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Properties

Label 4-2925e2-1.1-c0e2-0-2
Degree $4$
Conductor $8555625$
Sign $1$
Analytic cond. $2.13091$
Root an. cond. $1.20820$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s + 2·13-s − 16-s + 2·19-s + 2·31-s + 2·37-s + 2·49-s + 2·67-s + 2·73-s − 4·91-s − 2·97-s + 2·109-s + 2·112-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2·7-s + 2·13-s − 16-s + 2·19-s + 2·31-s + 2·37-s + 2·49-s + 2·67-s + 2·73-s − 4·91-s − 2·97-s + 2·109-s + 2·112-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8555625\)    =    \(3^{4} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(2.13091\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8555625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.286758884\)
\(L(\frac12)\) \(\approx\) \(1.286758884\)
\(L(1)\) 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 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
11$C_2^2$ \( 1 + T^{4} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
41$C_2^2$ \( 1 + T^{4} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2^2$ \( 1 + T^{4} \)
97$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
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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

−9.097407592431904067470331168552, −8.985993131654618376910504969253, −8.249313731272776710654606122550, −8.129456183186555261499199370028, −7.65393889534254582285729320616, −7.15485647259275612211134353752, −6.70561597173879844154223885900, −6.34361901833019919199888573346, −6.32056528747818534711733054607, −5.88636541586108532440335442379, −5.18744245549190367212731792590, −5.08973662254021009056302118022, −4.16427502390397744213224719821, −4.00413991492576385201963831895, −3.54135330926877451265211800089, −3.04201429419698191488151207029, −2.78208793556568982001997120457, −2.24686146542404139963555246619, −1.17424830026051623143488192957, −0.859533339486763757477469956919, 0.859533339486763757477469956919, 1.17424830026051623143488192957, 2.24686146542404139963555246619, 2.78208793556568982001997120457, 3.04201429419698191488151207029, 3.54135330926877451265211800089, 4.00413991492576385201963831895, 4.16427502390397744213224719821, 5.08973662254021009056302118022, 5.18744245549190367212731792590, 5.88636541586108532440335442379, 6.32056528747818534711733054607, 6.34361901833019919199888573346, 6.70561597173879844154223885900, 7.15485647259275612211134353752, 7.65393889534254582285729320616, 8.129456183186555261499199370028, 8.249313731272776710654606122550, 8.985993131654618376910504969253, 9.097407592431904067470331168552

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