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Properties

Label 4-777e2-1.1-c1e2-0-30
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4-s − 3·5-s + 4·7-s + 3·9-s + 3·11-s + 2·12-s − 9·13-s − 6·15-s − 3·16-s − 3·17-s + 3·19-s − 3·20-s + 8·21-s + 15·23-s + 25-s + 4·27-s + 4·28-s + 3·31-s + 6·33-s − 12·35-s + 3·36-s − 10·37-s − 18·39-s − 9·41-s + 3·44-s − 9·45-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s − 1.34·5-s + 1.51·7-s + 9-s + 0.904·11-s + 0.577·12-s − 2.49·13-s − 1.54·15-s − 3/4·16-s − 0.727·17-s + 0.688·19-s − 0.670·20-s + 1.74·21-s + 3.12·23-s + 1/5·25-s + 0.769·27-s + 0.755·28-s + 0.538·31-s + 1.04·33-s − 2.02·35-s + 1/2·36-s − 1.64·37-s − 2.88·39-s − 1.40·41-s + 0.452·44-s − 1.34·45-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

\(L(1)\) \(\approx\) \(3.002529759\)
\(L(\frac12)\) \(\approx\) \(3.002529759\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.2.a_ab
5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_i
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
13$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.j_bo
17$C_2^2$ \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_u
19$C_2^2$ \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.19.ad_w
23$C_2^2$ \( 1 - 15 T + 98 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.23.ap_du
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.ad_bi
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_bo
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2^2$ \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.47.aj_bi
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.53.j_bc
59$C_2^2$ \( 1 + 9 T + 86 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.59.j_di
61$C_2^2$ \( 1 - 3 T + 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.61.ad_cm
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2^2$ \( 1 - 9 T + 10 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.71.aj_k
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.73.h_ay
79$C_2^2$ \( 1 - 27 T + 322 T^{2} - 27 p T^{3} + p^{2} T^{4} \) 2.79.abb_mk
83$C_2^2$ \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.83.ap_fm
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.89.a_afa
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \) 2.97.a_afq
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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

−10.67374854323516347921736165202, −10.05898011469377144668178664386, −9.452801077343685450129494744849, −9.150877181537019773928210501423, −8.792959291498559513358310336714, −8.447081613247916558407245274188, −7.76472956366667918868060209831, −7.43133223557508539181114710396, −7.42477287891220397249124086922, −6.68656629510633089481245553761, −6.66024007930809264844545837944, −5.24129614062571416970034750933, −4.84173113936830748167255201973, −4.79335630998417527937261911664, −4.19523524083057090783805211931, −3.24572471517519653198445134456, −3.23647590713636402959424961405, −2.08458978215859483380174928232, −2.07773424736350368863779423294, −0.830914611290480651833758326347, 0.830914611290480651833758326347, 2.07773424736350368863779423294, 2.08458978215859483380174928232, 3.23647590713636402959424961405, 3.24572471517519653198445134456, 4.19523524083057090783805211931, 4.79335630998417527937261911664, 4.84173113936830748167255201973, 5.24129614062571416970034750933, 6.66024007930809264844545837944, 6.68656629510633089481245553761, 7.42477287891220397249124086922, 7.43133223557508539181114710396, 7.76472956366667918868060209831, 8.447081613247916558407245274188, 8.792959291498559513358310336714, 9.150877181537019773928210501423, 9.452801077343685450129494744849, 10.05898011469377144668178664386, 10.67374854323516347921736165202

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