Properties

Label 4-143e2-1.1-c1e2-0-0
Degree $4$
Conductor $20449$
Sign $-1$
Analytic cond. $1.30384$
Root an. cond. $1.06857$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·9-s + 4·13-s − 4·16-s − 4·17-s − 2·23-s − 9·25-s + 14·27-s − 8·39-s − 12·43-s + 8·48-s − 10·49-s + 8·51-s − 12·53-s + 24·61-s + 4·69-s + 18·75-s − 20·79-s − 4·81-s + 4·101-s − 32·103-s + 36·107-s + 18·113-s − 12·117-s + 121-s + 127-s + 24·129-s + ⋯
L(s)  = 1  − 1.15·3-s − 9-s + 1.10·13-s − 16-s − 0.970·17-s − 0.417·23-s − 9/5·25-s + 2.69·27-s − 1.28·39-s − 1.82·43-s + 1.15·48-s − 1.42·49-s + 1.12·51-s − 1.64·53-s + 3.07·61-s + 0.481·69-s + 2.07·75-s − 2.25·79-s − 4/9·81-s + 0.398·101-s − 3.15·103-s + 3.48·107-s + 1.69·113-s − 1.10·117-s + 1/11·121-s + 0.0887·127-s + 2.11·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(20449\)    =    \(11^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(1.30384\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{20449} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 20449,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−10.91113543903689003112366466403, −10.03550909718107888433464868208, −9.697960688174592536443811508987, −8.679083346749551349707825193313, −8.603539619290756001226038948684, −7.972094664118160382294418175959, −6.98822017919486224475963260290, −6.36261389471308870138602900888, −6.13551737647869362536665067533, −5.42467290200223196482028296291, −4.84072493234829119227722123670, −4.01156608984212096591011250141, −3.11506443817379691538930485019, −1.98576135654663457689533375342, 0, 1.98576135654663457689533375342, 3.11506443817379691538930485019, 4.01156608984212096591011250141, 4.84072493234829119227722123670, 5.42467290200223196482028296291, 6.13551737647869362536665067533, 6.36261389471308870138602900888, 6.98822017919486224475963260290, 7.972094664118160382294418175959, 8.603539619290756001226038948684, 8.679083346749551349707825193313, 9.697960688174592536443811508987, 10.03550909718107888433464868208, 10.91113543903689003112366466403

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