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Properties

Label 4-693e2-1.1-c1e2-0-23
Degree $4$
Conductor $480249$
Sign $-1$
Analytic cond. $30.6210$
Root an. cond. $2.35236$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 13-s − 3·16-s − 7·19-s + 4·25-s + 5·31-s − 4·37-s − 3·43-s − 7·49-s − 52-s + 4·61-s + 7·64-s + 5·67-s + 23·73-s + 7·76-s + 3·79-s + 12·97-s − 4·100-s − 5·103-s − 9·109-s − 121-s − 5·124-s + 127-s + 131-s + 137-s + 139-s + 4·148-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.277·13-s − 3/4·16-s − 1.60·19-s + 4/5·25-s + 0.898·31-s − 0.657·37-s − 0.457·43-s − 49-s − 0.138·52-s + 0.512·61-s + 7/8·64-s + 0.610·67-s + 2.69·73-s + 0.802·76-s + 0.337·79-s + 1.21·97-s − 2/5·100-s − 0.492·103-s − 0.862·109-s − 0.0909·121-s − 0.449·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.328·148-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 480249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480249 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(480249\)    =    \(3^{4} \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(30.6210\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 480249,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
7$C_2$ \( 1 + p T^{2} \)
11$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
5$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.5.a_ae
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.ab_y
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.17.a_q
19$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.h_bw
23$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.23.a_az
29$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.29.a_ao
31$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.af_ba
37$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.37.e_cr
41$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \) 2.41.a_y
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.43.d_ac
47$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.47.a_abm
53$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.53.a_ade
59$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.59.a_ck
61$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.61.ae_es
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.67.af_eg
71$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \) 2.71.a_aj
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) 2.73.ax_km
79$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.ad_fa
83$C_2^2$ \( 1 + 87 T^{2} + p^{2} T^{4} \) 2.83.a_dj
89$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.89.a_o
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.am_gk
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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

−8.384949703856756874942379834376, −8.037829615881562767659111341635, −7.41589294498630329879953960107, −6.78795216655047013063833103140, −6.46474749746792263003418565534, −6.21832963282134520906170794995, −5.34477991983511437267937915109, −4.88046846902455276202328240149, −4.63501604263519821658798531085, −3.81862908254803253399694091305, −3.59788033285030893128170448014, −2.57988176607845137820285931875, −2.18859021475581575920629672924, −1.17420759746258049161059586082, 0, 1.17420759746258049161059586082, 2.18859021475581575920629672924, 2.57988176607845137820285931875, 3.59788033285030893128170448014, 3.81862908254803253399694091305, 4.63501604263519821658798531085, 4.88046846902455276202328240149, 5.34477991983511437267937915109, 6.21832963282134520906170794995, 6.46474749746792263003418565534, 6.78795216655047013063833103140, 7.41589294498630329879953960107, 8.037829615881562767659111341635, 8.384949703856756874942379834376

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