Properties

Label 4-1881e2-1.1-c1e2-0-3
Degree $4$
Conductor $3538161$
Sign $1$
Analytic cond. $225.596$
Root an. cond. $3.87554$
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  + 6·5-s − 11-s − 4·16-s + 8·23-s + 17·25-s − 12·31-s + 18·47-s + 11·49-s − 20·53-s − 6·55-s + 16·59-s + 16·67-s + 24·71-s − 24·80-s + 12·89-s − 20·97-s − 4·103-s − 4·113-s + 48·115-s − 10·121-s + 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 2.68·5-s − 0.301·11-s − 16-s + 1.66·23-s + 17/5·25-s − 2.15·31-s + 2.62·47-s + 11/7·49-s − 2.74·53-s − 0.809·55-s + 2.08·59-s + 1.95·67-s + 2.84·71-s − 2.68·80-s + 1.27·89-s − 2.03·97-s − 0.394·103-s − 0.376·113-s + 4.47·115-s − 0.909·121-s + 1.60·125-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3538161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3538161 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3538161\)    =    \(3^{4} \cdot 11^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(225.596\)
Root analytic conductor: \(3.87554\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3538161,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.284391072\)
\(L(\frac12)\) \(\approx\) \(4.284391072\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_2$ \( 1 + T + p T^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−7.30232741284270530134583231435, −6.85073886350288365363080648917, −6.76279579633310090758813603712, −6.12241751609850681792173625996, −5.82070906602956189936825435616, −5.34668146252658900228619106104, −5.17678752821842570429938873970, −4.82498557247709479492897810782, −3.89094638986417660475021381976, −3.67840796781373175574076394185, −2.61687998381550726785939862479, −2.52520253713806450674658123012, −2.05270270116931634569567010251, −1.50764144758403640274585810032, −0.75861270564125663687760718182, 0.75861270564125663687760718182, 1.50764144758403640274585810032, 2.05270270116931634569567010251, 2.52520253713806450674658123012, 2.61687998381550726785939862479, 3.67840796781373175574076394185, 3.89094638986417660475021381976, 4.82498557247709479492897810782, 5.17678752821842570429938873970, 5.34668146252658900228619106104, 5.82070906602956189936825435616, 6.12241751609850681792173625996, 6.76279579633310090758813603712, 6.85073886350288365363080648917, 7.30232741284270530134583231435

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