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Properties

Label 4-199692-1.1-c1e2-0-10
Degree $4$
Conductor $199692$
Sign $-1$
Analytic cond. $12.7325$
Root an. cond. $1.88898$
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  + 3-s + 4-s − 4·7-s + 9-s + 12-s + 4·13-s + 16-s − 8·19-s − 4·21-s − 6·25-s + 27-s − 4·28-s − 8·31-s + 36-s − 16·37-s + 4·39-s + 2·43-s + 48-s − 2·49-s + 4·52-s − 8·57-s − 16·61-s − 4·63-s + 64-s + 8·67-s + 20·73-s − 6·75-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s − 1.51·7-s + 1/3·9-s + 0.288·12-s + 1.10·13-s + 1/4·16-s − 1.83·19-s − 0.872·21-s − 6/5·25-s + 0.192·27-s − 0.755·28-s − 1.43·31-s + 1/6·36-s − 2.63·37-s + 0.640·39-s + 0.304·43-s + 0.144·48-s − 2/7·49-s + 0.554·52-s − 1.05·57-s − 2.04·61-s − 0.503·63-s + 1/8·64-s + 0.977·67-s + 2.34·73-s − 0.692·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199692 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199692 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(199692\)    =    \(2^{2} \cdot 3^{3} \cdot 43^{2}\)
Sign: $-1$
Analytic conductor: \(12.7325\)
Root analytic conductor: \(1.88898\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 199692,\ (\ :1/2, 1/2),\ -1)\)

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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 - T \)
43$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.23.a_bq
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.37.q_fi
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.a_dm
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.a_abm
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.83.a_dy
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
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

−9.020476255077092786616671816203, −8.290506818723521192685804080324, −8.101467935576574417653248009955, −7.32096231625382667213308034502, −6.78357109229019949580867845825, −6.60179582841051422383113975893, −5.94294360088337250572036157519, −5.66859888652078003621682617285, −4.72770536442364898085524768277, −3.99939689788655104010649530295, −3.45261714591514304546140653309, −3.26495541761552880908110302411, −2.18519746266503578273129656850, −1.72854740657408103556538678005, 0, 1.72854740657408103556538678005, 2.18519746266503578273129656850, 3.26495541761552880908110302411, 3.45261714591514304546140653309, 3.99939689788655104010649530295, 4.72770536442364898085524768277, 5.66859888652078003621682617285, 5.94294360088337250572036157519, 6.60179582841051422383113975893, 6.78357109229019949580867845825, 7.32096231625382667213308034502, 8.101467935576574417653248009955, 8.290506818723521192685804080324, 9.020476255077092786616671816203

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