Properties

Label 4-231e2-1.1-c1e2-0-13
Degree $4$
Conductor $53361$
Sign $-1$
Analytic cond. $3.40234$
Root an. cond. $1.35814$
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  − 3-s + 2·5-s − 2·7-s − 2·9-s − 2·15-s − 4·16-s − 4·17-s + 2·21-s − 7·25-s + 5·27-s − 4·35-s + 6·37-s − 16·41-s − 12·43-s − 4·45-s + 16·47-s + 4·48-s − 3·49-s + 4·51-s + 10·59-s + 4·63-s − 14·67-s + 7·75-s − 20·79-s − 8·80-s + 81-s − 12·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.755·7-s − 2/3·9-s − 0.516·15-s − 16-s − 0.970·17-s + 0.436·21-s − 7/5·25-s + 0.962·27-s − 0.676·35-s + 0.986·37-s − 2.49·41-s − 1.82·43-s − 0.596·45-s + 2.33·47-s + 0.577·48-s − 3/7·49-s + 0.560·51-s + 1.30·59-s + 0.503·63-s − 1.71·67-s + 0.808·75-s − 2.25·79-s − 0.894·80-s + 1/9·81-s − 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(53361\)    =    \(3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(3.40234\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 53361,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T + p T^{2} \)
7$C_2$ \( 1 + 2 T + p T^{2} \)
11$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$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
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

−10.03550909718107888433464868208, −9.306254765286992650868144024138, −8.701526732425662214712394784166, −8.603539619290756001226038948684, −7.59708149153402218158879958117, −7.03477908276603020210779304212, −6.36261389471308870138602900888, −6.21101030419725238522849385481, −5.55294080899685406164716506758, −4.97685157209324421449824776256, −4.28111751945710160622264923994, −3.43654811333543968800336820680, −2.58331042621701060186581520596, −1.87316747898306017870215940641, 0, 1.87316747898306017870215940641, 2.58331042621701060186581520596, 3.43654811333543968800336820680, 4.28111751945710160622264923994, 4.97685157209324421449824776256, 5.55294080899685406164716506758, 6.21101030419725238522849385481, 6.36261389471308870138602900888, 7.03477908276603020210779304212, 7.59708149153402218158879958117, 8.603539619290756001226038948684, 8.701526732425662214712394784166, 9.306254765286992650868144024138, 10.03550909718107888433464868208

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