Properties

Label 4-76832-1.1-c1e2-0-3
Degree $4$
Conductor $76832$
Sign $-1$
Analytic cond. $4.89887$
Root an. cond. $1.48772$
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 − 8-s − 2·9-s + 8·13-s + 16-s − 12·17-s + 2·18-s − 10·25-s − 8·26-s − 12·29-s − 32-s + 12·34-s − 2·36-s + 4·37-s − 12·41-s + 10·50-s + 8·52-s + 12·53-s + 12·58-s − 16·61-s + 64-s − 12·68-s + 2·72-s − 4·73-s − 4·74-s − 5·81-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2/3·9-s + 2.21·13-s + 1/4·16-s − 2.91·17-s + 0.471·18-s − 2·25-s − 1.56·26-s − 2.22·29-s − 0.176·32-s + 2.05·34-s − 1/3·36-s + 0.657·37-s − 1.87·41-s + 1.41·50-s + 1.10·52-s + 1.64·53-s + 1.57·58-s − 2.04·61-s + 1/8·64-s − 1.45·68-s + 0.235·72-s − 0.468·73-s − 0.464·74-s − 5/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76832 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76832 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(76832\)    =    \(2^{5} \cdot 7^{4}\)
Sign: $-1$
Analytic conductor: \(4.89887\)
Root analytic conductor: \(1.48772\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{76832} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 76832,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
7 \( 1 \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−9.205467104844532229657218112988, −9.044637307272642732451675861332, −8.624667079434487291694904422005, −8.199638847074308569053560326614, −7.55563465371540871894331307120, −7.01851678171449461328017406897, −6.24500065333065006116855945568, −6.12986038892916091786195943044, −5.53331414913354333592453982899, −4.55372588498345584946220609716, −3.87985229263682105648190826671, −3.42554077807776045337947490220, −2.25140513775369021989127014661, −1.73289286183865826039844070287, 0, 1.73289286183865826039844070287, 2.25140513775369021989127014661, 3.42554077807776045337947490220, 3.87985229263682105648190826671, 4.55372588498345584946220609716, 5.53331414913354333592453982899, 6.12986038892916091786195943044, 6.24500065333065006116855945568, 7.01851678171449461328017406897, 7.55563465371540871894331307120, 8.199638847074308569053560326614, 8.624667079434487291694904422005, 9.044637307272642732451675861332, 9.205467104844532229657218112988

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