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Properties

Label 4-165e2-1.1-c0e2-0-0
Degree $4$
Conductor $27225$
Sign $1$
Analytic cond. $0.00678081$
Root an. cond. $0.286959$
Motivic weight $0$
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  − 2·3-s + 3·9-s − 16-s + 2·23-s − 25-s − 4·27-s + 2·37-s + 2·47-s + 2·48-s − 2·53-s − 4·59-s − 2·67-s − 4·69-s + 2·75-s + 5·81-s − 2·97-s − 2·103-s − 4·111-s + 2·113-s − 121-s + 127-s + 131-s + 137-s + 139-s − 4·141-s − 3·144-s + 149-s + ⋯
L(s)  = 1  − 2·3-s + 3·9-s − 16-s + 2·23-s − 25-s − 4·27-s + 2·37-s + 2·47-s + 2·48-s − 2·53-s − 4·59-s − 2·67-s − 4·69-s + 2·75-s + 5·81-s − 2·97-s − 2·103-s − 4·111-s + 2·113-s − 121-s + 127-s + 131-s + 137-s + 139-s − 4·141-s − 3·144-s + 149-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(27225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(0.00678081\)
Root analytic conductor: \(0.286959\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 27225,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2625871624\)
\(L(\frac12)\) \(\approx\) \(0.2625871624\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
11$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
59$C_1$ \( ( 1 + T )^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
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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

−13.04626153620637147925560241206, −12.80673196513357283655651651307, −12.14520824230585135033675814288, −11.89286158380736589932985726108, −11.16066340036910961031208830099, −10.95760162931962160367338185741, −10.68830450870167606505493878184, −9.886113795338133613131479961396, −9.212166649138198661442524128971, −9.206166716883822325673110415413, −7.83768670628207218172432513277, −7.52181963981657335249948391031, −6.88508674445321842543244130273, −6.29718563946434510881114313508, −5.92208702387101527522442263091, −5.22744274393448764590226605608, −4.50569616635841001221954384387, −4.28989820608368591912189901280, −2.92269017277789166739562281737, −1.47786184727082608092675371664, 1.47786184727082608092675371664, 2.92269017277789166739562281737, 4.28989820608368591912189901280, 4.50569616635841001221954384387, 5.22744274393448764590226605608, 5.92208702387101527522442263091, 6.29718563946434510881114313508, 6.88508674445321842543244130273, 7.52181963981657335249948391031, 7.83768670628207218172432513277, 9.206166716883822325673110415413, 9.212166649138198661442524128971, 9.886113795338133613131479961396, 10.68830450870167606505493878184, 10.95760162931962160367338185741, 11.16066340036910961031208830099, 11.89286158380736589932985726108, 12.14520824230585135033675814288, 12.80673196513357283655651651307, 13.04626153620637147925560241206

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