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Properties

Label 4-18e2-1.1-c39e2-0-0
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $30071.4$
Root an. cond. $13.1685$
Motivic weight $39$
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  + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+39/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(30071.4\)
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

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - p^{19} T )^{2} \)
3 \( 1 \)
good5$D_{4}$ \( 1 + 1374653713308 p T + \)\(87\!\cdots\!66\)\( p^{6} T^{2} + 1374653713308 p^{40} T^{3} + p^{78} T^{4} \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.42772969619329473209256273821, −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

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