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Properties

Label 4-1184e2-1.1-c1e2-0-6
Degree $4$
Conductor $1401856$
Sign $-1$
Analytic cond. $89.3835$
Root an. cond. $3.07478$
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  − 8·5-s + 3·9-s + 4·13-s + 16·17-s + 38·25-s − 2·37-s − 10·41-s − 24·45-s − 13·49-s + 18·53-s − 12·61-s − 32·65-s + 22·73-s − 128·85-s − 36·89-s − 4·97-s − 18·101-s + 4·109-s − 16·113-s + 12·117-s − 13·121-s − 136·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  − 3.57·5-s + 9-s + 1.10·13-s + 3.88·17-s + 38/5·25-s − 0.328·37-s − 1.56·41-s − 3.57·45-s − 1.85·49-s + 2.47·53-s − 1.53·61-s − 3.96·65-s + 2.57·73-s − 13.8·85-s − 3.81·89-s − 0.406·97-s − 1.79·101-s + 0.383·109-s − 1.50·113-s + 1.10·117-s − 1.18·121-s − 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1401856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1401856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1401856\)    =    \(2^{10} \cdot 37^{2}\)
Sign: $-1$
Analytic conductor: \(89.3835\)
Root analytic conductor: \(3.07478\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1401856,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
37$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.3.a_ad
5$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.5.i_ba
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.a_n
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.17.aq_du
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.41.k_ed
43$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.43.a_de
47$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.47.a_abb
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.53.as_hf
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.61.m_gc
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.71.a_en
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.73.aw_kh
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.83.a_fl
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \) 2.89.bk_ti
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_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

−7.63572304597459780773128038137, −7.53594316387635585160999233290, −6.98150467834209165101675556515, −6.80988404697903381946029941059, −5.98376476141732920027999353241, −5.20817284495310242430018493679, −5.15252674038388255907026116656, −4.27611685284512326935756014151, −3.99465509935121990247876564159, −3.62991491005395914988227234813, −3.29292662358312635823710209810, −2.97824748954438677044219419910, −1.19206411521547945511921007919, −1.17899122706032057657563867954, 0, 1.17899122706032057657563867954, 1.19206411521547945511921007919, 2.97824748954438677044219419910, 3.29292662358312635823710209810, 3.62991491005395914988227234813, 3.99465509935121990247876564159, 4.27611685284512326935756014151, 5.15252674038388255907026116656, 5.20817284495310242430018493679, 5.98376476141732920027999353241, 6.80988404697903381946029941059, 6.98150467834209165101675556515, 7.53594316387635585160999233290, 7.63572304597459780773128038137

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