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Properties

Label 4-84672-1.1-c1e2-0-31
Degree $4$
Conductor $84672$
Sign $-1$
Analytic cond. $5.39876$
Root an. cond. $1.52431$
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  + 3-s − 4·7-s + 9-s − 8·17-s − 4·21-s − 6·25-s + 27-s − 4·37-s − 8·41-s + 9·49-s − 8·51-s − 8·59-s − 4·63-s + 16·67-s − 6·75-s − 8·79-s + 81-s + 8·83-s − 24·89-s − 16·101-s + 12·109-s − 4·111-s + 32·119-s + 2·121-s − 8·123-s + 127-s + 131-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.94·17-s − 0.872·21-s − 6/5·25-s + 0.192·27-s − 0.657·37-s − 1.24·41-s + 9/7·49-s − 1.12·51-s − 1.04·59-s − 0.503·63-s + 1.95·67-s − 0.692·75-s − 0.900·79-s + 1/9·81-s + 0.878·83-s − 2.54·89-s − 1.59·101-s + 1.14·109-s − 0.379·111-s + 2.93·119-s + 2/11·121-s − 0.721·123-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(84672\)    =    \(2^{6} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(5.39876\)
Root analytic conductor: \(1.52431\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 84672,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$ \( 1 - T \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.i_bu
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.23.a_g
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.31.a_abi
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.i_dq
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.53.a_ak
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.61.a_ak
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.67.aq_ha
71$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.71.a_cs
73$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.73.a_ck
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.i_be
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.ai_eo
89$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.y_mg
97$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.97.a_da
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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

−9.599876921050449715693012105168, −8.962827877128938733443338280943, −8.527346408624314011571057188420, −8.090514919080159649094443071908, −7.29780808550045439764648383061, −6.84455019659191407667988543997, −6.52609009332375884121270381052, −5.93365554456607431560037312657, −5.24720386863760161640819538331, −4.43439341990892411245348529652, −3.90361914414778626691142442272, −3.29848666061953735636387585401, −2.59552920704093375581584943398, −1.84998880734961742604860171416, 0, 1.84998880734961742604860171416, 2.59552920704093375581584943398, 3.29848666061953735636387585401, 3.90361914414778626691142442272, 4.43439341990892411245348529652, 5.24720386863760161640819538331, 5.93365554456607431560037312657, 6.52609009332375884121270381052, 6.84455019659191407667988543997, 7.29780808550045439764648383061, 8.090514919080159649094443071908, 8.527346408624314011571057188420, 8.962827877128938733443338280943, 9.599876921050449715693012105168

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