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Properties

Label 4-735e2-1.1-c0e2-0-2
Degree $4$
Conductor $540225$
Sign $1$
Analytic cond. $0.134551$
Root an. cond. $0.605650$
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 − 2·5-s + 3·9-s − 4·15-s − 16-s + 3·25-s + 4·27-s − 6·45-s − 2·48-s + 6·75-s + 2·80-s + 5·81-s − 4·109-s + 2·121-s − 4·125-s + 127-s + 131-s − 8·135-s + 137-s + 139-s − 3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + ⋯
L(s)  = 1  + 2·3-s − 2·5-s + 3·9-s − 4·15-s − 16-s + 3·25-s + 4·27-s − 6·45-s − 2·48-s + 6·75-s + 2·80-s + 5·81-s − 4·109-s + 2·121-s − 4·125-s + 127-s + 131-s − 8·135-s + 137-s + 139-s − 3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(540225\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.134551\)
Root analytic conductor: \(0.605650\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 540225,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.270906935\)
\(L(\frac12)\) \(\approx\) \(1.270906935\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_2^2$ \( 1 + T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−10.77768093268297960775342011246, −10.40773855280116794576302295303, −9.752931178064457392572862024245, −9.363250945143462697860701583681, −8.978941710016274445341356389273, −8.595093423959007355677406077621, −8.097276034469993834674201595001, −8.060314826854830759034884489992, −7.43231271360043508287367320463, −6.96371513771519269028208804194, −6.94527195981563389294367829869, −6.14663827348177540740676776082, −5.05119757585646418650918676465, −4.69867315682297751960105926440, −4.14552515157552891623268802086, −3.86403498685224914303091667555, −3.32584452150696290349171029178, −2.81268713147475234175828835137, −2.27087558023038572895945376674, −1.26452296747627704830958768689, 1.26452296747627704830958768689, 2.27087558023038572895945376674, 2.81268713147475234175828835137, 3.32584452150696290349171029178, 3.86403498685224914303091667555, 4.14552515157552891623268802086, 4.69867315682297751960105926440, 5.05119757585646418650918676465, 6.14663827348177540740676776082, 6.94527195981563389294367829869, 6.96371513771519269028208804194, 7.43231271360043508287367320463, 8.060314826854830759034884489992, 8.097276034469993834674201595001, 8.595093423959007355677406077621, 8.978941710016274445341356389273, 9.363250945143462697860701583681, 9.752931178064457392572862024245, 10.40773855280116794576302295303, 10.77768093268297960775342011246

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