Properties

Label 4-209088-1.1-c1e2-0-35
Degree $4$
Conductor $209088$
Sign $-1$
Analytic cond. $13.3316$
Root an. cond. $1.91082$
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 − 3-s − 4-s − 4·5-s − 6-s − 3·8-s + 9-s − 4·10-s + 12-s + 4·15-s − 16-s + 18-s + 4·20-s + 16·23-s + 3·24-s + 2·25-s − 27-s − 12·29-s + 4·30-s + 5·32-s − 36-s + 12·40-s − 4·45-s + 16·46-s + 16·47-s + 48-s + 2·49-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 1.78·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 1.26·10-s + 0.288·12-s + 1.03·15-s − 1/4·16-s + 0.235·18-s + 0.894·20-s + 3.33·23-s + 0.612·24-s + 2/5·25-s − 0.192·27-s − 2.22·29-s + 0.730·30-s + 0.883·32-s − 1/6·36-s + 1.89·40-s − 0.596·45-s + 2.35·46-s + 2.33·47-s + 0.144·48-s + 2/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209088 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209088 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(209088\)    =    \(2^{6} \cdot 3^{3} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(13.3316\)
Root analytic conductor: \(1.91082\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 209088,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
3$C_1$ \( 1 + T \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−8.943562213091257213134062166584, −8.495584631629313203905977418995, −7.46995816184461358823952634116, −7.42275429983940735099135130430, −7.18896686257792985441250548424, −6.25377810228368305041472746290, −5.68721248179579351211919147224, −5.31622482563829752699303340190, −4.72191662284056988692815425652, −4.20407903283329927801590362766, −3.81064897716730687178513490998, −3.31425171933473999322603981332, −2.57961275990012009877837343882, −1.06341230628835129193123862023, 0, 1.06341230628835129193123862023, 2.57961275990012009877837343882, 3.31425171933473999322603981332, 3.81064897716730687178513490998, 4.20407903283329927801590362766, 4.72191662284056988692815425652, 5.31622482563829752699303340190, 5.68721248179579351211919147224, 6.25377810228368305041472746290, 7.18896686257792985441250548424, 7.42275429983940735099135130430, 7.46995816184461358823952634116, 8.495584631629313203905977418995, 8.943562213091257213134062166584

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