Properties

Label 4-432e2-1.1-c1e2-0-2
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·13-s + 10·25-s + 22·37-s − 13·49-s − 2·61-s + 14·73-s + 38·97-s − 4·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 49·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 2.77·13-s + 2·25-s + 3.61·37-s − 1.85·49-s − 0.256·61-s + 1.63·73-s + 3.85·97-s − 0.383·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-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 + 3.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-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: induced by $\chi_{432} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 186624,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.351438044\)
\(L(\frac12)\) \(\approx\) \(1.351438044\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )^{2} \)
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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

−11.58386422465684181661120154846, −10.92514051473156948034913745098, −10.40092110870909484542111328135, −10.01499262651823060439340749546, −9.460037139390836583475419601217, −9.377707646825199205050485379873, −8.747607751506310630572770261150, −7.86832013851592462962153361050, −7.83973002046380619543041672981, −7.32032889076460286744362461331, −6.60817338763161248111332032538, −6.44016159785880704463838000079, −5.58479572145862521463143700297, −4.84219169406570477186014901126, −4.83791216107130437830044645588, −4.17075201709690677704368048402, −3.10183920413156191538396756360, −2.70999451656945664170355507599, −2.08031965102448241621398582936, −0.74084876375293433558861635527, 0.74084876375293433558861635527, 2.08031965102448241621398582936, 2.70999451656945664170355507599, 3.10183920413156191538396756360, 4.17075201709690677704368048402, 4.83791216107130437830044645588, 4.84219169406570477186014901126, 5.58479572145862521463143700297, 6.44016159785880704463838000079, 6.60817338763161248111332032538, 7.32032889076460286744362461331, 7.83973002046380619543041672981, 7.86832013851592462962153361050, 8.747607751506310630572770261150, 9.377707646825199205050485379873, 9.460037139390836583475419601217, 10.01499262651823060439340749546, 10.40092110870909484542111328135, 10.92514051473156948034913745098, 11.58386422465684181661120154846

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