Properties

Label 4-219488-1.1-c1e2-0-4
Degree $4$
Conductor $219488$
Sign $-1$
Analytic cond. $13.9947$
Root an. cond. $1.93415$
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-s + 2·3-s + 4-s − 2·6-s − 8-s − 3·9-s + 2·12-s + 16-s + 6·17-s + 3·18-s + 19-s − 2·24-s − 10·25-s − 14·27-s − 8·31-s − 32-s − 6·34-s − 3·36-s − 38-s + 2·48-s − 13·49-s + 10·50-s + 12·51-s + 14·54-s + 2·57-s + 18·59-s − 20·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s − 9-s + 0.577·12-s + 1/4·16-s + 1.45·17-s + 0.707·18-s + 0.229·19-s − 0.408·24-s − 2·25-s − 2.69·27-s − 1.43·31-s − 0.176·32-s − 1.02·34-s − 1/2·36-s − 0.162·38-s + 0.288·48-s − 1.85·49-s + 1.41·50-s + 1.68·51-s + 1.90·54-s + 0.264·57-s + 2.34·59-s − 2.56·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 219488 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 219488 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(219488\)    =    \(2^{5} \cdot 19^{3}\)
Sign: $-1$
Analytic conductor: \(13.9947\)
Root analytic conductor: \(1.93415\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{219488} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 219488,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
19$C_1$ \( 1 - T \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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

−8.684413720393242419277779815238, −8.519556646017435527009405270191, −7.73591843742988285704805588989, −7.66995734298328287502507015405, −7.29904094327803276580864474565, −6.29607557013450001946456563759, −5.81027585652647529822118508915, −5.64317441971502851003917388776, −4.83778114166735308330677292657, −3.83932823582255789112395130179, −3.41239623655734813432880019975, −2.99432139105097413830284496966, −2.21851462150366027971745653477, −1.59049790592698824897087379644, 0, 1.59049790592698824897087379644, 2.21851462150366027971745653477, 2.99432139105097413830284496966, 3.41239623655734813432880019975, 3.83932823582255789112395130179, 4.83778114166735308330677292657, 5.64317441971502851003917388776, 5.81027585652647529822118508915, 6.29607557013450001946456563759, 7.29904094327803276580864474565, 7.66995734298328287502507015405, 7.73591843742988285704805588989, 8.519556646017435527009405270191, 8.684413720393242419277779815238

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