Properties

Label 4-2925e2-1.1-c1e2-0-2
Degree $4$
Conductor $8555625$
Sign $1$
Analytic cond. $545.514$
Root an. cond. $4.83282$
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  + 3·4-s − 8·11-s + 5·16-s − 20·29-s + 8·31-s − 12·41-s − 24·44-s − 2·49-s + 24·59-s − 4·61-s + 3·64-s − 16·79-s − 4·89-s + 36·101-s + 4·109-s − 60·116-s + 26·121-s + 24·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 36·164-s + ⋯
L(s)  = 1  + 3/2·4-s − 2.41·11-s + 5/4·16-s − 3.71·29-s + 1.43·31-s − 1.87·41-s − 3.61·44-s − 2/7·49-s + 3.12·59-s − 0.512·61-s + 3/8·64-s − 1.80·79-s − 0.423·89-s + 3.58·101-s + 0.383·109-s − 5.57·116-s + 2.36·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s − 2.81·164-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(8555625\)    =    \(3^{4} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(545.514\)
Root analytic conductor: \(4.83282\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8555625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.457947311\)
\(L(\frac12)\) \(\approx\) \(1.457947311\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−8.772690652692617962106754501795, −8.488049936849601662791879763476, −8.105137708122562591303314349168, −7.63295805216301077889711192275, −7.52215925676824301498297475112, −7.01704765163057923983705989247, −6.95199897208198663300635311659, −6.09488652561611503424574208184, −6.02446013151447915170303196880, −5.52897037324362149421164876483, −5.09618066963863900792939241798, −4.96285408740251922171154036978, −4.17087109576072520398871798955, −3.52300394618510796945620991644, −3.37538769670545714128455133228, −2.57855157966598001635863159004, −2.46681253267662140395938145967, −1.95573353710866215063471080347, −1.45075543654873564354733847921, −0.34372289651979761721810109214, 0.34372289651979761721810109214, 1.45075543654873564354733847921, 1.95573353710866215063471080347, 2.46681253267662140395938145967, 2.57855157966598001635863159004, 3.37538769670545714128455133228, 3.52300394618510796945620991644, 4.17087109576072520398871798955, 4.96285408740251922171154036978, 5.09618066963863900792939241798, 5.52897037324362149421164876483, 6.02446013151447915170303196880, 6.09488652561611503424574208184, 6.95199897208198663300635311659, 7.01704765163057923983705989247, 7.52215925676824301498297475112, 7.63295805216301077889711192275, 8.105137708122562591303314349168, 8.488049936849601662791879763476, 8.772690652692617962106754501795

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