Properties

Label 4-57e3-1.1-c1e2-0-6
Degree $4$
Conductor $185193$
Sign $-1$
Analytic cond. $11.8080$
Root an. cond. $1.85372$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·7-s + 9-s − 6·13-s − 4·16-s + 19-s − 2·21-s + 5·25-s + 27-s + 6·31-s − 4·37-s − 6·39-s − 6·43-s − 4·48-s − 7·49-s + 57-s − 14·61-s − 2·63-s − 12·67-s + 22·73-s + 5·75-s − 12·79-s + 81-s + 12·91-s + 6·93-s − 30·97-s + 2·103-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.66·13-s − 16-s + 0.229·19-s − 0.436·21-s + 25-s + 0.192·27-s + 1.07·31-s − 0.657·37-s − 0.960·39-s − 0.914·43-s − 0.577·48-s − 49-s + 0.132·57-s − 1.79·61-s − 0.251·63-s − 1.46·67-s + 2.57·73-s + 0.577·75-s − 1.35·79-s + 1/9·81-s + 1.25·91-s + 0.622·93-s − 3.04·97-s + 0.197·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(185193\)    =    \(3^{3} \cdot 19^{3}\)
Sign: $-1$
Analytic conductor: \(11.8080\)
Root analytic conductor: \(1.85372\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 185193,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
19$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 9 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 + 12 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.940115357310449446465701267899, −8.472219813091981302185925227949, −7.997392793595315494976449359677, −7.30702114428586488307370566559, −7.12799967982723123126786945595, −6.48456210962229854870652413173, −6.19683667007204581085395723445, −5.13420857796973012012772829293, −4.91985703645555358766641041275, −4.34165698013814923627044795587, −3.56569898627655092431905032622, −2.83860099231743616364780125887, −2.58067309650043255229038729097, −1.55907972181611260166091782813, 0, 1.55907972181611260166091782813, 2.58067309650043255229038729097, 2.83860099231743616364780125887, 3.56569898627655092431905032622, 4.34165698013814923627044795587, 4.91985703645555358766641041275, 5.13420857796973012012772829293, 6.19683667007204581085395723445, 6.48456210962229854870652413173, 7.12799967982723123126786945595, 7.30702114428586488307370566559, 7.997392793595315494976449359677, 8.472219813091981302185925227949, 8.940115357310449446465701267899

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