↑

Properties

Label 4-199692-1.1-c1e2-0-9
Degree $4$
Conductor $199692$
Sign $-1$
Analytic cond. $12.7325$
Root an. cond. $1.88898$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4-s + 9-s − 12-s − 6·13-s + 16-s − 4·19-s − 3·25-s + 27-s + 12·31-s − 36-s − 4·37-s − 6·39-s + 48-s − 13·49-s + 6·52-s − 4·57-s − 12·61-s − 64-s − 8·67-s − 12·73-s − 3·75-s + 4·76-s + 12·79-s + 81-s + 12·93-s − 18·97-s + ⋯
L(s)  = 1  + 0.577·3-s − 1/2·4-s + 1/3·9-s − 0.288·12-s − 1.66·13-s + 1/4·16-s − 0.917·19-s − 3/5·25-s + 0.192·27-s + 2.15·31-s − 1/6·36-s − 0.657·37-s − 0.960·39-s + 0.144·48-s − 1.85·49-s + 0.832·52-s − 0.529·57-s − 1.53·61-s − 1/8·64-s − 0.977·67-s − 1.40·73-s − 0.346·75-s + 0.458·76-s + 1.35·79-s + 1/9·81-s + 1.24·93-s − 1.82·97-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: yes
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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( 1 - T \)
43$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.5.a_d
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
11$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \) 2.11.a_ap
13$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.g_bf
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.e_bh
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \) 2.29.a_abt
31$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.31.am_de
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.e_bq
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
47$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.47.a_bt
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.a_g
59$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.59.a_g
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.61.m_gc
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.i_g
71$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.71.a_abi
73$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.73.m_gw
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.79.am_hi
83$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \) 2.83.a_ap
89$C_2^2$ \( 1 - 126 T^{2} + p^{2} T^{4} \) 2.89.a_aew
97$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.97.s_jz
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.794087344176606936923568500805, −8.399460047613123746517001623112, −7.966460710590193427933358013276, −7.51556821901647152371195088821, −7.05211631588547902253803126544, −6.37824176682823895122682721276, −6.05181136065730957462811593875, −5.17141384954938841752023439268, −4.67362741269112481658010539118, −4.44565762000985887619730823287, −3.63298581292141333806061808659, −2.91820165733202468640904882583, −2.40818465163262239309442830985, −1.52043177969289184111367795624, 0, 1.52043177969289184111367795624, 2.40818465163262239309442830985, 2.91820165733202468640904882583, 3.63298581292141333806061808659, 4.44565762000985887619730823287, 4.67362741269112481658010539118, 5.17141384954938841752023439268, 6.05181136065730957462811593875, 6.37824176682823895122682721276, 7.05211631588547902253803126544, 7.51556821901647152371195088821, 7.966460710590193427933358013276, 8.399460047613123746517001623112, 8.794087344176606936923568500805

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