Properties

Label 4-18e2-1.1-c37e2-0-0
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $24362.6$
Root an. cond. $12.4934$
Motivic weight $37$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+37/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

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

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + p^{18} T )^{2} \)
3 \( 1 \)
good5$D_{4}$ \( 1 + 33460806876 p^{3} T + 73668826362342135734 p^{9} T^{2} + 33460806876 p^{40} T^{3} + p^{74} T^{4} \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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

Graph of the $Z$-function along the critical line