Properties

Label 4-83232-1.1-c1e2-0-20
Degree $4$
Conductor $83232$
Sign $-1$
Analytic cond. $5.30694$
Root an. cond. $1.51778$
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 − 9-s − 4·13-s + 16-s + 2·17-s + 18-s − 6·25-s + 4·26-s − 8·29-s − 32-s − 2·34-s − 36-s − 4·41-s − 2·49-s + 6·50-s − 4·52-s − 12·53-s + 8·58-s + 64-s + 2·68-s + 72-s + 12·73-s + 81-s + 4·82-s − 20·89-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 − 6/5·25-s + 0.784·26-s − 1.48·29-s − 0.176·32-s − 0.342·34-s − 1/6·36-s − 0.624·41-s − 2/7·49-s + 0.848·50-s − 0.554·52-s − 1.64·53-s + 1.05·58-s + 1/8·64-s + 0.242·68-s + 0.117·72-s + 1.40·73-s + 1/9·81-s + 0.441·82-s − 2.11·89-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: yes
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$C_2$ \( 1 + T^{2} \)
17$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−9.477941905821665691626084112053, −9.151639055783464951459222603253, −8.370019906904940061920602443875, −7.930676237285212094472724061590, −7.62354597396408000948809901579, −6.99724737605352586376847722358, −6.51812678651441251860092785054, −5.76355851904107998660418967961, −5.40685100114979573546260066433, −4.69754971390797089185452424646, −3.87352021530535573359626181100, −3.21089717173670688697888300617, −2.39306829178122446919027837033, −1.62723977333978330803040771283, 0, 1.62723977333978330803040771283, 2.39306829178122446919027837033, 3.21089717173670688697888300617, 3.87352021530535573359626181100, 4.69754971390797089185452424646, 5.40685100114979573546260066433, 5.76355851904107998660418967961, 6.51812678651441251860092785054, 6.99724737605352586376847722358, 7.62354597396408000948809901579, 7.930676237285212094472724061590, 8.370019906904940061920602443875, 9.151639055783464951459222603253, 9.477941905821665691626084112053

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