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Properties

Label 4-95e2-1.1-c1e2-0-3
Degree $4$
Conductor $9025$
Sign $1$
Analytic cond. $0.575441$
Root an. cond. $0.870964$
Motivic weight $1$
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 + 2·4-s + 5-s − 8·7-s + 3·9-s + 6·11-s + 4·12-s − 2·13-s + 2·15-s − 6·17-s − 7·19-s + 2·20-s − 16·21-s + 10·27-s − 16·28-s + 3·29-s − 14·31-s + 12·33-s − 8·35-s + 6·36-s + 16·37-s − 4·39-s + 6·41-s + 4·43-s + 12·44-s + 3·45-s − 6·47-s + ⋯
L(s)  = 1  + 1.15·3-s + 4-s + 0.447·5-s − 3.02·7-s + 9-s + 1.80·11-s + 1.15·12-s − 0.554·13-s + 0.516·15-s − 1.45·17-s − 1.60·19-s + 0.447·20-s − 3.49·21-s + 1.92·27-s − 3.02·28-s + 0.557·29-s − 2.51·31-s + 2.08·33-s − 1.35·35-s + 36-s + 2.63·37-s − 0.640·39-s + 0.937·41-s + 0.609·43-s + 1.80·44-s + 0.447·45-s − 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9025\)    =    \(5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.575441\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.386773719\)
\(L(\frac12)\) \(\approx\) \(1.386773719\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
19$C_2$ \( 1 + 7 T + p T^{2} \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
3$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 8 T - 9 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−14.45727815431112247704379615372, −13.46187718623727369698285947222, −13.23941662520608353357695345489, −12.79406976587063842094663671018, −12.40959478732948953346559269490, −11.74685280654655406145570868760, −10.80327490361199919615475316806, −10.55286062112174331753025360208, −9.547593772168487256781364363024, −9.280394832524382960342480921885, −9.239927903917167007568240168427, −8.323530410914496101847811241859, −7.03575539981179466954492587892, −6.93957105375992591277788891737, −6.33318627269780152687854974749, −6.12009696796375254811739836851, −4.25245972936901947248198134348, −3.77337017171378786291803365910, −2.76150367300491270317309970663, −2.29937073169802967093938323200, 2.29937073169802967093938323200, 2.76150367300491270317309970663, 3.77337017171378786291803365910, 4.25245972936901947248198134348, 6.12009696796375254811739836851, 6.33318627269780152687854974749, 6.93957105375992591277788891737, 7.03575539981179466954492587892, 8.323530410914496101847811241859, 9.239927903917167007568240168427, 9.280394832524382960342480921885, 9.547593772168487256781364363024, 10.55286062112174331753025360208, 10.80327490361199919615475316806, 11.74685280654655406145570868760, 12.40959478732948953346559269490, 12.79406976587063842094663671018, 13.23941662520608353357695345489, 13.46187718623727369698285947222, 14.45727815431112247704379615372

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