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Properties

Label 4-259308-1.1-c1e2-0-6
Degree $4$
Conductor $259308$
Sign $1$
Analytic cond. $16.5337$
Root an. cond. $2.01647$
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 + 9-s − 12-s + 8·13-s + 16-s + 8·19-s − 25-s − 27-s + 2·31-s + 36-s + 16·37-s − 8·39-s − 20·43-s − 48-s + 8·52-s − 8·57-s + 20·61-s + 64-s − 20·67-s − 4·73-s + 75-s + 8·76-s − 2·79-s + 81-s − 2·93-s + 2·97-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s + 1/3·9-s − 0.288·12-s + 2.21·13-s + 1/4·16-s + 1.83·19-s − 1/5·25-s − 0.192·27-s + 0.359·31-s + 1/6·36-s + 2.63·37-s − 1.28·39-s − 3.04·43-s − 0.144·48-s + 1.10·52-s − 1.05·57-s + 2.56·61-s + 1/8·64-s − 2.44·67-s − 0.468·73-s + 0.115·75-s + 0.917·76-s − 0.225·79-s + 1/9·81-s − 0.207·93-s + 0.203·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 259308 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259308 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(259308\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(16.5337\)
Root analytic conductor: \(2.01647\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 259308,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.024435147\)
\(L(\frac12)\) \(\approx\) \(2.024435147\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
7 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.a_n
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.a_ax
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.31.ac_cl
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.37.aq_fi
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.43.u_he
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.53.a_dt
59$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.59.a_ef
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.67.u_ja
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.79.c_gd
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.83.a_dh
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.97.ac_hn
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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

−8.804654554318459390164507311566, −8.437574389959171996814247300132, −8.000167698268580301677238135628, −7.45538149578601412402973602617, −7.00728426330403345958805784350, −6.39409357283054887478798556947, −6.07864651997567107362244436260, −5.68536165312647493892897914533, −5.07800539018688221972389717941, −4.49123641406362856214796748304, −3.71959442230100202249236284366, −3.36779260852424710508563371535, −2.63841008431534209471451574051, −1.53413432618226811661848805906, −1.01560111630863456380812991130, 1.01560111630863456380812991130, 1.53413432618226811661848805906, 2.63841008431534209471451574051, 3.36779260852424710508563371535, 3.71959442230100202249236284366, 4.49123641406362856214796748304, 5.07800539018688221972389717941, 5.68536165312647493892897914533, 6.07864651997567107362244436260, 6.39409357283054887478798556947, 7.00728426330403345958805784350, 7.45538149578601412402973602617, 8.000167698268580301677238135628, 8.437574389959171996814247300132, 8.804654554318459390164507311566

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