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Properties

Label 4-777e2-1.1-c1e2-0-21
Degree $4$
Conductor $603729$
Sign $1$
Analytic cond. $38.4942$
Root an. cond. $2.49085$
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  + 2·2-s − 2·3-s − 4-s − 5-s − 4·6-s + 4·7-s − 8·8-s + 3·9-s − 2·10-s − 11-s + 2·12-s + 13-s + 8·14-s + 2·15-s − 7·16-s + 7·17-s + 6·18-s + 5·19-s + 20-s − 8·21-s − 2·22-s + 9·23-s + 16·24-s + 5·25-s + 2·26-s − 4·27-s − 4·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s − 1/2·4-s − 0.447·5-s − 1.63·6-s + 1.51·7-s − 2.82·8-s + 9-s − 0.632·10-s − 0.301·11-s + 0.577·12-s + 0.277·13-s + 2.13·14-s + 0.516·15-s − 7/4·16-s + 1.69·17-s + 1.41·18-s + 1.14·19-s + 0.223·20-s − 1.74·21-s − 0.426·22-s + 1.87·23-s + 3.26·24-s + 25-s + 0.392·26-s − 0.769·27-s − 0.755·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 603729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603729 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(603729\)    =    \(3^{2} \cdot 7^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(38.4942\)
Root analytic conductor: \(2.49085\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 603729,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.128361833\)
\(L(\frac12)\) \(\approx\) \(2.128361833\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 4 T + p T^{2} \)
37$C_2$ \( 1 + 10 T + p T^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.2.ac_f
5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ae
11$C_2^2$ \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) 2.11.b_ak
13$C_2^2$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_am
17$C_2^2$ \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.17.ah_bg
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.19.af_g
23$C_2^2$ \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.23.aj_cg
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2^2$ \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.31.d_aw
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_bo
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2^2$ \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.47.aj_bi
53$C_2^2$ \( 1 - 7 T - 4 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.53.ah_ae
59$C_2^2$ \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.59.h_ak
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.61.h_am
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2^2$ \( 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4} \) 2.71.ab_acs
73$C_2^2$ \( 1 - 5 T - 48 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.73.af_abw
79$C_2^2$ \( 1 + 7 T - 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.79.h_abe
83$C_2^2$ \( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.83.ad_acw
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \) 2.97.bk_ty
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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

−10.71664719470439907982455456655, −10.25686230924135372493712638361, −9.641400645933961367031416152900, −9.256516654006790789099899823358, −8.761801377996032898760240387270, −8.517386252248213674045446723527, −7.66496762347149062531864431830, −7.61580381688073259430506696342, −6.95882026092662170420232701332, −6.37965005436199603192150657154, −5.66117751432999274631941959346, −5.25199765680923410594787632849, −5.12609412827779148469881583415, −5.10618832614173302001877096546, −4.07715706766751645728375828985, −3.95100402296586704630565668414, −3.29205084061689595667561593473, −2.71294935819919177906305180503, −1.31299507869879350274697563132, −0.75857181276723355620098909606, 0.75857181276723355620098909606, 1.31299507869879350274697563132, 2.71294935819919177906305180503, 3.29205084061689595667561593473, 3.95100402296586704630565668414, 4.07715706766751645728375828985, 5.10618832614173302001877096546, 5.12609412827779148469881583415, 5.25199765680923410594787632849, 5.66117751432999274631941959346, 6.37965005436199603192150657154, 6.95882026092662170420232701332, 7.61580381688073259430506696342, 7.66496762347149062531864431830, 8.517386252248213674045446723527, 8.761801377996032898760240387270, 9.256516654006790789099899823358, 9.641400645933961367031416152900, 10.25686230924135372493712638361, 10.71664719470439907982455456655

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