Properties

Label 4-1575e2-1.1-c0e2-0-4
Degree $4$
Conductor $2480625$
Sign $1$
Analytic cond. $0.617839$
Root an. cond. $0.886581$
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  + 4-s + 2·7-s + 2·28-s − 2·37-s − 2·43-s + 3·49-s − 64-s + 2·67-s − 2·79-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s − 2·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 4-s + 2·7-s + 2·28-s − 2·37-s − 2·43-s + 3·49-s − 64-s + 2·67-s − 2·79-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s − 2·172-s + 173-s + 179-s + 181-s + 191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2480625\)    =    \(3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.617839\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1575} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2480625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.839449936\)
\(L(\frac12)\) \(\approx\) \(1.839449936\)
\(L(1)\) 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 \( 1 \)
5 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_2^2$ \( 1 - T^{2} + T^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2$ \( ( 1 + T + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 + T + T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_2$ \( ( 1 - T + T^{2} )^{2} \)
71$C_2^2$ \( 1 - T^{2} + T^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2$ \( ( 1 + T + T^{2} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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

−10.02944591943552765476781696696, −9.349615901642857061912383115461, −8.902902054120991952451699376794, −8.424769593532622569408273807934, −8.301686865379305276477974453446, −7.932734379283643434734382269335, −7.24880537699402714312724214322, −6.99940598940526418624129680332, −6.93558457106526796701733672629, −6.05763279934618195094641238203, −5.80869347817437297985492816577, −5.23210941933012756997585437601, −4.79922409943941392386911840550, −4.63449302403676709744177095760, −3.84928487291438368783595224287, −3.41769352103128106380981335470, −2.76184147861209524498395113499, −1.96985727032657027268391864894, −1.93960480328905945514688461578, −1.20585865558191633564296078501, 1.20585865558191633564296078501, 1.93960480328905945514688461578, 1.96985727032657027268391864894, 2.76184147861209524498395113499, 3.41769352103128106380981335470, 3.84928487291438368783595224287, 4.63449302403676709744177095760, 4.79922409943941392386911840550, 5.23210941933012756997585437601, 5.80869347817437297985492816577, 6.05763279934618195094641238203, 6.93558457106526796701733672629, 6.99940598940526418624129680332, 7.24880537699402714312724214322, 7.932734379283643434734382269335, 8.301686865379305276477974453446, 8.424769593532622569408273807934, 8.902902054120991952451699376794, 9.349615901642857061912383115461, 10.02944591943552765476781696696

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