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Properties

Label 4-1475e2-1.1-c0e2-0-0
Degree $4$
Conductor $2175625$
Sign $1$
Analytic cond. $0.541873$
Root an. cond. $0.857974$
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  − 2·9-s − 16-s − 2·49-s + 2·59-s + 4·79-s + 3·81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 2·9-s − 16-s − 2·49-s + 2·59-s + 4·79-s + 3·81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2175625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2175625\)    =    \(5^{4} \cdot 59^{2}\)
Sign: $1$
Analytic conductor: \(0.541873\)
Root analytic conductor: \(0.857974\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2175625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7410074265\)
\(L(\frac12)\) \(\approx\) \(0.7410074265\)
\(L(1)\) 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)$
bad5 \( 1 \)
59$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
3$C_2$ \( ( 1 + T^{2} )^{2} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_1$ \( ( 1 - T )^{4} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−9.661824760670272744223403996093, −9.526686414883737466004981706288, −9.079562784043829733195235290261, −8.648666571366913225727854546299, −8.337700370332096547381535784099, −8.019136638773275010020655443501, −7.63104556222187187241859609647, −6.82782471194764958260720039451, −6.77603651759951632737527942140, −6.17420791319497562684711211992, −5.83692501207181423016422091503, −5.34694528797564488199459493844, −4.94482706029995125024908061788, −4.57241829855701045579646929327, −3.82308343416056008318829456303, −3.35823307088419587551206029338, −2.96782634813814768949175756229, −2.26594889936624647880784999183, −1.98475609478405311410719508406, −0.67806292399861975845774580866, 0.67806292399861975845774580866, 1.98475609478405311410719508406, 2.26594889936624647880784999183, 2.96782634813814768949175756229, 3.35823307088419587551206029338, 3.82308343416056008318829456303, 4.57241829855701045579646929327, 4.94482706029995125024908061788, 5.34694528797564488199459493844, 5.83692501207181423016422091503, 6.17420791319497562684711211992, 6.77603651759951632737527942140, 6.82782471194764958260720039451, 7.63104556222187187241859609647, 8.019136638773275010020655443501, 8.337700370332096547381535784099, 8.648666571366913225727854546299, 9.079562784043829733195235290261, 9.526686414883737466004981706288, 9.661824760670272744223403996093

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