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Properties

Label 4-18e2-1.1-c31e2-0-0
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $12007.5$
Root an. cond. $10.4679$
Motivic weight $31$
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  − 6.55e4·2-s + 3.22e9·4-s + 9.78e10·5-s − 1.36e12·7-s − 1.40e14·8-s − 6.41e15·10-s − 2.11e16·11-s − 2.52e17·13-s + 8.94e16·14-s + 5.76e18·16-s − 3.43e18·17-s − 1.42e19·19-s + 3.15e20·20-s + 1.38e21·22-s − 8.35e20·23-s + 5.32e21·25-s + 1.65e22·26-s − 4.39e21·28-s + 6.17e22·29-s + 3.47e22·31-s − 2.26e23·32-s + 2.24e23·34-s − 1.33e23·35-s − 3.74e24·37-s + 9.35e23·38-s − 1.37e25·40-s − 6.86e24·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 1.43·5-s − 0.108·7-s − 1.41·8-s − 2.02·10-s − 1.52·11-s − 1.37·13-s + 0.153·14-s + 5/4·16-s − 0.290·17-s − 0.215·19-s + 2.15·20-s + 2.16·22-s − 0.653·23-s + 1.14·25-s + 1.93·26-s − 0.162·28-s + 1.32·29-s + 0.266·31-s − 1.06·32-s + 0.411·34-s − 0.155·35-s − 1.84·37-s + 0.305·38-s − 2.02·40-s − 0.689·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+31/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(12007.5\)
Root analytic conductor: \(10.4679\)
Motivic weight: \(31\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 324,\ (\ :31/2, 31/2),\ 1)\)

Particular Values

\(L(16)\) \(\approx\) \(0.5925846885\)
\(L(\frac12)\) \(\approx\) \(0.5925846885\)
\(L(\frac{33}{2})\) 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$C_1$ \( ( 1 + p^{15} T )^{2} \)
3 \( 1 \)
good5$D_{4}$ \( 1 - 19571806464 p T + 1361887021861040498 p^{5} T^{2} - 19571806464 p^{32} T^{3} + p^{62} T^{4} \)
7$D_{4}$ \( 1 + 194914795400 p T + \)\(95\!\cdots\!90\)\( p^{4} T^{2} + 194914795400 p^{32} T^{3} + p^{62} T^{4} \)
11$D_{4}$ \( 1 + 21183610881653760 T + \)\(41\!\cdots\!26\)\( p T^{2} + 21183610881653760 p^{31} T^{3} + p^{62} T^{4} \)
13$D_{4}$ \( 1 + 19451192193258140 p T + \)\(37\!\cdots\!70\)\( p^{3} T^{2} + 19451192193258140 p^{32} T^{3} + p^{62} T^{4} \)
17$D_{4}$ \( 1 + 3430905241741641216 T + \)\(46\!\cdots\!90\)\( p T^{2} + 3430905241741641216 p^{31} T^{3} + p^{62} T^{4} \)
19$D_{4}$ \( 1 + 14274217914732506192 T + \)\(37\!\cdots\!66\)\( p T^{2} + 14274217914732506192 p^{31} T^{3} + p^{62} T^{4} \)
23$D_{4}$ \( 1 + \)\(83\!\cdots\!00\)\( T + \)\(12\!\cdots\!98\)\( p T^{2} + \)\(83\!\cdots\!00\)\( p^{31} T^{3} + p^{62} T^{4} \)
29$D_{4}$ \( 1 - \)\(21\!\cdots\!00\)\( p T + \)\(62\!\cdots\!42\)\( p^{2} T^{2} - \)\(21\!\cdots\!00\)\( p^{32} T^{3} + p^{62} T^{4} \)
31$D_{4}$ \( 1 - \)\(11\!\cdots\!32\)\( p T + \)\(29\!\cdots\!78\)\( T^{2} - \)\(11\!\cdots\!32\)\( p^{32} T^{3} + p^{62} T^{4} \)
37$D_{4}$ \( 1 + \)\(37\!\cdots\!00\)\( T + \)\(94\!\cdots\!90\)\( T^{2} + \)\(37\!\cdots\!00\)\( p^{31} T^{3} + p^{62} T^{4} \)
41$D_{4}$ \( 1 + \)\(68\!\cdots\!80\)\( T + \)\(20\!\cdots\!82\)\( T^{2} + \)\(68\!\cdots\!80\)\( p^{31} T^{3} + p^{62} T^{4} \)
43$D_{4}$ \( 1 - \)\(23\!\cdots\!40\)\( T + \)\(51\!\cdots\!78\)\( T^{2} - \)\(23\!\cdots\!40\)\( p^{31} T^{3} + p^{62} T^{4} \)
47$D_{4}$ \( 1 - \)\(30\!\cdots\!00\)\( T + \)\(46\!\cdots\!06\)\( T^{2} - \)\(30\!\cdots\!00\)\( p^{31} T^{3} + p^{62} T^{4} \)
53$D_{4}$ \( 1 - \)\(82\!\cdots\!16\)\( T + \)\(73\!\cdots\!58\)\( T^{2} - \)\(82\!\cdots\!16\)\( p^{31} T^{3} + p^{62} T^{4} \)
59$D_{4}$ \( 1 - \)\(21\!\cdots\!80\)\( T + \)\(80\!\cdots\!82\)\( T^{2} - \)\(21\!\cdots\!80\)\( p^{31} T^{3} + p^{62} T^{4} \)
61$D_{4}$ \( 1 + \)\(44\!\cdots\!96\)\( T + \)\(26\!\cdots\!26\)\( T^{2} + \)\(44\!\cdots\!96\)\( p^{31} T^{3} + p^{62} T^{4} \)
67$D_{4}$ \( 1 + \)\(45\!\cdots\!00\)\( T + \)\(11\!\cdots\!82\)\( T^{2} + \)\(45\!\cdots\!00\)\( p^{31} T^{3} + p^{62} T^{4} \)
71$D_{4}$ \( 1 + \)\(42\!\cdots\!20\)\( T + \)\(51\!\cdots\!98\)\( T^{2} + \)\(42\!\cdots\!20\)\( p^{31} T^{3} + p^{62} T^{4} \)
73$D_{4}$ \( 1 + \)\(88\!\cdots\!00\)\( T + \)\(44\!\cdots\!54\)\( T^{2} + \)\(88\!\cdots\!00\)\( p^{31} T^{3} + p^{62} T^{4} \)
79$D_{4}$ \( 1 + \)\(17\!\cdots\!92\)\( T + \)\(13\!\cdots\!74\)\( T^{2} + \)\(17\!\cdots\!92\)\( p^{31} T^{3} + p^{62} T^{4} \)
83$D_{4}$ \( 1 - \)\(12\!\cdots\!68\)\( T + \)\(98\!\cdots\!90\)\( T^{2} - \)\(12\!\cdots\!68\)\( p^{31} T^{3} + p^{62} T^{4} \)
89$D_{4}$ \( 1 + \)\(83\!\cdots\!40\)\( T + \)\(36\!\cdots\!34\)\( T^{2} + \)\(83\!\cdots\!40\)\( p^{31} T^{3} + p^{62} T^{4} \)
97$D_{4}$ \( 1 - \)\(76\!\cdots\!40\)\( T + \)\(76\!\cdots\!10\)\( T^{2} - \)\(76\!\cdots\!40\)\( p^{31} T^{3} + p^{62} T^{4} \)
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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

−12.20770690074888278640493121255, −12.10805356090066750249370665306, −10.86925548764762535159275384259, −10.42393160114892316273428913055, −9.992522539720456733588484108072, −9.721878222787435544271552850753, −8.672038517565997205218925649466, −8.575658871511717962003217880702, −7.37080084316327406364392379855, −7.33642495078617472943073673126, −6.29818163668337981380310216692, −5.87299060985307031579791020874, −5.11894992738684832930591757554, −4.58534750741804910299348982470, −3.13405559375304399915101595182, −2.74105428693519537618858810633, −1.96068876980962422401862904958, −1.93764889995614386483096907858, −0.880729987282392406509728330577, −0.23561042889191889189889937904, 0.23561042889191889189889937904, 0.880729987282392406509728330577, 1.93764889995614386483096907858, 1.96068876980962422401862904958, 2.74105428693519537618858810633, 3.13405559375304399915101595182, 4.58534750741804910299348982470, 5.11894992738684832930591757554, 5.87299060985307031579791020874, 6.29818163668337981380310216692, 7.33642495078617472943073673126, 7.37080084316327406364392379855, 8.575658871511717962003217880702, 8.672038517565997205218925649466, 9.721878222787435544271552850753, 9.992522539720456733588484108072, 10.42393160114892316273428913055, 10.86925548764762535159275384259, 12.10805356090066750249370665306, 12.20770690074888278640493121255

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