Properties

Label 4-57e3-1.1-c1e2-0-7
Degree $4$
Conductor $185193$
Sign $-1$
Analytic cond. $11.8080$
Root an. cond. $1.85372$
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  + 2·2-s + 3-s − 4-s + 2·6-s − 8·8-s + 9-s − 12-s − 7·16-s + 2·18-s − 19-s − 8·24-s − 6·25-s + 27-s + 4·29-s + 14·32-s − 36-s − 2·38-s − 4·41-s − 8·43-s − 7·48-s − 14·49-s − 12·50-s − 12·53-s + 2·54-s − 57-s + 8·58-s − 24·59-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s − 1/2·4-s + 0.816·6-s − 2.82·8-s + 1/3·9-s − 0.288·12-s − 7/4·16-s + 0.471·18-s − 0.229·19-s − 1.63·24-s − 6/5·25-s + 0.192·27-s + 0.742·29-s + 2.47·32-s − 1/6·36-s − 0.324·38-s − 0.624·41-s − 1.21·43-s − 1.01·48-s − 2·49-s − 1.69·50-s − 1.64·53-s + 0.272·54-s − 0.132·57-s + 1.05·58-s − 3.12·59-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: no
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$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−9.032116632714081125535639739797, −8.263513432862467471567669978548, −8.149677875507257955940242249903, −7.61675340778570823466262486344, −6.50584246724891866846885766490, −6.46787589567348650226556085189, −5.82020929693792166281675138030, −5.18025942388109829342113033634, −4.59000163102556089132067894166, −4.56991321080597356703354311172, −3.52439250757221266883814274902, −3.44365518362789223021999993571, −2.73578806708278678142126219562, −1.66141629985644983818808667860, 0, 1.66141629985644983818808667860, 2.73578806708278678142126219562, 3.44365518362789223021999993571, 3.52439250757221266883814274902, 4.56991321080597356703354311172, 4.59000163102556089132067894166, 5.18025942388109829342113033634, 5.82020929693792166281675138030, 6.46787589567348650226556085189, 6.50584246724891866846885766490, 7.61675340778570823466262486344, 8.149677875507257955940242249903, 8.263513432862467471567669978548, 9.032116632714081125535639739797

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