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Properties

Label 4-842400-1.1-c1e2-0-9
Degree $4$
Conductor $842400$
Sign $-1$
Analytic cond. $53.7121$
Root an. cond. $2.70718$
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 + 3·13-s + 16-s + 25-s − 3·26-s − 32-s − 20·37-s + 12·41-s + 2·49-s − 50-s + 3·52-s − 12·53-s − 8·61-s + 64-s + 4·73-s + 20·74-s − 12·82-s + 4·97-s − 2·98-s + 100-s − 12·101-s − 3·104-s + 12·106-s + 4·109-s − 12·113-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.832·13-s + 1/4·16-s + 1/5·25-s − 0.588·26-s − 0.176·32-s − 3.28·37-s + 1.87·41-s + 2/7·49-s − 0.141·50-s + 0.416·52-s − 1.64·53-s − 1.02·61-s + 1/8·64-s + 0.468·73-s + 2.32·74-s − 1.32·82-s + 0.406·97-s − 0.202·98-s + 1/10·100-s − 1.19·101-s − 0.294·104-s + 1.16·106-s + 0.383·109-s − 1.12·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 842400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 842400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(842400\)    =    \(2^{5} \cdot 3^{4} \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(53.7121\)
Root analytic conductor: \(2.70718\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 842400,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
3 \( 1 \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.19.a_ao
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.43.a_bi
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.59.a_ack
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.i_dy
67$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.67.a_w
71$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.71.a_aba
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.83.a_aby
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.ae_cc
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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

−7.947702940252845088008567507832, −7.67666829490131639668218617832, −7.19833325535718218888044380889, −6.53102182473319516627255084362, −6.49943734486256449436779480570, −5.77956654292058324291746586226, −5.35410599581172484373573795407, −4.86597514948924627077433513568, −4.17117332354804603959532599295, −3.64938529514480753448340445304, −3.16120470743429024794389697316, −2.51243434711616276579048255334, −1.74544461582689143755568671235, −1.18966713341167874617601412059, 0, 1.18966713341167874617601412059, 1.74544461582689143755568671235, 2.51243434711616276579048255334, 3.16120470743429024794389697316, 3.64938529514480753448340445304, 4.17117332354804603959532599295, 4.86597514948924627077433513568, 5.35410599581172484373573795407, 5.77956654292058324291746586226, 6.49943734486256449436779480570, 6.53102182473319516627255084362, 7.19833325535718218888044380889, 7.67666829490131639668218617832, 7.947702940252845088008567507832

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