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Properties

Label 4-18e2-1.1-c39e2-0-1
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 − 5.36e13·5-s + 7.49e15·7-s + 5.76e17·8-s − 5.62e19·10-s + 4.32e20·11-s − 1.11e21·13-s + 7.86e21·14-s + 3.77e23·16-s + 3.61e23·17-s + 1.15e25·19-s − 4.42e25·20-s + 4.53e26·22-s − 7.33e26·23-s + 3.80e26·25-s − 1.16e27·26-s + 6.18e27·28-s + 6.62e28·29-s − 1.86e29·31-s + 2.37e29·32-s + 3.79e29·34-s − 4.02e29·35-s + 2.97e30·37-s + 1.20e31·38-s − 3.09e31·40-s − 1.67e31·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 1.25·5-s + 0.248·7-s + 1.41·8-s − 1.77·10-s + 2.13·11-s − 0.210·13-s + 0.351·14-s + 5/4·16-s + 0.366·17-s + 1.33·19-s − 1.88·20-s + 3.01·22-s − 2.04·23-s + 0.208·25-s − 0.298·26-s + 0.372·28-s + 2.01·29-s − 1.54·31-s + 1.06·32-s + 0.518·34-s − 0.312·35-s + 0.782·37-s + 1.89·38-s − 1.77·40-s − 0.596·41-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

\(L(20)\) \(\approx\) \(12.63563810\)
\(L(\frac12)\) \(\approx\) \(12.63563810\)
\(L(\frac{41}{2})\) 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$D_{4}$ \( 1 + 10724547633324 p T + \)\(31\!\cdots\!38\)\( p^{7} T^{2} + 10724547633324 p^{40} T^{3} + p^{78} T^{4} \)
7$D_{4}$ \( 1 - 1071141660835216 p T - \)\(52\!\cdots\!54\)\( p^{5} T^{2} - 1071141660835216 p^{40} T^{3} + p^{78} T^{4} \)
11$D_{4}$ \( 1 - \)\(43\!\cdots\!16\)\( T + \)\(97\!\cdots\!66\)\( p^{3} T^{2} - \)\(43\!\cdots\!16\)\( p^{39} T^{3} + p^{78} T^{4} \)
13$D_{4}$ \( 1 + 85439760336767723012 p T + \)\(30\!\cdots\!02\)\( p^{2} T^{2} + 85439760336767723012 p^{40} T^{3} + p^{78} T^{4} \)
17$D_{4}$ \( 1 - \)\(36\!\cdots\!08\)\( T + \)\(67\!\cdots\!98\)\( p^{2} T^{2} - \)\(36\!\cdots\!08\)\( p^{39} T^{3} + p^{78} T^{4} \)
19$D_{4}$ \( 1 - \)\(60\!\cdots\!40\)\( p T + \)\(43\!\cdots\!78\)\( p^{2} T^{2} - \)\(60\!\cdots\!40\)\( p^{40} T^{3} + p^{78} T^{4} \)
23$D_{4}$ \( 1 + \)\(31\!\cdots\!88\)\( p T + \)\(66\!\cdots\!42\)\( p^{2} T^{2} + \)\(31\!\cdots\!88\)\( p^{40} T^{3} + p^{78} T^{4} \)
29$D_{4}$ \( 1 - \)\(22\!\cdots\!40\)\( p T + \)\(38\!\cdots\!18\)\( p^{2} T^{2} - \)\(22\!\cdots\!40\)\( p^{40} T^{3} + p^{78} T^{4} \)
31$D_{4}$ \( 1 + \)\(60\!\cdots\!76\)\( p T + \)\(26\!\cdots\!66\)\( p^{2} T^{2} + \)\(60\!\cdots\!76\)\( p^{40} T^{3} + p^{78} T^{4} \)
37$D_{4}$ \( 1 - \)\(80\!\cdots\!96\)\( p T + \)\(11\!\cdots\!38\)\( p^{2} T^{2} - \)\(80\!\cdots\!96\)\( p^{40} T^{3} + p^{78} T^{4} \)
41$D_{4}$ \( 1 + \)\(16\!\cdots\!24\)\( T + \)\(89\!\cdots\!66\)\( T^{2} + \)\(16\!\cdots\!24\)\( p^{39} T^{3} + p^{78} T^{4} \)
43$D_{4}$ \( 1 + \)\(98\!\cdots\!16\)\( T + \)\(83\!\cdots\!78\)\( T^{2} + \)\(98\!\cdots\!16\)\( p^{39} T^{3} + p^{78} T^{4} \)
47$D_{4}$ \( 1 + \)\(67\!\cdots\!32\)\( T + \)\(42\!\cdots\!22\)\( T^{2} + \)\(67\!\cdots\!32\)\( p^{39} T^{3} + p^{78} T^{4} \)
53$D_{4}$ \( 1 - \)\(87\!\cdots\!36\)\( T + \)\(49\!\cdots\!58\)\( T^{2} - \)\(87\!\cdots\!36\)\( p^{39} T^{3} + p^{78} T^{4} \)
59$D_{4}$ \( 1 - \)\(12\!\cdots\!20\)\( T + \)\(23\!\cdots\!78\)\( T^{2} - \)\(12\!\cdots\!20\)\( p^{39} T^{3} + p^{78} T^{4} \)
61$D_{4}$ \( 1 + \)\(20\!\cdots\!16\)\( T + \)\(85\!\cdots\!46\)\( T^{2} + \)\(20\!\cdots\!16\)\( p^{39} T^{3} + p^{78} T^{4} \)
67$D_{4}$ \( 1 + \)\(43\!\cdots\!08\)\( T + \)\(24\!\cdots\!22\)\( T^{2} + \)\(43\!\cdots\!08\)\( p^{39} T^{3} + p^{78} T^{4} \)
71$D_{4}$ \( 1 - \)\(50\!\cdots\!36\)\( T + \)\(29\!\cdots\!86\)\( T^{2} - \)\(50\!\cdots\!36\)\( p^{39} T^{3} + p^{78} T^{4} \)
73$D_{4}$ \( 1 - \)\(52\!\cdots\!24\)\( T + \)\(15\!\cdots\!18\)\( T^{2} - \)\(52\!\cdots\!24\)\( p^{39} T^{3} + p^{78} T^{4} \)
79$D_{4}$ \( 1 - \)\(51\!\cdots\!40\)\( T - \)\(73\!\cdots\!62\)\( T^{2} - \)\(51\!\cdots\!40\)\( p^{39} T^{3} + p^{78} T^{4} \)
83$D_{4}$ \( 1 - \)\(64\!\cdots\!96\)\( T + \)\(24\!\cdots\!98\)\( T^{2} - \)\(64\!\cdots\!96\)\( p^{39} T^{3} + p^{78} T^{4} \)
89$D_{4}$ \( 1 - \)\(35\!\cdots\!80\)\( T + \)\(69\!\cdots\!18\)\( T^{2} - \)\(35\!\cdots\!80\)\( p^{39} T^{3} + p^{78} T^{4} \)
97$D_{4}$ \( 1 + \)\(14\!\cdots\!68\)\( T + \)\(46\!\cdots\!22\)\( T^{2} + \)\(14\!\cdots\!68\)\( p^{39} T^{3} + p^{78} T^{4} \)
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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

−11.73549984645772352438991319781, −11.69434834534394552565118628165, −10.72766870133211603997416996538, −10.00019832002683397785991921348, −9.379445268219434015420802815351, −8.543596929704107641460015589324, −7.86498386173423973535990834227, −7.49738205097333959212355683922, −6.72577641533247765987632497090, −6.37946143316669317631305282903, −5.65565957054125537146674507680, −5.03917749506250533412257181076, −4.29005042532876075892822706534, −4.05817793688756456861608436492, −3.44346238707500423909810527315, −3.23807196251118861605227918673, −2.07092378086911200825365683401, −1.81510695839005690619969959017, −0.810019919792221000466392361280, −0.68095453835325148527642597993, 0.68095453835325148527642597993, 0.810019919792221000466392361280, 1.81510695839005690619969959017, 2.07092378086911200825365683401, 3.23807196251118861605227918673, 3.44346238707500423909810527315, 4.05817793688756456861608436492, 4.29005042532876075892822706534, 5.03917749506250533412257181076, 5.65565957054125537146674507680, 6.37946143316669317631305282903, 6.72577641533247765987632497090, 7.49738205097333959212355683922, 7.86498386173423973535990834227, 8.543596929704107641460015589324, 9.379445268219434015420802815351, 10.00019832002683397785991921348, 10.72766870133211603997416996538, 11.69434834534394552565118628165, 11.73549984645772352438991319781

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