Properties

Label 4-165e2-1.1-c1e2-0-14
Degree $4$
Conductor $27225$
Sign $1$
Analytic cond. $1.73588$
Root an. cond. $1.14783$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 4-s − 2·5-s + 4·6-s − 4·7-s + 3·9-s + 4·10-s − 2·11-s − 2·12-s + 8·14-s + 4·15-s + 16-s − 8·17-s − 6·18-s − 8·19-s − 2·20-s + 8·21-s + 4·22-s − 8·23-s + 3·25-s − 4·27-s − 4·28-s − 4·29-s − 8·30-s + 2·32-s + 4·33-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s + 1.63·6-s − 1.51·7-s + 9-s + 1.26·10-s − 0.603·11-s − 0.577·12-s + 2.13·14-s + 1.03·15-s + 1/4·16-s − 1.94·17-s − 1.41·18-s − 1.83·19-s − 0.447·20-s + 1.74·21-s + 0.852·22-s − 1.66·23-s + 3/5·25-s − 0.769·27-s − 0.755·28-s − 0.742·29-s − 1.46·30-s + 0.353·32-s + 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1.73588\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{165} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 27225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$C_4$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−12.71356479536445702086775698581, −11.89236308217832953012385481959, −11.40122504862480372315304683882, −11.08616747244285173223687672393, −10.31600728048091969811308172255, −10.24270735616098881995544842593, −9.366705082037274225159796771471, −9.307790568978610657685973125159, −8.380360614730459104025703531425, −8.145492917234143663137211812727, −7.44551346438669193859360856384, −6.64325833235590056459070713525, −6.31227099398064812462775450942, −5.95319055097928357696185137670, −4.60397357310367788518780208858, −4.37694419147338609265442814529, −3.38533677853718646521597180185, −2.19032666796141313339072907284, 0, 0, 2.19032666796141313339072907284, 3.38533677853718646521597180185, 4.37694419147338609265442814529, 4.60397357310367788518780208858, 5.95319055097928357696185137670, 6.31227099398064812462775450942, 6.64325833235590056459070713525, 7.44551346438669193859360856384, 8.145492917234143663137211812727, 8.380360614730459104025703531425, 9.307790568978610657685973125159, 9.366705082037274225159796771471, 10.24270735616098881995544842593, 10.31600728048091969811308172255, 11.08616747244285173223687672393, 11.40122504862480372315304683882, 11.89236308217832953012385481959, 12.71356479536445702086775698581

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