Properties

Label 4-1881e2-1.1-c1e2-0-6
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 − 2·5-s − 5·11-s + 12·16-s + 8·20-s + 8·23-s − 7·25-s + 20·44-s − 26·47-s − 5·49-s + 10·55-s − 32·64-s − 24·80-s − 32·92-s + 28·100-s − 16·115-s + 14·121-s + 26·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 2·4-s − 0.894·5-s − 1.50·11-s + 3·16-s + 1.78·20-s + 1.66·23-s − 7/5·25-s + 3.01·44-s − 3.79·47-s − 5/7·49-s + 1.34·55-s − 4·64-s − 2.68·80-s − 3.33·92-s + 14/5·100-s − 1.49·115-s + 1.27·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-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$C_2$ \( 1 + 5 T + p T^{2} \)
19$C_2$ \( 1 + p T^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 - p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2$ \( ( 1 - p T^{2} )^{2} \)
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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

−8.738979976976201257853810631753, −8.725045427049835051731943339193, −8.357066735738094337650560605505, −7.72771058721078606758332559119, −7.72615454437306286116666309475, −7.43951567857818206553383057015, −6.50912467096663711582366281990, −6.28392277529063751415490332146, −5.59148906110287344542070161431, −5.04612041303317826896093672029, −5.04081955678973546134935018204, −4.68777304787031224524260433257, −3.86267349177480595066897407640, −3.84923680137051357943678206515, −3.10658555482041043539164480008, −2.88597296933107317108311339072, −1.82822659615438531971089097147, −1.08028915083979659362350929453, 0, 0, 1.08028915083979659362350929453, 1.82822659615438531971089097147, 2.88597296933107317108311339072, 3.10658555482041043539164480008, 3.84923680137051357943678206515, 3.86267349177480595066897407640, 4.68777304787031224524260433257, 5.04081955678973546134935018204, 5.04612041303317826896093672029, 5.59148906110287344542070161431, 6.28392277529063751415490332146, 6.50912467096663711582366281990, 7.43951567857818206553383057015, 7.72615454437306286116666309475, 7.72771058721078606758332559119, 8.357066735738094337650560605505, 8.725045427049835051731943339193, 8.738979976976201257853810631753

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