Properties

Label 4-1881e2-1.1-c1e2-0-4
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s − 6·5-s − 3·11-s + 12·16-s + 24·20-s + 17·25-s − 8·31-s + 4·37-s + 12·44-s + 6·47-s − 13·49-s − 24·53-s + 18·55-s + 12·59-s − 32·64-s − 8·67-s − 12·71-s − 72·80-s − 24·89-s + 16·97-s − 68·100-s + 28·103-s − 12·113-s − 2·121-s + 32·124-s − 18·125-s + 127-s + ⋯
L(s)  = 1  − 2·4-s − 2.68·5-s − 0.904·11-s + 3·16-s + 5.36·20-s + 17/5·25-s − 1.43·31-s + 0.657·37-s + 1.80·44-s + 0.875·47-s − 1.85·49-s − 3.29·53-s + 2.42·55-s + 1.56·59-s − 4·64-s − 0.977·67-s − 1.42·71-s − 8.04·80-s − 2.54·89-s + 1.62·97-s − 6.79·100-s + 2.75·103-s − 1.12·113-s − 0.181·121-s + 2.87·124-s − 1.60·125-s + 0.0887·127-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: \(2\)
Selberg data: \((4,\ 3538161,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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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

−7.43141403312000422638829977686, −6.81406384107615347926754313575, −6.18353669060809229826575345742, −5.51097905283736056328679977781, −5.36768332519191234744203052270, −4.61370997508791849125436584714, −4.41466206651142691425988858089, −4.28843211103852600447914619291, −3.59240242589109173846669898693, −3.23857552173567549926694025031, −3.07536984090222800043510549725, −1.80078999411668520307404119788, −0.862388966919604043351564526767, 0, 0, 0.862388966919604043351564526767, 1.80078999411668520307404119788, 3.07536984090222800043510549725, 3.23857552173567549926694025031, 3.59240242589109173846669898693, 4.28843211103852600447914619291, 4.41466206651142691425988858089, 4.61370997508791849125436584714, 5.36768332519191234744203052270, 5.51097905283736056328679977781, 6.18353669060809229826575345742, 6.81406384107615347926754313575, 7.43141403312000422638829977686

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