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Properties

Label 4-432e2-1.1-c1e2-0-33
Degree $4$
Conductor $186624$
Sign $1$
Analytic cond. $11.8993$
Root an. cond. $1.85729$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Related objects

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·7-s − 14·13-s + 2·19-s − 10·25-s + 8·31-s − 2·37-s − 16·43-s + 61·49-s − 26·61-s − 22·67-s + 34·73-s + 26·79-s + 140·91-s + 10·97-s + 14·103-s + 4·109-s − 22·121-s + 127-s + 131-s − 20·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 3.77·7-s − 3.88·13-s + 0.458·19-s − 2·25-s + 1.43·31-s − 0.328·37-s − 2.43·43-s + 61/7·49-s − 3.32·61-s − 2.68·67-s + 3.97·73-s + 2.92·79-s + 14.6·91-s + 1.01·97-s + 1.37·103-s + 0.383·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s − 1.73·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(186624\)    =    \(2^{8} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(11.8993\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 186624,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.7.k_bn
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.13.o_cx
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.19.ac_bn
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.37.c_cx
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.61.ba_lf
67$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.67.w_jv
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \) 2.73.abi_qt
79$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.79.aba_mp
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.97.ak_il
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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

−13.96193815188290577662344738835, −13.23223380215344977355129385908, −13.23223380215344977355129385908, −12.36374263312951862609136713881, −12.36374263312951862609136713881, −11.89058242352606683485837364744, −11.89058242352606683485837364744, −10.40150154645553667557841727686, −10.40150154645553667557841727686, −9.772574318258794189724223366751, −9.772574318258794189724223366751, −9.220031324604069206185525205678, −9.220031324604069206185525205678, −7.78745335695197737171573573692, −7.78745335695197737171573573692, −6.91823062220437394787576967524, −6.91823062220437394787576967524, −6.08584189960381722201120484886, −6.08584189960381722201120484886, −4.87832652992641646209905365152, −4.87832652992641646209905365152, −3.51757494847066237517697434263, −3.51757494847066237517697434263, −2.49929826070171115553006367440, −2.49929826070171115553006367440, 0, 0, 2.49929826070171115553006367440, 2.49929826070171115553006367440, 3.51757494847066237517697434263, 3.51757494847066237517697434263, 4.87832652992641646209905365152, 4.87832652992641646209905365152, 6.08584189960381722201120484886, 6.08584189960381722201120484886, 6.91823062220437394787576967524, 6.91823062220437394787576967524, 7.78745335695197737171573573692, 7.78745335695197737171573573692, 9.220031324604069206185525205678, 9.220031324604069206185525205678, 9.772574318258794189724223366751, 9.772574318258794189724223366751, 10.40150154645553667557841727686, 10.40150154645553667557841727686, 11.89058242352606683485837364744, 11.89058242352606683485837364744, 12.36374263312951862609136713881, 12.36374263312951862609136713881, 13.23223380215344977355129385908, 13.23223380215344977355129385908, 13.96193815188290577662344738835

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