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Properties

Label 4-95e2-1.1-c0e2-0-0
Degree $4$
Conductor $9025$
Sign $1$
Analytic cond. $0.00224781$
Root an. cond. $0.217741$
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  − 2·5-s − 16-s − 2·19-s + 3·25-s + 2·49-s + 2·80-s − 81-s + 4·95-s − 2·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-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 + ⋯
L(s)  = 1  − 2·5-s − 16-s − 2·19-s + 3·25-s + 2·49-s + 2·80-s − 81-s + 4·95-s − 2·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-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 + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(9025\)    =    \(5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.00224781\)
Root analytic conductor: \(0.217741\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9025,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2489762189\)
\(L(\frac12)\) \(\approx\) \(0.2489762189\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
3$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2^2$ \( 1 + T^{4} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2^2$ \( 1 + T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2^2$ \( 1 + T^{4} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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$C_2^2$ \( 1 + T^{4} \)
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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

−14.61957954625341714164664019560, −14.13944646882214871532234905578, −13.27042701489436960383904648520, −12.93875799995861054376414921190, −12.22315444399224238023174054159, −11.97440438418630739376367126084, −11.35478032774755308995244767083, −10.72883796175334370510374909405, −10.64837931859429375334750285042, −9.566156585394324383794480118050, −8.691956235481506596317612803994, −8.588273787705263002746602261300, −7.88660575820453984407934978439, −7.14472277853861817875692871455, −6.82424315532089692950431228435, −5.92355506824304704240004536396, −4.75709052287312937184999225316, −4.29396961792893118367281974002, −3.66881192896887252833915429072, −2.51518215952811900062369427304, 2.51518215952811900062369427304, 3.66881192896887252833915429072, 4.29396961792893118367281974002, 4.75709052287312937184999225316, 5.92355506824304704240004536396, 6.82424315532089692950431228435, 7.14472277853861817875692871455, 7.88660575820453984407934978439, 8.588273787705263002746602261300, 8.691956235481506596317612803994, 9.566156585394324383794480118050, 10.64837931859429375334750285042, 10.72883796175334370510374909405, 11.35478032774755308995244767083, 11.97440438418630739376367126084, 12.22315444399224238023174054159, 12.93875799995861054376414921190, 13.27042701489436960383904648520, 14.13944646882214871532234905578, 14.61957954625341714164664019560

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