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Properties

Label 4-405e2-1.1-c1e2-0-11
Degree $4$
Conductor $164025$
Sign $-1$
Analytic cond. $10.4583$
Root an. cond. $1.79831$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·13-s − 4·16-s − 10·19-s + 25-s − 18·31-s − 20·37-s − 4·43-s − 14·49-s − 4·61-s + 12·67-s − 16·73-s + 24·79-s + 28·97-s + 4·103-s − 2·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + ⋯
L(s)  = 1  + 2.21·13-s − 16-s − 2.29·19-s + 1/5·25-s − 3.23·31-s − 3.28·37-s − 0.609·43-s − 2·49-s − 0.512·61-s + 1.46·67-s − 1.87·73-s + 2.70·79-s + 2.84·97-s + 0.394·103-s − 0.191·109-s + 3/11·121-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 + 1.69·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(164025\)    =    \(3^{8} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(10.4583\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 164025,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad3 \( 1 \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.11.a_ad
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.19.k_cl
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.29.a_bh
31$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.31.s_fn
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.41.a_bh
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.43.e_dm
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.a_dm
53$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.53.a_bq
59$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.59.a_en
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.67.am_go
71$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.71.a_fl
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.73.q_ic
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.79.ay_lq
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.89.a_dt
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.97.abc_pa
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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

−8.824991276786141939896068673100, −8.740838314696191226965539549612, −8.203253537981750073119106447227, −7.50006448856980317673435327311, −6.94257636940465251066789252595, −6.35526976205125359438346080603, −6.31354273414722554277544867988, −5.41659698665533312416876873683, −4.99212686725716379469941020875, −4.23466268327432431147155666300, −3.50061572991346427363382995646, −3.49410387761142976656546809271, −1.89799498723585767421033230965, −1.84657171814318698813806363337, 0, 1.84657171814318698813806363337, 1.89799498723585767421033230965, 3.49410387761142976656546809271, 3.50061572991346427363382995646, 4.23466268327432431147155666300, 4.99212686725716379469941020875, 5.41659698665533312416876873683, 6.31354273414722554277544867988, 6.35526976205125359438346080603, 6.94257636940465251066789252595, 7.50006448856980317673435327311, 8.203253537981750073119106447227, 8.740838314696191226965539549612, 8.824991276786141939896068673100

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