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Properties

Label 4-55e2-1.1-c1e2-0-0
Degree $4$
Conductor $3025$
Sign $1$
Analytic cond. $0.192876$
Root an. cond. $0.662704$
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  + 5-s − 5·9-s + 2·11-s − 4·16-s − 4·25-s + 14·31-s − 16·41-s − 5·45-s − 10·49-s + 2·55-s + 10·59-s + 24·61-s − 6·71-s − 20·79-s − 4·80-s + 16·81-s + 30·89-s − 10·99-s + 4·101-s + 20·109-s + 3·121-s − 9·125-s + 127-s + 131-s + 137-s + 139-s + 20·144-s + ⋯
L(s)  = 1  + 0.447·5-s − 5/3·9-s + 0.603·11-s − 16-s − 4/5·25-s + 2.51·31-s − 2.49·41-s − 0.745·45-s − 1.42·49-s + 0.269·55-s + 1.30·59-s + 3.07·61-s − 0.712·71-s − 2.25·79-s − 0.447·80-s + 16/9·81-s + 3.17·89-s − 1.00·99-s + 0.398·101-s + 1.91·109-s + 3/11·121-s − 0.804·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/3·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(0.192876\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7204129186\)
\(L(\frac12)\) \(\approx\) \(0.7204129186\)
\(L(\frac{3}{2})\) 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)$
bad5$C_2$ \( 1 - T + p T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
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

−12.97276368073609917691874306128, −11.91373644505446088258662904950, −11.50442818083742402892485916796, −11.45125861034521058353374083886, −10.07566606066947823589561871516, −10.03550909718107888433464868208, −8.906461952934258973434563843342, −8.603539619290756001226038948684, −7.952532350978681778925001629133, −6.74210091587050441798111318378, −6.36261389471308870138602900888, −5.49322091026488161466653442724, −4.70136976599285716284507090528, −3.45386803680567664069181238677, −2.35074848930377289705338533775, 2.35074848930377289705338533775, 3.45386803680567664069181238677, 4.70136976599285716284507090528, 5.49322091026488161466653442724, 6.36261389471308870138602900888, 6.74210091587050441798111318378, 7.952532350978681778925001629133, 8.603539619290756001226038948684, 8.906461952934258973434563843342, 10.03550909718107888433464868208, 10.07566606066947823589561871516, 11.45125861034521058353374083886, 11.50442818083742402892485916796, 11.91373644505446088258662904950, 12.97276368073609917691874306128

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