Properties

Label 4-1881e2-1.1-c1e2-0-0
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  − 2·5-s + 3·11-s − 4·16-s − 8·23-s − 7·25-s + 4·31-s + 16·37-s − 6·47-s − 5·49-s + 12·53-s − 6·55-s + 16·67-s − 24·71-s + 8·80-s − 20·89-s − 4·97-s + 28·103-s + 12·113-s + 16·115-s − 2·121-s + 26·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.904·11-s − 16-s − 1.66·23-s − 7/5·25-s + 0.718·31-s + 2.63·37-s − 0.875·47-s − 5/7·49-s + 1.64·53-s − 0.809·55-s + 1.95·67-s − 2.84·71-s + 0.894·80-s − 2.11·89-s − 0.406·97-s + 2.75·103-s + 1.12·113-s + 1.49·115-s − 0.181·121-s + 2.32·125-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

\(L(1)\) \(\approx\) \(1.092539744\)
\(L(\frac12)\) \(\approx\) \(1.092539744\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 3 T + p T^{2} \)
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$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−7.33179869161333338136909211767, −7.29707898629694381108395138586, −6.60752827051636486522149579414, −6.12831923497636436478284774657, −6.05225069915822473275862465295, −5.48074142728575470832715294214, −4.75227634922127063518685270995, −4.26443638337971987135583135453, −4.20883425493377341015603476651, −3.71796474960251568579923305667, −3.14721318815960475902542972767, −2.44031011733442394942262196334, −2.05755062836700898947844296367, −1.29269868638492858267482468590, −0.40068131495792376184564288152, 0.40068131495792376184564288152, 1.29269868638492858267482468590, 2.05755062836700898947844296367, 2.44031011733442394942262196334, 3.14721318815960475902542972767, 3.71796474960251568579923305667, 4.20883425493377341015603476651, 4.26443638337971987135583135453, 4.75227634922127063518685270995, 5.48074142728575470832715294214, 6.05225069915822473275862465295, 6.12831923497636436478284774657, 6.60752827051636486522149579414, 7.29707898629694381108395138586, 7.33179869161333338136909211767

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