Properties

Label 4-18e2-1.1-c18e2-0-0
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $1366.74$
Root an. cond. $6.08025$
Motivic weight $18$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.31e5·4-s − 7.04e5·7-s + 9.70e9·13-s + 1.71e10·16-s − 9.09e10·19-s + 7.20e12·25-s + 9.23e10·28-s − 3.78e13·31-s − 3.33e14·37-s − 1.46e15·43-s − 3.25e15·49-s − 1.27e15·52-s − 1.38e16·61-s − 2.25e15·64-s + 2.45e16·67-s + 1.90e17·73-s + 1.19e16·76-s + 4.50e17·79-s − 6.83e15·91-s − 8.51e17·97-s − 9.44e17·100-s − 1.51e18·103-s + 1.05e18·109-s − 1.21e16·112-s + 1.02e19·121-s + 4.96e18·124-s + 127-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.0174·7-s + 0.915·13-s + 1/4·16-s − 0.281·19-s + 1.88·25-s + 0.00872·28-s − 1.43·31-s − 2.56·37-s − 2.90·43-s − 1.99·49-s − 0.457·52-s − 1.18·61-s − 1/8·64-s + 0.903·67-s + 3.23·73-s + 0.140·76-s + 3.75·79-s − 0.0159·91-s − 1.11·97-s − 0.944·100-s − 1.15·103-s + 0.483·109-s − 0.00436·112-s + 1.84·121-s + 0.716·124-s + 0.00491·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+9)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1366.74\)
Root analytic conductor: \(6.08025\)
Motivic weight: \(18\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 324,\ (\ :9, 9),\ 1)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(1.345906913\)
\(L(\frac12)\) \(\approx\) \(1.345906913\)
\(L(10)\) 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$C_2$ \( 1 + p^{17} T^{2} \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 11531308928 p^{4} T^{2} + p^{36} T^{4} \)
7$C_2$ \( ( 1 + 352276 T + p^{18} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 10235889071916562514 T^{2} + p^{36} T^{4} \)
13$C_2$ \( ( 1 - 373207064 p T + p^{18} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 61540613423526697184 p^{2} T^{2} + p^{36} T^{4} \)
19$C_2$ \( ( 1 + 45462084496 T + p^{18} T^{2} )^{2} \)
23$C_2^2$ \( 1 - \)\(32\!\cdots\!90\)\( T^{2} + p^{36} T^{4} \)
29$C_2^2$ \( 1 - \)\(36\!\cdots\!04\)\( T^{2} + p^{36} T^{4} \)
31$C_2$ \( ( 1 + 18943596568996 T + p^{18} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 166886471089186 T + p^{18} T^{2} )^{2} \)
41$C_2^2$ \( 1 - \)\(19\!\cdots\!84\)\( T^{2} + p^{36} T^{4} \)
43$C_2$ \( ( 1 + 730556755353160 T + p^{18} T^{2} )^{2} \)
47$C_2^2$ \( 1 + \)\(41\!\cdots\!22\)\( T^{2} + p^{36} T^{4} \)
53$C_2^2$ \( 1 - \)\(91\!\cdots\!80\)\( T^{2} + p^{36} T^{4} \)
59$C_2^2$ \( 1 - \)\(14\!\cdots\!90\)\( T^{2} + p^{36} T^{4} \)
61$C_2$ \( ( 1 + 6927847017559630 T + p^{18} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12289128589408232 T + p^{18} T^{2} )^{2} \)
71$C_2^2$ \( 1 - \)\(37\!\cdots\!70\)\( T^{2} + p^{36} T^{4} \)
73$C_2$ \( ( 1 - 95119031558434736 T + p^{18} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 225037391450064644 T + p^{18} T^{2} )^{2} \)
83$C_2^2$ \( 1 - \)\(48\!\cdots\!46\)\( T^{2} + p^{36} T^{4} \)
89$C_2^2$ \( 1 - \)\(14\!\cdots\!40\)\( T^{2} + p^{36} T^{4} \)
97$C_2$ \( ( 1 + 425589333450778096 T + p^{18} T^{2} )^{2} \)
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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

−15.03175397167232432771708484754, −13.73149719427354199688736963149, −13.73126119388606724368470108897, −12.55205409793819038241942697098, −12.43786047838454099405547073324, −11.17061636116092340925188640707, −10.83856949771865856809733701952, −9.990350357854369899987429494137, −9.188741971764477595774743185259, −8.566585188982975339605036675028, −8.031933836453856949101132045543, −6.82819293845238976885637730989, −6.47608283722096441712418219654, −5.13684326771282464154784097889, −4.98101607698471126849341797032, −3.54096213677051747611122838602, −3.41517275044166097817268267507, −2.00088030014253334547811176577, −1.33294331744059374531782457721, −0.35816941344330883538541614417, 0.35816941344330883538541614417, 1.33294331744059374531782457721, 2.00088030014253334547811176577, 3.41517275044166097817268267507, 3.54096213677051747611122838602, 4.98101607698471126849341797032, 5.13684326771282464154784097889, 6.47608283722096441712418219654, 6.82819293845238976885637730989, 8.031933836453856949101132045543, 8.566585188982975339605036675028, 9.188741971764477595774743185259, 9.990350357854369899987429494137, 10.83856949771865856809733701952, 11.17061636116092340925188640707, 12.43786047838454099405547073324, 12.55205409793819038241942697098, 13.73126119388606724368470108897, 13.73149719427354199688736963149, 15.03175397167232432771708484754

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