↑

Properties

Label 4-6032-1.1-c1e2-0-0
Degree $4$
Conductor $6032$
Sign $1$
Analytic cond. $0.384605$
Root an. cond. $0.787506$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 3·5-s − 2·9-s + 6·13-s + 4·16-s − 3·17-s − 6·20-s − 25-s + 2·29-s + 4·36-s − 14·37-s − 12·41-s − 6·45-s − 4·49-s − 12·52-s + 18·53-s + 7·61-s − 8·64-s + 18·65-s + 6·68-s − 14·73-s + 12·80-s − 5·81-s − 9·85-s + 3·89-s + 7·97-s + 2·100-s + ⋯
L(s)  = 1  − 4-s + 1.34·5-s − 2/3·9-s + 1.66·13-s + 16-s − 0.727·17-s − 1.34·20-s − 1/5·25-s + 0.371·29-s + 2/3·36-s − 2.30·37-s − 1.87·41-s − 0.894·45-s − 4/7·49-s − 1.66·52-s + 2.47·53-s + 0.896·61-s − 64-s + 2.23·65-s + 0.727·68-s − 1.63·73-s + 1.34·80-s − 5/9·81-s − 0.976·85-s + 0.317·89-s + 0.710·97-s + 1/5·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6032\)    =    \(2^{4} \cdot 13 \cdot 29\)
Sign: $1$
Analytic conductor: \(0.384605\)
Root analytic conductor: \(0.787506\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6032,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8638295230\)
\(L(\frac12)\) \(\approx\) \(0.8638295230\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 5 T + p T^{2} ) \)
29$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 3 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.5.ad_k
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.11.a_af
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.19.a_z
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.23.a_aba
31$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.31.a_abj
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.37.o_et
41$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.41.m_ef
43$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.43.a_e
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.47.a_ao
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.53.as_gw
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.59.a_bu
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.ah_ek
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.a_ack
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.o_ek
79$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.79.a_abg
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.83.a_cg
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.89.ad_ge
97$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.ah_y
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

−11.92452692408347322735269236987, −11.61273287990120787720776932942, −10.62421186740291358053509556515, −10.31641110926210258202069372428, −9.752412479191050908402096379946, −8.903582143199867768728542514666, −8.710666107935939056437956579395, −8.224123461231545256743607430396, −7.02160059752068185550241650652, −6.31781713162194054768167062204, −5.63567096495448927563133329445, −5.25059973700108178017427319326, −4.11356927924001896534175820796, −3.28232247297491903361532119637, −1.80243843586560891979518056842, 1.80243843586560891979518056842, 3.28232247297491903361532119637, 4.11356927924001896534175820796, 5.25059973700108178017427319326, 5.63567096495448927563133329445, 6.31781713162194054768167062204, 7.02160059752068185550241650652, 8.224123461231545256743607430396, 8.710666107935939056437956579395, 8.903582143199867768728542514666, 9.752412479191050908402096379946, 10.31641110926210258202069372428, 10.62421186740291358053509556515, 11.61273287990120787720776932942, 11.92452692408347322735269236987

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