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Properties

Label 4-728e2-1.1-c1e2-0-13
Degree $4$
Conductor $529984$
Sign $1$
Analytic cond. $33.7922$
Root an. cond. $2.41103$
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 + 3·5-s − 4·7-s + 3·9-s + 3·11-s − 2·13-s + 3·15-s − 7·17-s + 19-s − 4·21-s − 23-s + 5·25-s + 8·27-s − 4·29-s − 9·31-s + 3·33-s − 12·35-s + 3·37-s − 2·39-s + 20·41-s + 8·43-s + 9·45-s − 3·47-s + 9·49-s − 7·51-s + 53-s + 9·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s − 1.51·7-s + 9-s + 0.904·11-s − 0.554·13-s + 0.774·15-s − 1.69·17-s + 0.229·19-s − 0.872·21-s − 0.208·23-s + 25-s + 1.53·27-s − 0.742·29-s − 1.61·31-s + 0.522·33-s − 2.02·35-s + 0.493·37-s − 0.320·39-s + 3.12·41-s + 1.21·43-s + 1.34·45-s − 0.437·47-s + 9/7·49-s − 0.980·51-s + 0.137·53-s + 1.21·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(529984\)    =    \(2^{6} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(33.7922\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 529984,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.716100195\)
\(L(\frac12)\) \(\approx\) \(2.716100195\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_ac
5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_e
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
17$C_2^2$ \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.17.h_bg
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.ab_as
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) 2.23.b_aw
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2^2$ \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.31.j_by
37$C_2^2$ \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.37.ad_abc
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.41.au_ha
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.47.d_abm
53$C_2^2$ \( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} \) 2.53.ab_aca
59$C_2^2$ \( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.59.al_ck
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.ab_aci
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.67.ah_as
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.ah_ay
79$C_2^2$ \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.79.al_bq
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.83.i_ha
89$C_2^2$ \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) 2.89.b_adk
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−10.63093029142436353287112069061, −9.737235304556369366034061668449, −9.731865444256721017122381481268, −9.465413251542432290872909183943, −9.021570395243144228664604523765, −8.840313978190087346445080687555, −7.969090184181778678623087228087, −7.52924624440617576593640281936, −6.79033916091804252130232647924, −6.78172719457631538530772791062, −6.40277559528180129545465932433, −5.66526736330495188913803836571, −5.44419339073611806752710635074, −4.56076225428856058876833923460, −3.96437978248850991825437624709, −3.81015905313302725879341380622, −2.74511818347429415360205142110, −2.45128867925479836039053771455, −1.87788654386522647609214175329, −0.837591603142460961702535837250, 0.837591603142460961702535837250, 1.87788654386522647609214175329, 2.45128867925479836039053771455, 2.74511818347429415360205142110, 3.81015905313302725879341380622, 3.96437978248850991825437624709, 4.56076225428856058876833923460, 5.44419339073611806752710635074, 5.66526736330495188913803836571, 6.40277559528180129545465932433, 6.78172719457631538530772791062, 6.79033916091804252130232647924, 7.52924624440617576593640281936, 7.969090184181778678623087228087, 8.840313978190087346445080687555, 9.021570395243144228664604523765, 9.465413251542432290872909183943, 9.731865444256721017122381481268, 9.737235304556369366034061668449, 10.63093029142436353287112069061

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