Dirichlet series
L(s) = 1 | + 1.04e6·2-s + 8.24e11·4-s − 6.87e12·5-s + 1.28e16·7-s + 5.76e17·8-s − 7.20e18·10-s − 3.38e20·11-s + 4.61e21·13-s + 1.35e22·14-s + 3.77e23·16-s − 1.27e24·17-s − 7.87e24·19-s − 5.66e24·20-s − 3.55e26·22-s + 9.28e26·23-s − 1.32e27·25-s + 4.83e27·26-s + 1.06e28·28-s − 3.22e27·29-s − 1.09e29·31-s + 2.37e29·32-s − 1.33e30·34-s − 8.86e28·35-s − 2.96e30·37-s − 8.25e30·38-s − 3.96e30·40-s + 3.18e30·41-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.161·5-s + 0.427·7-s + 1.41·8-s − 0.227·10-s − 1.66·11-s + 0.875·13-s + 0.604·14-s + 5/4·16-s − 1.29·17-s − 0.912·19-s − 0.241·20-s − 2.36·22-s + 2.59·23-s − 0.727·25-s + 1.23·26-s + 0.641·28-s − 0.0980·29-s − 0.904·31-s + 1.06·32-s − 1.82·34-s − 0.0688·35-s − 0.780·37-s − 1.29·38-s − 0.227·40-s + 0.113·41-s + ⋯ |
Functional equation
Invariants
Degree: | \(4\) |
Conductor: | \(324\) = \(2^{2} \cdot 3^{4}\) |
Sign: | $1$ |
Analytic conductor: | \(30071.4\) |
Root analytic conductor: | \(13.1685\) |
Motivic weight: | \(39\) |
Rational: | yes |
Arithmetic: | yes |
Character: | Trivial |
Primitive: | no |
Self-dual: | yes |
Analytic rank: | \(0\) |
Selberg data: | \((4,\ 324,\ (\ :39/2, 39/2),\ 1)\) |
Particular Values
\(L(20)\) | \(\approx\) | \(7.638314406\) |
\(L(\frac12)\) | \(\approx\) | \(7.638314406\) |
\(L(\frac{41}{2})\) | not available | |
\(L(1)\) | not available |
Euler product
$p$ | $\Gal(F_p)$ | $F_p(T)$ | |
---|---|---|---|
bad | 2 | $C_1$ | \( ( 1 - p^{19} T )^{2} \) |
3 | \( 1 \) | ||
good | 5 | $D_{4}$ | \( 1 + 1374653713308 p T + \)\(87\!\cdots\!66\)\( p^{6} T^{2} + 1374653713308 p^{40} T^{3} + p^{78} T^{4} \) |
7 | $D_{4}$ | \( 1 - 12892539328650112 T + \)\(31\!\cdots\!54\)\( p^{3} T^{2} - 12892539328650112 p^{39} T^{3} + p^{78} T^{4} \) | |
11 | $D_{4}$ | \( 1 + 30790427290839391944 p T + \)\(68\!\cdots\!66\)\( p^{3} T^{2} + 30790427290839391944 p^{40} T^{3} + p^{78} T^{4} \) | |
13 | $D_{4}$ | \( 1 - \)\(35\!\cdots\!88\)\( p T + \)\(10\!\cdots\!02\)\( p^{2} T^{2} - \)\(35\!\cdots\!88\)\( p^{40} T^{3} + p^{78} T^{4} \) | |
17 | $D_{4}$ | \( 1 + \)\(74\!\cdots\!76\)\( p T + \)\(47\!\cdots\!94\)\( p^{3} T^{2} + \)\(74\!\cdots\!76\)\( p^{40} T^{3} + p^{78} T^{4} \) | |
19 | $D_{4}$ | \( 1 + \)\(78\!\cdots\!40\)\( T + \)\(48\!\cdots\!82\)\( p T^{2} + \)\(78\!\cdots\!40\)\( p^{39} T^{3} + p^{78} T^{4} \) | |
23 | $D_{4}$ | \( 1 - \)\(92\!\cdots\!76\)\( T + \)\(18\!\cdots\!66\)\( p T^{2} - \)\(92\!\cdots\!76\)\( p^{39} T^{3} + p^{78} T^{4} \) | |
29 | $D_{4}$ | \( 1 + \)\(32\!\cdots\!40\)\( T - \)\(16\!\cdots\!78\)\( p T^{2} + \)\(32\!\cdots\!40\)\( p^{39} T^{3} + p^{78} T^{4} \) | |
31 | $D_{4}$ | \( 1 + \)\(10\!\cdots\!56\)\( T + \)\(54\!\cdots\!46\)\( p T^{2} + \)\(10\!\cdots\!56\)\( p^{39} T^{3} + p^{78} T^{4} \) | |
37 | $D_{4}$ | \( 1 + \)\(80\!\cdots\!04\)\( p T + \)\(21\!\cdots\!38\)\( p^{2} T^{2} + \)\(80\!\cdots\!04\)\( p^{40} T^{3} + p^{78} T^{4} \) | |
41 | $D_{4}$ | \( 1 - \)\(31\!\cdots\!76\)\( T + \)\(13\!\cdots\!66\)\( T^{2} - \)\(31\!\cdots\!76\)\( p^{39} T^{3} + p^{78} T^{4} \) | |
43 | $D_{4}$ | \( 1 - \)\(12\!\cdots\!84\)\( T + \)\(13\!\cdots\!78\)\( T^{2} - \)\(12\!\cdots\!84\)\( p^{39} T^{3} + p^{78} T^{4} \) | |
47 | $D_{4}$ | \( 1 + \)\(16\!\cdots\!32\)\( T + \)\(31\!\cdots\!22\)\( T^{2} + \)\(16\!\cdots\!32\)\( p^{39} T^{3} + p^{78} T^{4} \) | |
53 | $D_{4}$ | \( 1 - \)\(84\!\cdots\!36\)\( T + \)\(82\!\cdots\!58\)\( T^{2} - \)\(84\!\cdots\!36\)\( p^{39} T^{3} + p^{78} T^{4} \) | |
59 | $D_{4}$ | \( 1 + \)\(57\!\cdots\!80\)\( T + \)\(26\!\cdots\!78\)\( T^{2} + \)\(57\!\cdots\!80\)\( p^{39} T^{3} + p^{78} T^{4} \) | |
61 | $D_{4}$ | \( 1 - \)\(13\!\cdots\!84\)\( T + \)\(12\!\cdots\!46\)\( T^{2} - \)\(13\!\cdots\!84\)\( p^{39} T^{3} + p^{78} T^{4} \) | |
67 | $D_{4}$ | \( 1 - \)\(78\!\cdots\!92\)\( T + \)\(35\!\cdots\!22\)\( T^{2} - \)\(78\!\cdots\!92\)\( p^{39} T^{3} + p^{78} T^{4} \) | |
71 | $D_{4}$ | \( 1 + \)\(10\!\cdots\!64\)\( T + \)\(34\!\cdots\!86\)\( T^{2} + \)\(10\!\cdots\!64\)\( p^{39} T^{3} + p^{78} T^{4} \) | |
73 | $D_{4}$ | \( 1 + \)\(13\!\cdots\!76\)\( T - \)\(14\!\cdots\!82\)\( T^{2} + \)\(13\!\cdots\!76\)\( p^{39} T^{3} + p^{78} T^{4} \) | |
79 | $D_{4}$ | \( 1 - \)\(25\!\cdots\!40\)\( T + \)\(35\!\cdots\!38\)\( T^{2} - \)\(25\!\cdots\!40\)\( p^{39} T^{3} + p^{78} T^{4} \) | |
83 | $D_{4}$ | \( 1 - \)\(15\!\cdots\!96\)\( T + \)\(14\!\cdots\!98\)\( T^{2} - \)\(15\!\cdots\!96\)\( p^{39} T^{3} + p^{78} T^{4} \) | |
89 | $D_{4}$ | \( 1 - \)\(24\!\cdots\!80\)\( T + \)\(34\!\cdots\!18\)\( T^{2} - \)\(24\!\cdots\!80\)\( p^{39} T^{3} + p^{78} T^{4} \) | |
97 | $D_{4}$ | \( 1 - \)\(14\!\cdots\!32\)\( T + \)\(57\!\cdots\!22\)\( T^{2} - \)\(14\!\cdots\!32\)\( p^{39} T^{3} + p^{78} T^{4} \) | |
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Imaginary part of the first few zeros on the critical line
−11.42772969619329473209256273821, −11.26167970718971865444616904458, −10.69717866347713974648914521303, −10.36459206526916034541946510473, −9.076376821561588759311596862289, −8.784602388225537825109001255929, −7.73109710323686626187994407362, −7.64239228634608843106435072496, −6.55775703787829966354156914532, −6.47068715208445817217711378326, −5.39705328527498909281532233613, −5.18167667983784248357068684841, −4.64853738597781161985535674713, −4.00587282973604198420668666453, −3.40856130810147491454404468151, −2.88465160470278870218355860703, −2.09325177303673762636184934230, −2.02611279086804891931915199090, −0.972815937303718972931383208673, −0.42235371706694081186651957397, 0.42235371706694081186651957397, 0.972815937303718972931383208673, 2.02611279086804891931915199090, 2.09325177303673762636184934230, 2.88465160470278870218355860703, 3.40856130810147491454404468151, 4.00587282973604198420668666453, 4.64853738597781161985535674713, 5.18167667983784248357068684841, 5.39705328527498909281532233613, 6.47068715208445817217711378326, 6.55775703787829966354156914532, 7.64239228634608843106435072496, 7.73109710323686626187994407362, 8.784602388225537825109001255929, 9.076376821561588759311596862289, 10.36459206526916034541946510473, 10.69717866347713974648914521303, 11.26167970718971865444616904458, 11.42772969619329473209256273821