Properties

Label 4-209952-1.1-c1e2-0-5
Degree $4$
Conductor $209952$
Sign $1$
Analytic cond. $13.3867$
Root an. cond. $1.91279$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s + 4·13-s + 16-s − 6·17-s − 10·25-s + 4·26-s + 12·29-s + 32-s − 6·34-s − 8·37-s + 18·41-s − 10·49-s − 10·50-s + 4·52-s + 24·53-s + 12·58-s + 16·61-s + 64-s − 6·68-s + 22·73-s − 8·74-s + 18·82-s + 12·89-s + 10·97-s − 10·98-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.10·13-s + 1/4·16-s − 1.45·17-s − 2·25-s + 0.784·26-s + 2.22·29-s + 0.176·32-s − 1.02·34-s − 1.31·37-s + 2.81·41-s − 1.42·49-s − 1.41·50-s + 0.554·52-s + 3.29·53-s + 1.57·58-s + 2.04·61-s + 1/8·64-s − 0.727·68-s + 2.57·73-s − 0.929·74-s + 1.98·82-s + 1.27·89-s + 1.01·97-s − 1.01·98-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(209952\)    =    \(2^{5} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(13.3867\)
Root analytic conductor: \(1.91279\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 209952,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.722931025\)
\(L(\frac12)\) \(\approx\) \(2.722931025\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
3 \( 1 \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
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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

−8.982636192572195812922115743561, −8.543319215757720981308384286559, −8.131333319129826356153375895616, −7.63671497466111343831458484773, −6.97337389606279696915121240099, −6.48609808834188223043361623015, −6.25936593777468405983226275094, −5.57014449737897797729246275255, −5.13805326110939411578475614907, −4.39002474802971059462229776672, −3.94847330019587176084970982031, −3.57376731663222433353058317817, −2.48468545989121014995168597575, −2.19879385611801234222671009191, −0.967496248641159004586773303229, 0.967496248641159004586773303229, 2.19879385611801234222671009191, 2.48468545989121014995168597575, 3.57376731663222433353058317817, 3.94847330019587176084970982031, 4.39002474802971059462229776672, 5.13805326110939411578475614907, 5.57014449737897797729246275255, 6.25936593777468405983226275094, 6.48609808834188223043361623015, 6.97337389606279696915121240099, 7.63671497466111343831458484773, 8.131333319129826356153375895616, 8.543319215757720981308384286559, 8.982636192572195812922115743561

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