Properties

Label 4-898425-1.1-c1e2-0-3
Degree $4$
Conductor $898425$
Sign $1$
Analytic cond. $57.2843$
Root an. cond. $2.75111$
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-s − 3·4-s − 2·5-s − 8·7-s + 9-s − 11-s + 3·12-s + 4·13-s + 2·15-s + 5·16-s + 6·20-s + 8·21-s + 16·23-s − 25-s − 27-s + 24·28-s + 12·29-s − 16·31-s + 33-s + 16·35-s − 3·36-s − 4·39-s + 4·41-s + 3·44-s − 2·45-s + 16·47-s − 5·48-s + ⋯
L(s)  = 1  − 0.577·3-s − 3/2·4-s − 0.894·5-s − 3.02·7-s + 1/3·9-s − 0.301·11-s + 0.866·12-s + 1.10·13-s + 0.516·15-s + 5/4·16-s + 1.34·20-s + 1.74·21-s + 3.33·23-s − 1/5·25-s − 0.192·27-s + 4.53·28-s + 2.22·29-s − 2.87·31-s + 0.174·33-s + 2.70·35-s − 1/2·36-s − 0.640·39-s + 0.624·41-s + 0.452·44-s − 0.298·45-s + 2.33·47-s − 0.721·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 898425 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 898425 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(898425\)    =    \(3^{3} \cdot 5^{2} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(57.2843\)
Root analytic conductor: \(2.75111\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 898425,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5864442096\)
\(L(\frac12)\) \(\approx\) \(0.5864442096\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
11$C_1$ \( 1 + T \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−8.360742840269281397590237858671, −7.67376126887882242787787369865, −7.06746602683221101194508851954, −6.85553577597705614902881023761, −6.57667809882336453792833363635, −5.68297890149735059120081895574, −5.66464594246847012421851082372, −5.05334108529123054060716186457, −4.33065348032451946577135661956, −3.97182275742999473711448669060, −3.37409785551698507568515532805, −3.31119773648628104035349562784, −2.55490898649220854449117581353, −0.71891279628917143719551815370, −0.66299652490358276934219865411, 0.66299652490358276934219865411, 0.71891279628917143719551815370, 2.55490898649220854449117581353, 3.31119773648628104035349562784, 3.37409785551698507568515532805, 3.97182275742999473711448669060, 4.33065348032451946577135661956, 5.05334108529123054060716186457, 5.66464594246847012421851082372, 5.68297890149735059120081895574, 6.57667809882336453792833363635, 6.85553577597705614902881023761, 7.06746602683221101194508851954, 7.67376126887882242787787369865, 8.360742840269281397590237858671

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