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Properties

Label 4-417e2-1.1-c1e2-0-2
Degree $4$
Conductor $173889$
Sign $-1$
Analytic cond. $11.0873$
Root an. cond. $1.82476$
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 − 3·4-s + 6·7-s + 9-s − 6·12-s − 14·13-s + 5·16-s − 4·19-s + 12·21-s − 9·25-s − 4·27-s − 18·28-s + 18·31-s − 3·36-s + 4·37-s − 28·39-s − 8·43-s + 10·48-s + 13·49-s + 42·52-s − 8·57-s + 8·61-s + 6·63-s − 3·64-s + 10·67-s − 12·73-s − 18·75-s + ⋯
L(s)  = 1  + 1.15·3-s − 3/2·4-s + 2.26·7-s + 1/3·9-s − 1.73·12-s − 3.88·13-s + 5/4·16-s − 0.917·19-s + 2.61·21-s − 9/5·25-s − 0.769·27-s − 3.40·28-s + 3.23·31-s − 1/2·36-s + 0.657·37-s − 4.48·39-s − 1.21·43-s + 1.44·48-s + 13/7·49-s + 5.82·52-s − 1.05·57-s + 1.02·61-s + 0.755·63-s − 3/8·64-s + 1.22·67-s − 1.40·73-s − 2.07·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173889 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173889 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(173889\)    =    \(3^{2} \cdot 139^{2}\)
Sign: $-1$
Analytic conductor: \(11.0873\)
Root analytic conductor: \(1.82476\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 173889,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - 2 T + p T^{2} \)
139$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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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

−8.811884136384314893571593295473, −8.190203038931783584337942681835, −8.146982879470191624687775779965, −7.930144070522670332690586305239, −7.28764668532871403210342256953, −6.72800610302640283520955097645, −5.63026345899883128085004455451, −5.10543189027719410386585952401, −4.81914860705714327342053379347, −4.32495336461058370225288751855, −4.11185559790172093630296195648, −2.67890315671211610262367145613, −2.51194611483183451312862090581, −1.66332296267541803877349447055, 0, 1.66332296267541803877349447055, 2.51194611483183451312862090581, 2.67890315671211610262367145613, 4.11185559790172093630296195648, 4.32495336461058370225288751855, 4.81914860705714327342053379347, 5.10543189027719410386585952401, 5.63026345899883128085004455451, 6.72800610302640283520955097645, 7.28764668532871403210342256953, 7.930144070522670332690586305239, 8.146982879470191624687775779965, 8.190203038931783584337942681835, 8.811884136384314893571593295473

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