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Properties

Label 4-728e2-1.1-c1e2-0-21
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 $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 2·9-s + 2·13-s − 4·17-s − 7·25-s − 10·29-s − 4·37-s + 4·41-s − 4·45-s + 49-s + 6·53-s + 28·61-s + 4·65-s + 6·73-s − 5·81-s − 8·85-s − 18·89-s − 2·97-s − 24·101-s − 8·109-s + 2·113-s − 4·117-s − 6·121-s − 26·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.894·5-s − 2/3·9-s + 0.554·13-s − 0.970·17-s − 7/5·25-s − 1.85·29-s − 0.657·37-s + 0.624·41-s − 0.596·45-s + 1/7·49-s + 0.824·53-s + 3.58·61-s + 0.496·65-s + 0.702·73-s − 5/9·81-s − 0.867·85-s − 1.90·89-s − 0.203·97-s − 2.38·101-s − 0.766·109-s + 0.188·113-s − 0.369·117-s − 0.545·121-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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: \(1\)
Selberg data: \((4,\ 529984,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad2 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.5.ac_l
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.a_bl
23$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.23.a_ad
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.29.k_df
31$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.31.a_at
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.43.a_dh
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.a_n
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.53.ag_el
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.61.abc_mg
67$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.67.a_bi
71$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.71.a_acc
73$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.73.ag_fz
79$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.79.a_fd
83$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.83.a_fl
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.89.s_jz
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.97.c_hn
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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

−8.144388668476396997222119983247, −8.083973154130150661606579272593, −7.14918000366524593178728481657, −6.96069631485786572428198608010, −6.37684020436893069227778189405, −5.82577942199793673176864493442, −5.44176260711429074122925536322, −5.34298608831346054320252005808, −4.23622923362218866686267765866, −3.96547517025535220039746450659, −3.39334742454479681717934942712, −2.38361565588352713517870291579, −2.23637196597265827007393133486, −1.35836937709627763557977801195, 0, 1.35836937709627763557977801195, 2.23637196597265827007393133486, 2.38361565588352713517870291579, 3.39334742454479681717934942712, 3.96547517025535220039746450659, 4.23622923362218866686267765866, 5.34298608831346054320252005808, 5.44176260711429074122925536322, 5.82577942199793673176864493442, 6.37684020436893069227778189405, 6.96069631485786572428198608010, 7.14918000366524593178728481657, 8.083973154130150661606579272593, 8.144388668476396997222119983247

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