Properties

Label 4-894348-1.1-c1e2-0-36
Degree $4$
Conductor $894348$
Sign $-1$
Analytic cond. $57.0244$
Root an. cond. $2.74799$
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 − 4-s + 9-s − 12-s − 6·13-s + 16-s + 6·25-s + 27-s − 36-s − 6·39-s − 8·43-s + 48-s + 49-s + 6·52-s − 12·61-s − 64-s + 6·75-s + 81-s − 6·100-s + 16·103-s − 108-s − 6·117-s + 6·121-s + 127-s − 8·129-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.577·3-s − 1/2·4-s + 1/3·9-s − 0.288·12-s − 1.66·13-s + 1/4·16-s + 6/5·25-s + 0.192·27-s − 1/6·36-s − 0.960·39-s − 1.21·43-s + 0.144·48-s + 1/7·49-s + 0.832·52-s − 1.53·61-s − 1/8·64-s + 0.692·75-s + 1/9·81-s − 3/5·100-s + 1.57·103-s − 0.0962·108-s − 0.554·117-s + 6/11·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 894348 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 894348 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(894348\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(57.0244\)
Root analytic conductor: \(2.74799\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{894348} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 894348,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2$C_2$ \( 1 + T^{2} \)
3$C_1$ \( 1 - T \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_2$ \( 1 + 6 T + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−8.025688309168167905022269870458, −7.41024547576845205147702456890, −7.29897087109296957472268982667, −6.72513032249390455481214147424, −6.24592091240545789421425441517, −5.63620093622988544848667175177, −5.06857161439399297527291846350, −4.69582305853857169532061766063, −4.44004501872011223263262636066, −3.58903262929681548446051761425, −3.19604100345843133398004727348, −2.58874498006366720886251421530, −2.04062794059719293544223922165, −1.15644919730322647591299718428, 0, 1.15644919730322647591299718428, 2.04062794059719293544223922165, 2.58874498006366720886251421530, 3.19604100345843133398004727348, 3.58903262929681548446051761425, 4.44004501872011223263262636066, 4.69582305853857169532061766063, 5.06857161439399297527291846350, 5.63620093622988544848667175177, 6.24592091240545789421425441517, 6.72513032249390455481214147424, 7.29897087109296957472268982667, 7.41024547576845205147702456890, 8.025688309168167905022269870458

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