Properties

Label 4-209952-1.1-c1e2-0-19
Degree $4$
Conductor $209952$
Sign $-1$
Analytic cond. $13.3867$
Root an. cond. $1.91279$
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  + 2-s + 4-s + 8-s − 6·11-s − 5·13-s + 16-s − 6·22-s − 3·23-s + 8·25-s − 5·26-s + 32-s + 37-s − 6·44-s − 3·46-s − 12·47-s − 10·49-s + 8·50-s − 5·52-s − 3·59-s − 11·61-s + 64-s + 3·71-s + 4·73-s + 74-s − 12·83-s − 6·88-s − 3·92-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.80·11-s − 1.38·13-s + 1/4·16-s − 1.27·22-s − 0.625·23-s + 8/5·25-s − 0.980·26-s + 0.176·32-s + 0.164·37-s − 0.904·44-s − 0.442·46-s − 1.75·47-s − 1.42·49-s + 1.13·50-s − 0.693·52-s − 0.390·59-s − 1.40·61-s + 1/8·64-s + 0.356·71-s + 0.468·73-s + 0.116·74-s − 1.31·83-s − 0.639·88-s − 0.312·92-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: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 209952,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
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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

−8.670274077605398649960751251650, −8.240747879232697714406583035573, −7.77529422590753632701004175077, −7.42748451339827369800413996854, −6.88149477971273576832520832473, −6.37127546589436624514450600258, −5.80282719471908383140327704296, −5.04999835347998172852912181029, −4.98480653433086448867763565630, −4.48214866038882183870748168335, −3.55024271220538671458349359859, −2.84037249440615096555350895894, −2.59658973066534259191127744193, −1.64914251233014776706705464060, 0, 1.64914251233014776706705464060, 2.59658973066534259191127744193, 2.84037249440615096555350895894, 3.55024271220538671458349359859, 4.48214866038882183870748168335, 4.98480653433086448867763565630, 5.04999835347998172852912181029, 5.80282719471908383140327704296, 6.37127546589436624514450600258, 6.88149477971273576832520832473, 7.42748451339827369800413996854, 7.77529422590753632701004175077, 8.240747879232697714406583035573, 8.670274077605398649960751251650

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