Dirichlet series
L(s) = 1 | − 2.05e3·3-s + 1.24e5·4-s − 3.14e6·7-s + 1.14e7·9-s − 2.56e8·12-s − 1.58e9·13-s + 3.94e9·16-s − 5.61e10·19-s + 6.44e9·21-s + 3.01e11·25-s − 1.26e11·27-s − 3.92e11·28-s + 2.47e12·31-s + 1.42e12·36-s + 3.70e11·37-s + 3.24e12·39-s − 2.80e13·43-s − 8.10e12·48-s − 4.67e13·49-s − 1.97e14·52-s + 1.15e14·57-s + 3.62e14·61-s − 3.58e13·63-s − 4.23e14·64-s − 2.67e13·67-s + 3.17e14·73-s − 6.17e14·75-s + ⋯ |
L(s) = 1 | − 0.312·3-s + 1.90·4-s − 0.544·7-s + 0.265·9-s − 0.595·12-s − 1.93·13-s + 0.919·16-s − 3.30·19-s + 0.170·21-s + 1.97·25-s − 0.448·27-s − 1.03·28-s + 2.89·31-s + 0.505·36-s + 0.105·37-s + 0.606·39-s − 2.40·43-s − 0.287·48-s − 1.40·49-s − 3.69·52-s + 1.03·57-s + 1.88·61-s − 0.144·63-s − 1.50·64-s − 0.0659·67-s + 0.393·73-s − 0.617·75-s + ⋯ |
Functional equation
Invariants
Degree: | \(8\) |
Conductor: | \(81\) = \(3^{4}\) |
Sign: | $1$ |
Analytic conductor: | \(562.369\) |
Root analytic conductor: | \(2.20674\) |
Motivic weight: | \(16\) |
Rational: | yes |
Arithmetic: | yes |
Character: | Trivial |
Primitive: | no |
Self-dual: | yes |
Analytic rank: | \(0\) |
Selberg data: | \((8,\ 81,\ (\ :8, 8, 8, 8),\ 1)\) |
Particular Values
\(L(\frac{17}{2})\) | \(\approx\) | \(1.830039155\) |
\(L(\frac12)\) | \(\approx\) | \(1.830039155\) |
\(L(9)\) | not available | |
\(L(1)\) | not available |
Euler product
$p$ | $\Gal(F_p)$ | $F_p(T)$ | |
---|---|---|---|
bad | 3 | $D_{4}$ | \( 1 + 76 p^{3} T - 122 p^{10} T^{2} + 76 p^{19} T^{3} + p^{32} T^{4} \) |
good | 2 | $C_2^2 \wr C_2$ | \( 1 - 15605 p^{3} T^{2} + 11364297 p^{10} T^{4} - 15605 p^{35} T^{6} + p^{64} T^{8} \) |
5 | $C_2^2 \wr C_2$ | \( 1 - 12045550276 p^{2} T^{2} + 20208485552534687982 p^{5} T^{4} - 12045550276 p^{34} T^{6} + p^{64} T^{8} \) | |
7 | $D_{4}$ | \( ( 1 + 224396 p T + 79000079514 p^{3} T^{2} + 224396 p^{17} T^{3} + p^{32} T^{4} )^{2} \) | |
11 | $C_2^2 \wr C_2$ | \( 1 - 5004710841163244 p T^{2} + \)\(15\!\cdots\!26\)\( p^{3} T^{4} - 5004710841163244 p^{33} T^{6} + p^{64} T^{8} \) | |
13 | $D_{4}$ | \( ( 1 + 60797324 p T + 8193950692930518 p^{2} T^{2} + 60797324 p^{17} T^{3} + p^{32} T^{4} )^{2} \) | |
17 | $C_2^2 \wr C_2$ | \( 1 - \)\(13\!\cdots\!60\)\( T^{2} + \)\(81\!\cdots\!58\)\( T^{4} - \)\(13\!\cdots\!60\)\( p^{32} T^{6} + p^{64} T^{8} \) | |
19 | $D_{4}$ | \( ( 1 + 1476766220 p T + \)\(42\!\cdots\!98\)\( T^{2} + 1476766220 p^{17} T^{3} + p^{32} T^{4} )^{2} \) | |
23 | $C_2^2 \wr C_2$ | \( 1 - \)\(90\!\cdots\!80\)\( p T^{2} + \)\(18\!\cdots\!78\)\( T^{4} - \)\(90\!\cdots\!80\)\( p^{33} T^{6} + p^{64} T^{8} \) | |
29 | $C_2^2 \wr C_2$ | \( 1 - \)\(48\!\cdots\!44\)\( T^{2} + \)\(17\!\cdots\!66\)\( T^{4} - \)\(48\!\cdots\!44\)\( p^{32} T^{6} + p^{64} T^{8} \) | |
31 | $D_{4}$ | \( ( 1 - 39867438004 p T + \)\(18\!\cdots\!06\)\( T^{2} - 39867438004 p^{17} T^{3} + p^{32} T^{4} )^{2} \) | |
37 | $D_{4}$ | \( ( 1 - 185281606948 T + \)\(24\!\cdots\!22\)\( T^{2} - 185281606948 p^{16} T^{3} + p^{32} T^{4} )^{2} \) | |
41 | $C_2^2 \wr C_2$ | \( 1 - \)\(23\!\cdots\!04\)\( T^{2} + \)\(21\!\cdots\!66\)\( T^{4} - \)\(23\!\cdots\!04\)\( p^{32} T^{6} + p^{64} T^{8} \) | |
43 | $D_{4}$ | \( ( 1 + 14032511031332 T + \)\(22\!\cdots\!02\)\( T^{2} + 14032511031332 p^{16} T^{3} + p^{32} T^{4} )^{2} \) | |
47 | $C_2^2 \wr C_2$ | \( 1 - \)\(18\!\cdots\!60\)\( T^{2} + \)\(14\!\cdots\!98\)\( T^{4} - \)\(18\!\cdots\!60\)\( p^{32} T^{6} + p^{64} T^{8} \) | |
53 | $C_2^2 \wr C_2$ | \( 1 - \)\(12\!\cdots\!60\)\( T^{2} + \)\(71\!\cdots\!98\)\( T^{4} - \)\(12\!\cdots\!60\)\( p^{32} T^{6} + p^{64} T^{8} \) | |
59 | $C_2^2 \wr C_2$ | \( 1 - \)\(16\!\cdots\!04\)\( T^{2} + \)\(93\!\cdots\!66\)\( T^{4} - \)\(16\!\cdots\!04\)\( p^{32} T^{6} + p^{64} T^{8} \) | |
61 | $D_{4}$ | \( ( 1 - 181134896541604 T + \)\(81\!\cdots\!26\)\( T^{2} - 181134896541604 p^{16} T^{3} + p^{32} T^{4} )^{2} \) | |
67 | $D_{4}$ | \( ( 1 + 13387202681732 T + \)\(30\!\cdots\!42\)\( T^{2} + 13387202681732 p^{16} T^{3} + p^{32} T^{4} )^{2} \) | |
71 | $C_2^2 \wr C_2$ | \( 1 - \)\(12\!\cdots\!44\)\( T^{2} + \)\(66\!\cdots\!66\)\( T^{4} - \)\(12\!\cdots\!44\)\( p^{32} T^{6} + p^{64} T^{8} \) | |
73 | $D_{4}$ | \( ( 1 - 158814443769988 T + \)\(37\!\cdots\!62\)\( T^{2} - 158814443769988 p^{16} T^{3} + p^{32} T^{4} )^{2} \) | |
79 | $D_{4}$ | \( ( 1 - 2397008582592460 T + \)\(59\!\cdots\!22\)\( p T^{2} - 2397008582592460 p^{16} T^{3} + p^{32} T^{4} )^{2} \) | |
83 | $C_2^2 \wr C_2$ | \( 1 - \)\(46\!\cdots\!60\)\( T^{2} - \)\(18\!\cdots\!42\)\( T^{4} - \)\(46\!\cdots\!60\)\( p^{32} T^{6} + p^{64} T^{8} \) | |
89 | $C_2^2 \wr C_2$ | \( 1 - \)\(45\!\cdots\!84\)\( T^{2} + \)\(22\!\cdots\!06\)\( T^{4} - \)\(45\!\cdots\!84\)\( p^{32} T^{6} + p^{64} T^{8} \) | |
97 | $D_{4}$ | \( ( 1 + 6916897501300892 T + \)\(13\!\cdots\!62\)\( T^{2} + 6916897501300892 p^{16} T^{3} + p^{32} T^{4} )^{2} \) | |
show more | |||
show less |
Imaginary part of the first few zeros on the critical line
−16.38522057107798428245705012450, −16.37787049749053713916100802538, −15.20664748669160562869486037550, −15.14068280125796332056697862769, −15.04846567527132458154960033715, −14.12165222353993348217533209508, −13.26976432337153163029235510906, −12.64276244970839965742851531576, −12.42351768649475129769376715329, −11.71164296627906324522090579035, −11.31120669693049391981952110807, −10.59660824468351979174914713129, −10.27199668072309756602977930882, −9.641806859839399850193019481738, −8.558340964066520449583168421918, −8.000104043634496999618136882987, −6.84783688008767126190920179119, −6.56313577358796467533377958539, −6.53455730748199774289560544157, −4.99031573645298447056423255859, −4.40448525074930780880881606505, −2.90750237650572410083716922065, −2.41699071672461749929038480110, −1.83488451980409632271054600575, −0.42218955733838694584384415923, 0.42218955733838694584384415923, 1.83488451980409632271054600575, 2.41699071672461749929038480110, 2.90750237650572410083716922065, 4.40448525074930780880881606505, 4.99031573645298447056423255859, 6.53455730748199774289560544157, 6.56313577358796467533377958539, 6.84783688008767126190920179119, 8.000104043634496999618136882987, 8.558340964066520449583168421918, 9.641806859839399850193019481738, 10.27199668072309756602977930882, 10.59660824468351979174914713129, 11.31120669693049391981952110807, 11.71164296627906324522090579035, 12.42351768649475129769376715329, 12.64276244970839965742851531576, 13.26976432337153163029235510906, 14.12165222353993348217533209508, 15.04846567527132458154960033715, 15.14068280125796332056697862769, 15.20664748669160562869486037550, 16.37787049749053713916100802538, 16.38522057107798428245705012450