Properties

Label 4-1881e2-1.1-c1e2-0-1
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s + 4·5-s + 5·16-s − 12·20-s − 8·23-s + 2·25-s + 16·31-s − 20·37-s − 24·47-s − 14·49-s + 12·53-s + 24·59-s − 3·64-s − 8·67-s + 20·80-s + 4·89-s + 24·92-s + 20·97-s − 6·100-s + 16·103-s − 12·113-s − 32·115-s − 11·121-s − 48·124-s − 28·125-s + 127-s + 131-s + ⋯
L(s)  = 1  − 3/2·4-s + 1.78·5-s + 5/4·16-s − 2.68·20-s − 1.66·23-s + 2/5·25-s + 2.87·31-s − 3.28·37-s − 3.50·47-s − 2·49-s + 1.64·53-s + 3.12·59-s − 3/8·64-s − 0.977·67-s + 2.23·80-s + 0.423·89-s + 2.50·92-s + 2.03·97-s − 3/5·100-s + 1.57·103-s − 1.12·113-s − 2.98·115-s − 121-s − 4.31·124-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-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

\(L(1)\) \(\approx\) \(1.312454359\)
\(L(\frac12)\) \(\approx\) \(1.312454359\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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

−7.70753672790277479184810469564, −6.80620935754201724697544605747, −6.43588567247677360674191056356, −6.37820814688737049661034336254, −5.71365556174897493669616908213, −5.30285359031178015236684552035, −5.01983353040794858781277453483, −4.72172884745122847521792429694, −3.99863452731206564038285060374, −3.69904169102384607719527879903, −3.12155744805932295591521959890, −2.40533336339214878263035341483, −1.84627514401494290368794564616, −1.48488050469889056862198933364, −0.42531230386250425831126688255, 0.42531230386250425831126688255, 1.48488050469889056862198933364, 1.84627514401494290368794564616, 2.40533336339214878263035341483, 3.12155744805932295591521959890, 3.69904169102384607719527879903, 3.99863452731206564038285060374, 4.72172884745122847521792429694, 5.01983353040794858781277453483, 5.30285359031178015236684552035, 5.71365556174897493669616908213, 6.37820814688737049661034336254, 6.43588567247677360674191056356, 6.80620935754201724697544605747, 7.70753672790277479184810469564

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