Properties

Label 4-83232-1.1-c1e2-0-23
Degree $4$
Conductor $83232$
Sign $-1$
Analytic cond. $5.30694$
Root an. cond. $1.51778$
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 + 4-s − 4·5-s + 8-s + 9-s − 4·10-s − 4·13-s + 16-s + 2·17-s + 18-s − 4·20-s + 2·25-s − 4·26-s − 20·29-s + 32-s + 2·34-s + 36-s − 4·37-s − 4·40-s + 20·41-s − 4·45-s − 14·49-s + 2·50-s − 4·52-s + 12·53-s − 20·58-s − 20·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.353·8-s + 1/3·9-s − 1.26·10-s − 1.10·13-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.894·20-s + 2/5·25-s − 0.784·26-s − 3.71·29-s + 0.176·32-s + 0.342·34-s + 1/6·36-s − 0.657·37-s − 0.632·40-s + 3.12·41-s − 0.596·45-s − 2·49-s + 0.282·50-s − 0.554·52-s + 1.64·53-s − 2.62·58-s − 2.56·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83232 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83232 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(83232\)    =    \(2^{5} \cdot 3^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(5.30694\)
Root analytic conductor: \(1.51778\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{83232} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 83232,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
17$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−9.570394810938145814587700544156, −9.080746654144001493159861980940, −7.972624440390951709752181868622, −7.958882066985845549410362974675, −7.36440720073199929079936076774, −7.22142790948264379459326388401, −6.37869009161194820234692371426, −5.51457882926074302977186886250, −5.36244298721167160718725340278, −4.24541437163957543556604196065, −4.13875090005906653574798019221, −3.56656081546400251035965478912, −2.76309813409348720224135008444, −1.78155952274398559703453953653, 0, 1.78155952274398559703453953653, 2.76309813409348720224135008444, 3.56656081546400251035965478912, 4.13875090005906653574798019221, 4.24541437163957543556604196065, 5.36244298721167160718725340278, 5.51457882926074302977186886250, 6.37869009161194820234692371426, 7.22142790948264379459326388401, 7.36440720073199929079936076774, 7.958882066985845549410362974675, 7.972624440390951709752181868622, 9.080746654144001493159861980940, 9.570394810938145814587700544156

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