Dirichlet series
| L(s) = 1 | − 1.26e3·4-s + 6.92e5·16-s + 2.63e5·19-s + 5.62e5·25-s − 4.31e6·31-s + 4.24e7·49-s + 2.60e7·61-s − 2.23e8·64-s − 3.32e8·76-s − 2.62e8·79-s − 7.09e8·100-s − 1.18e9·109-s + 1.88e9·121-s + 5.44e9·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.19e9·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 4.92·4-s + 10.5·16-s + 2.02·19-s + 1.43·25-s − 4.67·31-s + 7.36·49-s + 1.88·61-s − 13.3·64-s − 9.97·76-s − 6.74·79-s − 7.09·100-s − 8.36·109-s + 8.77·121-s + 23.0·124-s + 2.69·169-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+4)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{32} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.62694\times 10^{20}\) |
| Root analytic conductor: | \(4.28159\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{32} \cdot 5^{16} ,\ ( \ : [4]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(0.2800999844\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2800999844\) |
| \(L(5)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 112486 p T^{2} + 2986481104 p^{2} T^{4} + 145562157406 p^{8} T^{6} - 6883550242706 p^{14} T^{8} + 145562157406 p^{24} T^{10} + 2986481104 p^{34} T^{12} - 112486 p^{49} T^{14} + p^{64} T^{16} \) | |
| good | 2 | \( ( 1 + 631 T^{2} + 125495 p T^{4} + 2644637 p^{5} T^{6} + 185102783 p^{7} T^{8} + 2644637 p^{21} T^{10} + 125495 p^{33} T^{12} + 631 p^{48} T^{14} + p^{64} T^{16} )^{2} \) |
| 7 | \( ( 1 - 3032066 p T^{2} + 851726098996 p^{3} T^{4} - 1085277749483797514 p^{4} T^{6} + \)\(15\!\cdots\!10\)\( p^{6} T^{8} - 1085277749483797514 p^{20} T^{10} + 851726098996 p^{35} T^{12} - 3032066 p^{49} T^{14} + p^{64} T^{16} )^{2} \) | |
| 11 | \( ( 1 - 940409906 T^{2} + 438884142460995940 T^{4} - \)\(13\!\cdots\!34\)\( T^{6} + \)\(33\!\cdots\!74\)\( T^{8} - \)\(13\!\cdots\!34\)\( p^{16} T^{10} + 438884142460995940 p^{32} T^{12} - 940409906 p^{48} T^{14} + p^{64} T^{16} )^{2} \) | |
| 13 | \( ( 1 - 1099572122 T^{2} + 1709772296384261308 T^{4} - \)\(18\!\cdots\!34\)\( T^{6} + \)\(15\!\cdots\!70\)\( T^{8} - \)\(18\!\cdots\!34\)\( p^{16} T^{10} + 1709772296384261308 p^{32} T^{12} - 1099572122 p^{48} T^{14} + p^{64} T^{16} )^{2} \) | |
| 17 | \( ( 1 + 19352339926 T^{2} + \)\(20\!\cdots\!20\)\( T^{4} + \)\(19\!\cdots\!94\)\( T^{6} + \)\(16\!\cdots\!94\)\( T^{8} + \)\(19\!\cdots\!94\)\( p^{16} T^{10} + \)\(20\!\cdots\!20\)\( p^{32} T^{12} + 19352339926 p^{48} T^{14} + p^{64} T^{16} )^{2} \) | |
| 19 | \( ( 1 - 65900 T + 47814330484 T^{2} - 1769229107898500 T^{3} + \)\(10\!\cdots\!26\)\( T^{4} - 1769229107898500 p^{8} T^{5} + 47814330484 p^{16} T^{6} - 65900 p^{24} T^{7} + p^{32} T^{8} )^{4} \) | |
| 23 | \( ( 1 + 347301623380 T^{2} + \)\(64\!\cdots\!24\)\( T^{4} + \)\(80\!\cdots\!40\)\( T^{6} + \)\(73\!\cdots\!86\)\( T^{8} + \)\(80\!\cdots\!40\)\( p^{16} T^{10} + \)\(64\!\cdots\!24\)\( p^{32} T^{12} + 347301623380 p^{48} T^{14} + p^{64} T^{16} )^{2} \) | |
| 29 | \( ( 1 - 1227286359230 T^{2} + \)\(12\!\cdots\!84\)\( T^{4} - \)\(78\!\cdots\!90\)\( T^{6} + \)\(45\!\cdots\!46\)\( T^{8} - \)\(78\!\cdots\!90\)\( p^{16} T^{10} + \)\(12\!\cdots\!84\)\( p^{32} T^{12} - 1227286359230 p^{48} T^{14} + p^{64} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 1078246 T + 391778114200 T^{2} + 240972717036783994 T^{3} + \)\(27\!\cdots\!94\)\( T^{4} + 240972717036783994 p^{8} T^{5} + 391778114200 p^{16} T^{6} + 1078246 p^{24} T^{7} + p^{32} T^{8} )^{4} \) | |
| 37 | \( ( 1 - 15557933552174 T^{2} + \)\(13\!\cdots\!60\)\( T^{4} - \)\(77\!\cdots\!26\)\( T^{6} + \)\(31\!\cdots\!34\)\( T^{8} - \)\(77\!\cdots\!26\)\( p^{16} T^{10} + \)\(13\!\cdots\!60\)\( p^{32} T^{12} - 15557933552174 p^{48} T^{14} + p^{64} T^{16} )^{2} \) | |
| 41 | \( ( 1 - 21354671689760 T^{2} + \)\(29\!\cdots\!44\)\( T^{4} - \)\(26\!\cdots\!80\)\( T^{6} + \)\(22\!\cdots\!46\)\( T^{8} - \)\(26\!\cdots\!80\)\( p^{16} T^{10} + \)\(29\!\cdots\!44\)\( p^{32} T^{12} - 21354671689760 p^{48} T^{14} + p^{64} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 72681018581984 T^{2} + \)\(24\!\cdots\!00\)\( T^{4} - \)\(49\!\cdots\!96\)\( T^{6} + \)\(69\!\cdots\!54\)\( T^{8} - \)\(49\!\cdots\!96\)\( p^{16} T^{10} + \)\(24\!\cdots\!00\)\( p^{32} T^{12} - 72681018581984 p^{48} T^{14} + p^{64} T^{16} )^{2} \) | |
| 47 | \( ( 1 + 104148876545320 T^{2} + \)\(57\!\cdots\!64\)\( T^{4} + \)\(21\!\cdots\!60\)\( T^{6} + \)\(60\!\cdots\!06\)\( T^{8} + \)\(21\!\cdots\!60\)\( p^{16} T^{10} + \)\(57\!\cdots\!64\)\( p^{32} T^{12} + 104148876545320 p^{48} T^{14} + p^{64} T^{16} )^{2} \) | |
| 53 | \( ( 1 + 269149172632630 T^{2} + \)\(40\!\cdots\!04\)\( T^{4} + \)\(40\!\cdots\!90\)\( T^{6} + \)\(29\!\cdots\!86\)\( T^{8} + \)\(40\!\cdots\!90\)\( p^{16} T^{10} + \)\(40\!\cdots\!04\)\( p^{32} T^{12} + 269149172632630 p^{48} T^{14} + p^{64} T^{16} )^{2} \) | |
| 59 | \( ( 1 - 526314458120546 T^{2} + \)\(16\!\cdots\!40\)\( T^{4} - \)\(37\!\cdots\!14\)\( T^{6} + \)\(62\!\cdots\!14\)\( T^{8} - \)\(37\!\cdots\!14\)\( p^{16} T^{10} + \)\(16\!\cdots\!40\)\( p^{32} T^{12} - 526314458120546 p^{48} T^{14} + p^{64} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 6524492 T + 332651202340468 T^{2} - \)\(49\!\cdots\!44\)\( T^{3} + \)\(70\!\cdots\!90\)\( T^{4} - \)\(49\!\cdots\!44\)\( p^{8} T^{5} + 332651202340468 p^{16} T^{6} - 6524492 p^{24} T^{7} + p^{32} T^{8} )^{4} \) | |
| 67 | \( ( 1 - 1247762453384 T^{2} + \)\(18\!\cdots\!40\)\( T^{4} - \)\(63\!\cdots\!96\)\( T^{6} + \)\(58\!\cdots\!34\)\( T^{8} - \)\(63\!\cdots\!96\)\( p^{16} T^{10} + \)\(18\!\cdots\!40\)\( p^{32} T^{12} - 1247762453384 p^{48} T^{14} + p^{64} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 2328093268169480 T^{2} + \)\(22\!\cdots\!04\)\( T^{4} - \)\(10\!\cdots\!40\)\( T^{6} + \)\(43\!\cdots\!86\)\( T^{8} - \)\(10\!\cdots\!40\)\( p^{16} T^{10} + \)\(22\!\cdots\!04\)\( p^{32} T^{12} - 2328093268169480 p^{48} T^{14} + p^{64} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 4305407522018012 T^{2} + \)\(81\!\cdots\!28\)\( T^{4} - \)\(93\!\cdots\!84\)\( T^{6} + \)\(82\!\cdots\!50\)\( T^{8} - \)\(93\!\cdots\!84\)\( p^{16} T^{10} + \)\(81\!\cdots\!28\)\( p^{32} T^{12} - 4305407522018012 p^{48} T^{14} + p^{64} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 65661466 T + 5570756090390920 T^{2} + \)\(26\!\cdots\!74\)\( T^{3} + \)\(12\!\cdots\!34\)\( T^{4} + \)\(26\!\cdots\!74\)\( p^{8} T^{5} + 5570756090390920 p^{16} T^{6} + 65661466 p^{24} T^{7} + p^{32} T^{8} )^{4} \) | |
| 83 | \( ( 1 + 8401048270456600 T^{2} + \)\(34\!\cdots\!04\)\( T^{4} + \)\(10\!\cdots\!00\)\( T^{6} + \)\(25\!\cdots\!26\)\( T^{8} + \)\(10\!\cdots\!00\)\( p^{16} T^{10} + \)\(34\!\cdots\!04\)\( p^{32} T^{12} + 8401048270456600 p^{48} T^{14} + p^{64} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 6689081416446656 T^{2} + \)\(10\!\cdots\!40\)\( T^{4} - \)\(44\!\cdots\!84\)\( T^{6} + \)\(38\!\cdots\!74\)\( T^{8} - \)\(44\!\cdots\!84\)\( p^{16} T^{10} + \)\(10\!\cdots\!40\)\( p^{32} T^{12} - 6689081416446656 p^{48} T^{14} + p^{64} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 14758840863584924 T^{2} + \)\(23\!\cdots\!20\)\( T^{4} - \)\(23\!\cdots\!16\)\( T^{6} + \)\(21\!\cdots\!74\)\( T^{8} - \)\(23\!\cdots\!16\)\( p^{16} T^{10} + \)\(23\!\cdots\!20\)\( p^{32} T^{12} - 14758840863584924 p^{48} T^{14} + p^{64} T^{16} )^{2} \) | |
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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
−3.26897065511622367653288999665, −3.05137295355658262271073980030, −3.03106487823860302670965121786, −2.84613105597223003981876243463, −2.71724436003652328531344664762, −2.64584755610038354457230994636, −2.37485070323099338802702135929, −2.12886019638768392778644014347, −2.07507895462703371375221827635, −2.05319998659777187114681369931, −2.00652822278911143451755046842, −1.89440887175025238170809843337, −1.52902153510009792189136080041, −1.29728616445176111751717325763, −1.14278406013133868636054310495, −1.08140381357914179064281426690, −1.07586704418648688781591115528, −1.01383499012276767357984654475, −0.972600027481697840477908271310, −0.73434621802923869788240859668, −0.48082746447867660408786845267, −0.21937406716994631117707092375, −0.21774197639378280846897904282, −0.21687205350860189661917381043, −0.13670725515516320464235695274, 0.13670725515516320464235695274, 0.21687205350860189661917381043, 0.21774197639378280846897904282, 0.21937406716994631117707092375, 0.48082746447867660408786845267, 0.73434621802923869788240859668, 0.972600027481697840477908271310, 1.01383499012276767357984654475, 1.07586704418648688781591115528, 1.08140381357914179064281426690, 1.14278406013133868636054310495, 1.29728616445176111751717325763, 1.52902153510009792189136080041, 1.89440887175025238170809843337, 2.00652822278911143451755046842, 2.05319998659777187114681369931, 2.07507895462703371375221827635, 2.12886019638768392778644014347, 2.37485070323099338802702135929, 2.64584755610038354457230994636, 2.71724436003652328531344664762, 2.84613105597223003981876243463, 3.03106487823860302670965121786, 3.05137295355658262271073980030, 3.26897065511622367653288999665
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.