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Properties

Label 4-483e2-1.1-c1e2-0-7
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  − 2-s − 2·3-s − 2·4-s + 5-s + 2·6-s − 2·7-s + 3·8-s + 3·9-s − 10-s + 4·12-s − 7·13-s + 2·14-s − 2·15-s + 16-s − 3·18-s + 2·19-s − 2·20-s + 4·21-s − 2·23-s − 6·24-s − 8·25-s + 7·26-s − 4·27-s + 4·28-s − 12·29-s + 2·30-s − 2·32-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 4-s + 0.447·5-s + 0.816·6-s − 0.755·7-s + 1.06·8-s + 9-s − 0.316·10-s + 1.15·12-s − 1.94·13-s + 0.534·14-s − 0.516·15-s + 1/4·16-s − 0.707·18-s + 0.458·19-s − 0.447·20-s + 0.872·21-s − 0.417·23-s − 1.22·24-s − 8/5·25-s + 1.37·26-s − 0.769·27-s + 0.755·28-s − 2.22·29-s + 0.365·30-s − 0.353·32-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$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) 2.2.b_d
5$D_{4}$ \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_j
11$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.11.a_r
13$D_{4}$ \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.13.h_bl
17$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \) 2.17.a_al
19$D_{4}$ \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_t
29$D_{4}$ \( 1 + 12 T + 89 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.29.m_dl
31$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.31.a_r
37$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.37.w_hn
41$D_{4}$ \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_ct
43$D_{4}$ \( 1 + T + 85 T^{2} + p T^{3} + p^{2} T^{4} \) 2.43.b_dh
47$D_{4}$ \( 1 + 10 T + 114 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.47.k_ek
53$D_{4}$ \( 1 + 15 T + 131 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.53.p_fb
59$D_{4}$ \( 1 - 21 T + 227 T^{2} - 21 p T^{3} + p^{2} T^{4} \) 2.59.av_it
61$D_{4}$ \( 1 + 3 T + 113 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.61.d_ej
67$D_{4}$ \( 1 - T + 103 T^{2} - p T^{3} + p^{2} T^{4} \) 2.67.ab_dz
71$D_{4}$ \( 1 - 11 T + 171 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.71.al_gp
73$D_{4}$ \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.73.m_fh
79$D_{4}$ \( 1 + 10 T + 163 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.79.k_gh
83$D_{4}$ \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_bt
89$D_{4}$ \( 1 - 21 T + 287 T^{2} - 21 p T^{3} + p^{2} T^{4} \) 2.89.av_lb
97$D_{4}$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_x
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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

−10.43456108302839949315555401966, −10.14537550371901374733916161598, −9.863206365776709529947375400318, −9.619949858350752214505865533901, −9.075972103146551047298056541198, −8.820775860580404723590433718057, −7.85811733320914678684367073479, −7.67789560682717400194453991327, −7.12127925952419627726429323147, −6.65306542256103135596342541307, −6.02296917538655672157109057355, −5.48259355845956839893593043248, −5.06715251370191373343897543879, −4.81033452073727822321584879007, −3.79473129818647837080905150654, −3.60949947899386907940474602157, −2.30587730490221570561396220636, −1.63316623178825698113233874014, 0, 0, 1.63316623178825698113233874014, 2.30587730490221570561396220636, 3.60949947899386907940474602157, 3.79473129818647837080905150654, 4.81033452073727822321584879007, 5.06715251370191373343897543879, 5.48259355845956839893593043248, 6.02296917538655672157109057355, 6.65306542256103135596342541307, 7.12127925952419627726429323147, 7.67789560682717400194453991327, 7.85811733320914678684367073479, 8.820775860580404723590433718057, 9.075972103146551047298056541198, 9.619949858350752214505865533901, 9.863206365776709529947375400318, 10.14537550371901374733916161598, 10.43456108302839949315555401966

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