↑

Properties

Label 4-1184e2-1.1-c1e2-0-15
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 $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5·9-s − 4·13-s − 8·17-s − 10·25-s − 2·37-s − 10·41-s − 13·49-s + 10·53-s − 12·61-s − 10·73-s + 16·81-s − 28·89-s − 4·97-s + 6·101-s − 4·109-s − 16·113-s + 20·117-s − 21·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 40·153-s + 157-s + 163-s + ⋯
L(s)  = 1  − 5/3·9-s − 1.10·13-s − 1.94·17-s − 2·25-s − 0.328·37-s − 1.56·41-s − 1.85·49-s + 1.37·53-s − 1.53·61-s − 1.17·73-s + 16/9·81-s − 2.96·89-s − 0.406·97-s + 0.597·101-s − 0.383·109-s − 1.50·113-s + 1.84·117-s − 1.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 3.23·153-s + 0.0798·157-s + 0.0783·163-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: \(2\)
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$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
11$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.11.a_v
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.17.i_by
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.23.a_bq
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$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.47.a_acx
53$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.53.ak_fb
59$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.59.a_ek
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$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.71.a_abb
73$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.73.k_gp
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
83$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.83.a_gj
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.89.bc_ok
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.63038328444641392108662540104, −6.87128996838340928428011955102, −6.80384633776451202812807515048, −6.12030903110471654805823588274, −5.82178883581689735565373783633, −5.23349472450029666027504407425, −4.99215332346230383019854002961, −4.21376201769914582932538220157, −4.02829710280020734935673890824, −3.03029375981211947401207493097, −2.89850403835078783059203894645, −2.11434405903642855125926303730, −1.73636861123368070695440785373, 0, 0, 1.73636861123368070695440785373, 2.11434405903642855125926303730, 2.89850403835078783059203894645, 3.03029375981211947401207493097, 4.02829710280020734935673890824, 4.21376201769914582932538220157, 4.99215332346230383019854002961, 5.23349472450029666027504407425, 5.82178883581689735565373783633, 6.12030903110471654805823588274, 6.80384633776451202812807515048, 6.87128996838340928428011955102, 7.63038328444641392108662540104

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