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Properties

Label 4-416000-1.1-c1e2-0-14
Degree $4$
Conductor $416000$
Sign $-1$
Analytic cond. $26.5245$
Root an. cond. $2.26940$
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 + 3·13-s − 8·17-s + 25-s + 12·29-s + 8·37-s − 12·41-s + 2·45-s − 2·49-s − 8·53-s + 4·61-s − 3·65-s − 24·73-s − 5·81-s + 8·85-s − 12·89-s + 16·97-s − 4·101-s + 4·109-s + 8·113-s − 6·117-s + 10·121-s − 125-s + ⋯
L(s)  = 1  − 0.447·5-s − 2/3·9-s + 0.832·13-s − 1.94·17-s + 1/5·25-s + 2.22·29-s + 1.31·37-s − 1.87·41-s + 0.298·45-s − 2/7·49-s − 1.09·53-s + 0.512·61-s − 0.372·65-s − 2.80·73-s − 5/9·81-s + 0.867·85-s − 1.27·89-s + 1.62·97-s − 0.398·101-s + 0.383·109-s + 0.752·113-s − 0.554·117-s + 0.909·121-s − 0.0894·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(416000\)    =    \(2^{8} \cdot 5^{3} \cdot 13\)
Sign: $-1$
Analytic conductor: \(26.5245\)
Root analytic conductor: \(2.26940\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 416000,\ (\ :1/2, 1/2),\ -1)\)

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$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.i_bu
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.23.a_ba
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
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$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.m_dy
43$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.43.a_ack
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.47.a_ao
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.i_di
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.67.a_bq
71$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.71.a_adu
73$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.y_la
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.79.a_be
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.83.a_cg
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.aq_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

−8.511183384398520400634152013055, −8.147906605345422515796396397244, −7.48287837553522276378431275075, −6.90068953913652748353761438800, −6.57530553589108334800293637845, −6.12492545534768472375921421815, −5.70377839711143215857774659281, −4.77712328711877504943139128512, −4.64236200797655702798549643471, −4.07630043473536051787453734246, −3.28252230827184677635122836462, −2.87454005358621247880217941455, −2.16315325629092576959096640650, −1.21608772357707047001025040269, 0, 1.21608772357707047001025040269, 2.16315325629092576959096640650, 2.87454005358621247880217941455, 3.28252230827184677635122836462, 4.07630043473536051787453734246, 4.64236200797655702798549643471, 4.77712328711877504943139128512, 5.70377839711143215857774659281, 6.12492545534768472375921421815, 6.57530553589108334800293637845, 6.90068953913652748353761438800, 7.48287837553522276378431275075, 8.147906605345422515796396397244, 8.511183384398520400634152013055

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