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Properties

Label 4-832000-1.1-c1e2-0-8
Degree $4$
Conductor $832000$
Sign $-1$
Analytic cond. $53.0490$
Root an. cond. $2.69879$
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  − 5-s − 2·9-s − 5·13-s + 25-s − 4·29-s + 8·37-s + 4·41-s + 2·45-s + 6·49-s + 16·53-s − 4·61-s + 5·65-s − 8·73-s − 5·81-s + 4·89-s − 16·97-s − 4·101-s + 4·109-s + 24·113-s + 10·117-s − 22·121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + ⋯
L(s)  = 1  − 0.447·5-s − 2/3·9-s − 1.38·13-s + 1/5·25-s − 0.742·29-s + 1.31·37-s + 0.624·41-s + 0.298·45-s + 6/7·49-s + 2.19·53-s − 0.512·61-s + 0.620·65-s − 0.936·73-s − 5/9·81-s + 0.423·89-s − 1.62·97-s − 0.398·101-s + 0.383·109-s + 2.25·113-s + 0.924·117-s − 2·121-s − 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.332·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(832000\)    =    \(2^{9} \cdot 5^{3} \cdot 13\)
Sign: $-1$
Analytic conductor: \(53.0490\)
Root analytic conductor: \(2.69879\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 832000,\ (\ :1/2, 1/2),\ -1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
5$C_1$ \( 1 + T \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 6 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.31.a_aby
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.ai_di
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ae_cs
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.43.a_by
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.53.aq_gk
59$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.59.a_acw
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.e_eg
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.67.a_acs
71$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.71.a_abi
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.i_gc
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.a_adu
83$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.83.a_ba
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ae_eo
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.q_io
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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

−7.975399996317021761637177682752, −7.53765996729605541234671425132, −7.16800477830639816267957712402, −6.85189703976895145026688091300, −6.12054051706452994069531459906, −5.63743168093476106014695569988, −5.44221623781686212238876016268, −4.63747710335269148615823781490, −4.36855800878218907812451552614, −3.78472189027544575246841050514, −3.11356961994890955574924273646, −2.55912744475119478531056657642, −2.15021984628723773006933737670, −1.00628232339531988541145195448, 0, 1.00628232339531988541145195448, 2.15021984628723773006933737670, 2.55912744475119478531056657642, 3.11356961994890955574924273646, 3.78472189027544575246841050514, 4.36855800878218907812451552614, 4.63747710335269148615823781490, 5.44221623781686212238876016268, 5.63743168093476106014695569988, 6.12054051706452994069531459906, 6.85189703976895145026688091300, 7.16800477830639816267957712402, 7.53765996729605541234671425132, 7.975399996317021761637177682752

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