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Properties

Label 4-165e2-1.1-c0e2-0-1
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 9-s − 16-s − 2·23-s − 25-s + 2·37-s − 2·47-s + 2·53-s + 4·59-s − 2·67-s + 81-s − 2·97-s − 2·103-s − 2·113-s − 121-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 9-s − 16-s − 2·23-s − 25-s + 2·37-s − 2·47-s + 2·53-s + 4·59-s − 2·67-s + 81-s − 2·97-s − 2·103-s − 2·113-s − 121-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-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

\(L(\frac{1}{2})\) \(\approx\) \(0.4091979955\)
\(L(\frac12)\) \(\approx\) \(0.4091979955\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
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

−13.58387184974898777730455684846, −12.99948646994780214470491825403, −12.08663318021246051986637663717, −11.71565787232289332281999171781, −11.57875433332852445614781140229, −10.91510297876007671294013685694, −10.28540261466507186940224189090, −9.728772459251592395121557522403, −9.416118499399610651969579898083, −8.616384978317592918256578488262, −8.187516102217810345877015148760, −7.83096366481836823171257604871, −6.93281965314840913390216512087, −6.46389700689867982798864646199, −5.70988805836185351027648661991, −5.42246871145185096342319339715, −4.27941801431036193310475328141, −3.95015737579757376172805773740, −2.78769295406376910388561306770, −2.09761890952711296516119355548, 2.09761890952711296516119355548, 2.78769295406376910388561306770, 3.95015737579757376172805773740, 4.27941801431036193310475328141, 5.42246871145185096342319339715, 5.70988805836185351027648661991, 6.46389700689867982798864646199, 6.93281965314840913390216512087, 7.83096366481836823171257604871, 8.187516102217810345877015148760, 8.616384978317592918256578488262, 9.416118499399610651969579898083, 9.728772459251592395121557522403, 10.28540261466507186940224189090, 10.91510297876007671294013685694, 11.57875433332852445614781140229, 11.71565787232289332281999171781, 12.08663318021246051986637663717, 12.99948646994780214470491825403, 13.58387184974898777730455684846

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