48·95-s − 52·101-s − 16·109-s − 96·116-s + 52·121-s − 64·124-s + ⋯ |
| L(s) = 1 | − 2·4-s + 0.894·5-s + 4/3·9-s + 3·16-s − 5.50·19-s − 1.78·20-s + 4.45·29-s + 2.87·31-s − 8/3·36-s + 2.49·41-s + 1.19·45-s − 2.85·49-s + 0.520·59-s + 2.04·61-s − 3·64-s − 6.64·71-s + 11.0·76-s − 1.80·79-s + 2.68·80-s + 2/3·81-s + 1.69·89-s − 4.92·95-s − 5.17·101-s − 1.53·109-s − 8.91·116-s + 4.72·121-s − 5.74·124-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{16} \cdot 5^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0596296\) |
| Root analytic conductor: | \(0.915657\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{16} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.6399247299\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6399247299\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 5 | \( 1 - 2 T + 4 T^{2} - 4 p T^{3} + 62 T^{4} - 118 T^{5} + 8 p^{2} T^{6} - 134 p T^{7} + 1639 T^{8} - 134 p^{2} T^{9} + 8 p^{4} T^{10} - 118 p^{3} T^{11} + 62 p^{4} T^{12} - 4 p^{6} T^{13} + 4 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 + 20 T^{2} + 202 T^{4} + 1244 T^{6} + 7267 T^{8} + 1244 p^{2} T^{10} + 202 p^{4} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} \) | |
| good | 2 | \( 1 + p^{2} T^{2} + p^{2} T^{4} - p^{3} T^{6} - 5 p^{2} T^{8} + p^{5} T^{10} + 3 p^{5} T^{12} - 13 p^{4} T^{14} - 59 p^{4} T^{16} - 13 p^{6} T^{18} + 3 p^{9} T^{20} + p^{11} T^{22} - 5 p^{10} T^{24} - p^{13} T^{26} + p^{14} T^{28} + p^{16} T^{30} + p^{16} T^{32} \) |
| 11 | \( ( 1 - 26 T^{2} - 28 T^{3} + 316 T^{4} + 518 T^{5} - 2872 T^{6} - 2968 T^{7} + 31007 T^{8} - 2968 p T^{9} - 2872 p^{2} T^{10} + 518 p^{3} T^{11} + 316 p^{4} T^{12} - 28 p^{5} T^{13} - 26 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 13 | \( ( 1 - 44 T^{2} + 1234 T^{4} - 23988 T^{6} + 358939 T^{8} - 23988 p^{2} T^{10} + 1234 p^{4} T^{12} - 44 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 + 76 T^{2} + 2664 T^{4} + 67392 T^{6} + 1583990 T^{8} + 33141472 T^{10} + 35499008 p T^{12} + 11039990476 T^{14} + 198790801859 T^{16} + 11039990476 p^{2} T^{18} + 35499008 p^{5} T^{20} + 33141472 p^{6} T^{22} + 1583990 p^{8} T^{24} + 67392 p^{10} T^{26} + 2664 p^{12} T^{28} + 76 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 + 12 T + 30 T^{2} - 80 T^{3} + 689 T^{4} + 9436 T^{5} + 28846 T^{6} + 59944 T^{7} + 291332 T^{8} + 59944 p T^{9} + 28846 p^{2} T^{10} + 9436 p^{3} T^{11} + 689 p^{4} T^{12} - 80 p^{5} T^{13} + 30 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 + 20 T^{2} - 1512 T^{4} - 18448 T^{6} + 1612902 T^{8} + 10149224 T^{10} - 54279328 p T^{12} - 1840459676 T^{14} + 770565238915 T^{16} - 1840459676 p^{2} T^{18} - 54279328 p^{5} T^{20} + 10149224 p^{6} T^{22} + 1612902 p^{8} T^{24} - 18448 p^{10} T^{26} - 1512 p^{12} T^{28} + 20 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 - 6 T + 78 T^{2} - 332 T^{3} + 2820 T^{4} - 332 p T^{5} + 78 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 31 | \( ( 1 - 8 T - 54 T^{2} + 352 T^{3} + 3285 T^{4} - 9272 T^{5} - 157694 T^{6} + 60416 T^{7} + 6248764 T^{8} + 60416 p T^{9} - 157694 p^{2} T^{10} - 9272 p^{3} T^{11} + 3285 p^{4} T^{12} + 352 p^{5} T^{13} - 54 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 + 128 T^{2} + 6574 T^{4} + 187656 T^{6} + 5183961 T^{8} + 158877780 T^{10} + 5690568750 T^{12} + 459641012708 T^{14} + 25029766954564 T^{16} + 459641012708 p^{2} T^{18} + 5690568750 p^{4} T^{20} + 158877780 p^{6} T^{22} + 5183961 p^{8} T^{24} + 187656 p^{10} T^{26} + 6574 p^{12} T^{28} + 128 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 4 T + 114 T^{2} - 346 T^{3} + 5976 T^{4} - 346 p T^{5} + 114 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 43 | \( ( 1 - 216 T^{2} + 22138 T^{4} - 1464812 T^{6} + 71619331 T^{8} - 1464812 p^{2} T^{10} + 22138 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 + 308 T^{2} + 50892 T^{4} + 5950136 T^{6} + 547353834 T^{8} + 41768130380 T^{10} + 2726163659344 T^{12} + 155042751818668 T^{14} + 7757667166613347 T^{16} + 155042751818668 p^{2} T^{18} + 2726163659344 p^{4} T^{20} + 41768130380 p^{6} T^{22} + 547353834 p^{8} T^{24} + 5950136 p^{10} T^{26} + 50892 p^{12} T^{28} + 308 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 + 264 T^{2} + 35652 T^{4} + 3177712 T^{6} + 212585578 T^{8} + 12279652216 T^{10} + 710817442960 T^{12} + 42611108403128 T^{14} + 2398437218467955 T^{16} + 42611108403128 p^{2} T^{18} + 710817442960 p^{4} T^{20} + 12279652216 p^{6} T^{22} + 212585578 p^{8} T^{24} + 3177712 p^{10} T^{26} + 35652 p^{12} T^{28} + 264 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 - 2 T - 226 T^{2} + 4 p T^{3} + 32016 T^{4} - 19820 T^{5} - 2961940 T^{6} + 407278 T^{7} + 205790979 T^{8} + 407278 p T^{9} - 2961940 p^{2} T^{10} - 19820 p^{3} T^{11} + 32016 p^{4} T^{12} + 4 p^{6} T^{13} - 226 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 8 T - 80 T^{2} + 376 T^{3} + 3998 T^{4} + 13396 T^{5} - 247632 T^{6} - 798752 T^{7} + 15355491 T^{8} - 798752 p T^{9} - 247632 p^{2} T^{10} + 13396 p^{3} T^{11} + 3998 p^{4} T^{12} + 376 p^{5} T^{13} - 80 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 + 356 T^{2} + 63502 T^{4} + 8090208 T^{6} + 858425049 T^{8} + 80231005476 T^{10} + 6774300927054 T^{12} + 525083738078264 T^{14} + 37066264561921156 T^{16} + 525083738078264 p^{2} T^{18} + 6774300927054 p^{4} T^{20} + 80231005476 p^{6} T^{22} + 858425049 p^{8} T^{24} + 8090208 p^{10} T^{26} + 63502 p^{12} T^{28} + 356 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 + 14 T + 194 T^{2} + 1648 T^{3} + 14264 T^{4} + 1648 p T^{5} + 194 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 73 | \( 1 + 312 T^{2} + 46894 T^{4} + 4662008 T^{6} + 358067385 T^{8} + 22613492772 T^{10} + 1186535858958 T^{12} + 57781939528716 T^{14} + 3463839236542756 T^{16} + 57781939528716 p^{2} T^{18} + 1186535858958 p^{4} T^{20} + 22613492772 p^{6} T^{22} + 358067385 p^{8} T^{24} + 4662008 p^{10} T^{26} + 46894 p^{12} T^{28} + 312 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 + 8 T - 98 T^{2} - 1128 T^{3} - 3647 T^{4} - 31028 T^{5} - 546546 T^{6} + 5486636 T^{7} + 152161556 T^{8} + 5486636 p T^{9} - 546546 p^{2} T^{10} - 31028 p^{3} T^{11} - 3647 p^{4} T^{12} - 1128 p^{5} T^{13} - 98 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 - 516 T^{2} + 125904 T^{4} - 18859016 T^{6} + 1891189930 T^{8} - 18859016 p^{2} T^{10} + 125904 p^{4} T^{12} - 516 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 8 T - 98 T^{2} + 1004 T^{3} + 4584 T^{4} - 89290 T^{5} + 1051584 T^{6} + 32888 p T^{7} - 1604897 p T^{8} + 32888 p^{2} T^{9} + 1051584 p^{2} T^{10} - 89290 p^{3} T^{11} + 4584 p^{4} T^{12} + 1004 p^{5} T^{13} - 98 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 756 T^{2} + 251932 T^{4} - 48333356 T^{6} + 5838717718 T^{8} - 48333356 p^{2} T^{10} + 251932 p^{4} T^{12} - 756 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
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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
−4.26635399155184993567323056758, −4.26532488847148437602068840128, −4.24566316324407391329028298964, −4.18813261495358227722058174304, −4.02584850869289944909730788564, −3.85827958121493631181405105693, −3.77312069482274301519274612710, −3.73047955803302805269561149076, −3.28267578199884670696270001220, −3.20362675685399739141785184270, −3.12050933430970824980945030730, −3.01438337874772800765663162593, −2.97583385594089171947943659525, −2.94360586972849811274325871876, −2.73014667258975010211173949989, −2.59554988575533808721808078263, −2.53877908495604778044000369050, −2.13593747941654210264596408384, −2.07868218420331142358449993098, −1.86548052668024560962857103361, −1.83203030046010410540688880013, −1.76346001999280288833826342344, −1.30292915341268219899589273766, −0.971534393631094972741733641938, −0.898060164990240354628423823197, 0.898060164990240354628423823197, 0.971534393631094972741733641938, 1.30292915341268219899589273766, 1.76346001999280288833826342344, 1.83203030046010410540688880013, 1.86548052668024560962857103361, 2.07868218420331142358449993098, 2.13593747941654210264596408384, 2.53877908495604778044000369050, 2.59554988575533808721808078263, 2.73014667258975010211173949989, 2.94360586972849811274325871876, 2.97583385594089171947943659525, 3.01438337874772800765663162593, 3.12050933430970824980945030730, 3.20362675685399739141785184270, 3.28267578199884670696270001220, 3.73047955803302805269561149076, 3.77312069482274301519274612710, 3.85827958121493631181405105693, 4.02584850869289944909730788564, 4.18813261495358227722058174304, 4.24566316324407391329028298964, 4.26532488847148437602068840128, 4.26635399155184993567323056758
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.