Properties

Label 4-483e2-1.1-c1e2-0-8
Degree $4$
Conductor $233289$
Sign $1$
Analytic cond. $14.8747$
Root an. cond. $1.96386$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 2·3-s + 4·4-s − 5·5-s − 6·6-s + 2·7-s − 3·8-s + 3·9-s + 15·10-s − 2·11-s + 8·12-s − 7·13-s − 6·14-s − 10·15-s + 3·16-s − 2·17-s − 9·18-s − 6·19-s − 20·20-s + 4·21-s + 6·22-s − 2·23-s − 6·24-s + 10·25-s + 21·26-s + 4·27-s + 8·28-s + ⋯
L(s)  = 1  − 2.12·2-s + 1.15·3-s + 2·4-s − 2.23·5-s − 2.44·6-s + 0.755·7-s − 1.06·8-s + 9-s + 4.74·10-s − 0.603·11-s + 2.30·12-s − 1.94·13-s − 1.60·14-s − 2.58·15-s + 3/4·16-s − 0.485·17-s − 2.12·18-s − 1.37·19-s − 4.47·20-s + 0.872·21-s + 1.27·22-s − 0.417·23-s − 1.22·24-s + 2·25-s + 4.11·26-s + 0.769·27-s + 1.51·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 233289 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 233289 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(233289\)    =    \(3^{2} \cdot 7^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(14.8747\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 233289,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.2.d_f
5$C_4$ \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.5.f_p
11$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_d
13$D_{4}$ \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.13.h_bl
17$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.17.c_bj
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.19.g_bv
29$D_{4}$ \( 1 + 6 T + 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_bv
31$D_{4}$ \( 1 + 12 T + 3 p T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.31.m_dp
37$D_{4}$ \( 1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_cl
41$D_{4}$ \( 1 - 4 T + p T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.41.ae_bp
43$D_{4}$ \( 1 - 5 T + 61 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.43.af_cj
47$D_{4}$ \( 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.47.k_cw
53$D_{4}$ \( 1 + 9 T + 125 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.53.j_ev
59$D_{4}$ \( 1 + 19 T + 197 T^{2} + 19 p T^{3} + p^{2} T^{4} \) 2.59.t_hp
61$D_{4}$ \( 1 - 3 T - 27 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.61.ad_abb
67$D_{4}$ \( 1 - 15 T + 179 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.67.ap_gx
71$C_4$ \( 1 + 19 T + 201 T^{2} + 19 p T^{3} + p^{2} T^{4} \) 2.71.t_ht
73$D_{4}$ \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_en
79$D_{4}$ \( 1 - 10 T + 163 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.79.ak_gh
83$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.83.g_gt
89$D_{4}$ \( 1 - 13 T + 209 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.89.an_ib
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.97.ak_il
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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

−10.43343526796892694157864761346, −10.39823081149817439046632654717, −9.578180479679418414146930744942, −9.327444753066146962134264057681, −8.777510517566518098706293840748, −8.639076508177464065000002641327, −7.87526213274663625327873522313, −7.86935605681606903657724655349, −7.42769423515147453862951066724, −7.40093200900812343193993508106, −6.56845343610710827421950839293, −5.57563154157219914651930065485, −4.66043629966342784974621871937, −4.50426503132686146127872626357, −3.69656774805218198534431793330, −3.24229584203906335485827551561, −2.23088730330174361858629621432, −1.78219363577744059942077016856, 0, 0, 1.78219363577744059942077016856, 2.23088730330174361858629621432, 3.24229584203906335485827551561, 3.69656774805218198534431793330, 4.50426503132686146127872626357, 4.66043629966342784974621871937, 5.57563154157219914651930065485, 6.56845343610710827421950839293, 7.40093200900812343193993508106, 7.42769423515147453862951066724, 7.86935605681606903657724655349, 7.87526213274663625327873522313, 8.639076508177464065000002641327, 8.777510517566518098706293840748, 9.327444753066146962134264057681, 9.578180479679418414146930744942, 10.39823081149817439046632654717, 10.43343526796892694157864761346

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