Dirichlet series
L(s) = 1 | − 5.24e5·2-s + 2.06e11·4-s − 4.18e12·5-s − 3.51e15·7-s − 7.20e16·8-s + 2.19e18·10-s − 2.56e19·11-s − 1.50e20·13-s + 1.84e21·14-s + 2.36e22·16-s − 1.65e23·17-s + 3.15e23·19-s − 8.62e23·20-s + 1.34e25·22-s − 2.34e25·23-s − 1.26e26·25-s + 7.88e25·26-s − 7.24e26·28-s + 1.54e26·29-s + 4.84e26·31-s − 7.42e27·32-s + 8.65e28·34-s + 1.46e28·35-s − 1.28e29·37-s − 1.65e29·38-s + 3.01e29·40-s + 1.11e30·41-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.490·5-s − 0.815·7-s − 1.41·8-s + 0.693·10-s − 1.39·11-s − 0.370·13-s + 1.15·14-s + 5/4·16-s − 2.84·17-s + 0.695·19-s − 0.735·20-s + 1.96·22-s − 1.50·23-s − 1.73·25-s + 0.524·26-s − 1.22·28-s + 0.136·29-s + 0.124·31-s − 1.06·32-s + 4.02·34-s + 0.399·35-s − 1.24·37-s − 0.982·38-s + 0.693·40-s + 1.61·41-s + ⋯ |
Functional equation
Invariants
Degree: | \(4\) |
Conductor: | \(324\) = \(2^{2} \cdot 3^{4}\) |
Sign: | $1$ |
Analytic conductor: | \(24362.6\) |
Root analytic conductor: | \(12.4934\) |
Motivic weight: | \(37\) |
Rational: | yes |
Arithmetic: | yes |
Character: | Trivial |
Primitive: | no |
Self-dual: | yes |
Analytic rank: | \(0\) |
Selberg data: | \((4,\ 324,\ (\ :37/2, 37/2),\ 1)\) |
Particular Values
\(L(19)\) | \(\approx\) | \(0.2382225529\) |
\(L(\frac12)\) | \(\approx\) | \(0.2382225529\) |
\(L(\frac{39}{2})\) | not available | |
\(L(1)\) | not available |
Euler product
$p$ | $\Gal(F_p)$ | $F_p(T)$ | |
---|---|---|---|
bad | 2 | $C_1$ | \( ( 1 + p^{18} T )^{2} \) |
3 | \( 1 \) | ||
good | 5 | $D_{4}$ | \( 1 + 33460806876 p^{3} T + 73668826362342135734 p^{9} T^{2} + 33460806876 p^{40} T^{3} + p^{74} T^{4} \) |
7 | $D_{4}$ | \( 1 + 501811279648208 p T + \)\(70\!\cdots\!98\)\( p^{4} T^{2} + 501811279648208 p^{38} T^{3} + p^{74} T^{4} \) | |
11 | $D_{4}$ | \( 1 + 25659945722560373304 T + \)\(60\!\cdots\!66\)\( p^{3} T^{2} + 25659945722560373304 p^{37} T^{3} + p^{74} T^{4} \) | |
13 | $D_{4}$ | \( 1 + \)\(15\!\cdots\!28\)\( T - \)\(38\!\cdots\!02\)\( p^{2} T^{2} + \)\(15\!\cdots\!28\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
17 | $D_{4}$ | \( 1 + \)\(97\!\cdots\!92\)\( p T + \)\(27\!\cdots\!06\)\( p^{3} T^{2} + \)\(97\!\cdots\!92\)\( p^{38} T^{3} + p^{74} T^{4} \) | |
19 | $D_{4}$ | \( 1 - \)\(31\!\cdots\!00\)\( T + \)\(95\!\cdots\!62\)\( p T^{2} - \)\(31\!\cdots\!00\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
23 | $D_{4}$ | \( 1 + \)\(44\!\cdots\!48\)\( p^{2} T + \)\(11\!\cdots\!18\)\( p^{2} T^{2} + \)\(44\!\cdots\!48\)\( p^{39} T^{3} + p^{74} T^{4} \) | |
29 | $D_{4}$ | \( 1 - \)\(53\!\cdots\!80\)\( p T + \)\(30\!\cdots\!98\)\( p^{2} T^{2} - \)\(53\!\cdots\!80\)\( p^{38} T^{3} + p^{74} T^{4} \) | |
31 | $D_{4}$ | \( 1 - \)\(15\!\cdots\!24\)\( p T + \)\(21\!\cdots\!46\)\( p^{2} T^{2} - \)\(15\!\cdots\!24\)\( p^{38} T^{3} + p^{74} T^{4} \) | |
37 | $D_{4}$ | \( 1 + \)\(12\!\cdots\!56\)\( T + \)\(21\!\cdots\!18\)\( T^{2} + \)\(12\!\cdots\!56\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
41 | $D_{4}$ | \( 1 - \)\(11\!\cdots\!56\)\( T + \)\(10\!\cdots\!46\)\( T^{2} - \)\(11\!\cdots\!56\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
43 | $D_{4}$ | \( 1 - \)\(10\!\cdots\!52\)\( T + \)\(41\!\cdots\!62\)\( T^{2} - \)\(10\!\cdots\!52\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
47 | $D_{4}$ | \( 1 + \)\(16\!\cdots\!04\)\( T + \)\(13\!\cdots\!78\)\( T^{2} + \)\(16\!\cdots\!04\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
53 | $D_{4}$ | \( 1 + \)\(37\!\cdots\!72\)\( T + \)\(21\!\cdots\!22\)\( T^{2} + \)\(37\!\cdots\!72\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
59 | $D_{4}$ | \( 1 + \)\(12\!\cdots\!60\)\( T + \)\(10\!\cdots\!38\)\( T^{2} + \)\(12\!\cdots\!60\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
61 | $D_{4}$ | \( 1 + \)\(13\!\cdots\!16\)\( T + \)\(23\!\cdots\!06\)\( T^{2} + \)\(13\!\cdots\!16\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
67 | $D_{4}$ | \( 1 - \)\(38\!\cdots\!04\)\( T + \)\(31\!\cdots\!58\)\( T^{2} - \)\(38\!\cdots\!04\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
71 | $D_{4}$ | \( 1 + \)\(14\!\cdots\!24\)\( T + \)\(67\!\cdots\!26\)\( T^{2} + \)\(14\!\cdots\!24\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
73 | $D_{4}$ | \( 1 - \)\(58\!\cdots\!72\)\( T + \)\(13\!\cdots\!02\)\( T^{2} - \)\(58\!\cdots\!72\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
79 | $D_{4}$ | \( 1 + \)\(16\!\cdots\!80\)\( T + \)\(39\!\cdots\!18\)\( T^{2} + \)\(16\!\cdots\!80\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
83 | $D_{4}$ | \( 1 + \)\(65\!\cdots\!12\)\( T + \)\(22\!\cdots\!82\)\( T^{2} + \)\(65\!\cdots\!12\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
89 | $D_{4}$ | \( 1 + \)\(69\!\cdots\!80\)\( T + \)\(91\!\cdots\!58\)\( T^{2} + \)\(69\!\cdots\!80\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
97 | $D_{4}$ | \( 1 - \)\(58\!\cdots\!44\)\( T + \)\(69\!\cdots\!58\)\( T^{2} - \)\(58\!\cdots\!44\)\( p^{37} T^{3} + p^{74} T^{4} \) | |
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Imaginary part of the first few zeros on the critical line
−11.44236170444361441211934321468, −11.29599305019974795900989705272, −10.33089430927003284430456191435, −10.23212671646968673789869680674, −9.206240423554905733182375861856, −9.176858077142549205423064317309, −8.197163011441869784412752281526, −7.79068836463704988390825738607, −7.33998576551168542470958791022, −6.62975178736435025042779589956, −6.08533104691339227763171060066, −5.52877125384186495422010726749, −4.42817528932784140927856596137, −4.11852415099835871698995337004, −2.96528742182739222621047450798, −2.79237404775140205240757562946, −1.85324912831040102283040349509, −1.79772565272002941276480950791, −0.34010847477710646194482060595, −0.31582783756790976311025942558, 0.31582783756790976311025942558, 0.34010847477710646194482060595, 1.79772565272002941276480950791, 1.85324912831040102283040349509, 2.79237404775140205240757562946, 2.96528742182739222621047450798, 4.11852415099835871698995337004, 4.42817528932784140927856596137, 5.52877125384186495422010726749, 6.08533104691339227763171060066, 6.62975178736435025042779589956, 7.33998576551168542470958791022, 7.79068836463704988390825738607, 8.197163011441869784412752281526, 9.176858077142549205423064317309, 9.206240423554905733182375861856, 10.23212671646968673789869680674, 10.33089430927003284430456191435, 11.29599305019974795900989705272, 11.44236170444361441211934321468