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Properties

Label 4-199692-1.1-c1e2-0-0
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s − 2·7-s + 9-s − 12-s + 2·13-s + 16-s + 2·19-s + 2·21-s − 25-s − 27-s − 2·28-s + 4·31-s + 36-s + 4·37-s − 2·39-s − 2·43-s − 48-s − 11·49-s + 2·52-s − 2·57-s − 2·63-s + 64-s + 4·67-s + 8·73-s + 75-s + 2·76-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s − 0.755·7-s + 1/3·9-s − 0.288·12-s + 0.554·13-s + 1/4·16-s + 0.458·19-s + 0.436·21-s − 1/5·25-s − 0.192·27-s − 0.377·28-s + 0.718·31-s + 1/6·36-s + 0.657·37-s − 0.320·39-s − 0.304·43-s − 0.144·48-s − 1.57·49-s + 0.277·52-s − 0.264·57-s − 0.251·63-s + 1/8·64-s + 0.488·67-s + 0.936·73-s + 0.115·75-s + 0.229·76-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: \(0\)
Selberg data: \((4,\ 199692,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.409979710\)
\(L(\frac12)\) \(\approx\) \(1.409979710\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.7.c_p
11$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.11.a_v
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.13.ac_bb
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.19.ac_bn
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.a_ax
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
47$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.47.a_abb
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.a_dm
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.73.ai_gg
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.79.abc_nq
83$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.83.a_gb
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \) 2.97.abi_sp
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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

−9.288371615522552197687168792212, −8.611793261135204485682877336763, −7.898097193807439752541133446541, −7.82875648118174425172353120079, −6.99859709293611633937351736353, −6.57965006759949692624968813085, −6.24159684860106097326095404180, −5.83597222229059038041503973729, −5.05944331396204869879222544336, −4.73770457943369648572673951323, −3.77373272120592059566750657059, −3.44011472926701267604846728864, −2.66866902051497631952347453875, −1.84248871297006401888237448812, −0.794520863410663961905847445914, 0.794520863410663961905847445914, 1.84248871297006401888237448812, 2.66866902051497631952347453875, 3.44011472926701267604846728864, 3.77373272120592059566750657059, 4.73770457943369648572673951323, 5.05944331396204869879222544336, 5.83597222229059038041503973729, 6.24159684860106097326095404180, 6.57965006759949692624968813085, 6.99859709293611633937351736353, 7.82875648118174425172353120079, 7.898097193807439752541133446541, 8.611793261135204485682877336763, 9.288371615522552197687168792212

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