Properties

Label 4-83232-1.1-c1e2-0-4
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s + 9-s + 4·13-s + 16-s − 2·17-s − 18-s − 10·25-s − 4·26-s − 32-s + 2·34-s + 36-s + 16·37-s + 12·41-s − 10·49-s + 10·50-s + 4·52-s + 12·53-s + 16·61-s + 64-s − 2·68-s − 72-s + 4·73-s − 16·74-s + 81-s − 12·82-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1/3·9-s + 1.10·13-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 2·25-s − 0.784·26-s − 0.176·32-s + 0.342·34-s + 1/6·36-s + 2.63·37-s + 1.87·41-s − 1.42·49-s + 1.41·50-s + 0.554·52-s + 1.64·53-s + 2.04·61-s + 1/8·64-s − 0.242·68-s − 0.117·72-s + 0.468·73-s − 1.85·74-s + 1/9·81-s − 1.32·82-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: \(0\)
Selberg data: \((4,\ 83232,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.108981159\)
\(L(\frac12)\) \(\approx\) \(1.108981159\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
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

−9.629005176149752438633105274587, −9.361746587069330588975575945669, −8.584868551115419667042947592500, −8.367508603382134314146293072792, −7.57323216500839527256914149425, −7.50641789037179894281138679970, −6.62762171992257665130856519803, −5.96956680213085492365415051586, −5.95967651264422152670977788910, −4.94376122657450226002634205632, −4.05372696693836488551925909574, −3.84609690192901140186989283204, −2.71372337045158087575655535987, −2.04126125287275460099535053657, −0.943136411131521382953657369703, 0.943136411131521382953657369703, 2.04126125287275460099535053657, 2.71372337045158087575655535987, 3.84609690192901140186989283204, 4.05372696693836488551925909574, 4.94376122657450226002634205632, 5.95967651264422152670977788910, 5.96956680213085492365415051586, 6.62762171992257665130856519803, 7.50641789037179894281138679970, 7.57323216500839527256914149425, 8.367508603382134314146293072792, 8.584868551115419667042947592500, 9.361746587069330588975575945669, 9.629005176149752438633105274587

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