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Properties

Label 4-2230272-1.1-c1e2-0-38
Degree $4$
Conductor $2230272$
Sign $-1$
Analytic cond. $142.204$
Root an. cond. $3.45324$
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  + 2·3-s + 3·9-s + 2·11-s − 4·17-s − 10·25-s + 4·27-s + 4·33-s − 12·41-s + 24·43-s − 10·49-s − 8·51-s − 8·67-s + 4·73-s − 20·75-s + 5·81-s + 8·83-s − 28·89-s − 4·97-s + 6·99-s − 8·107-s − 28·113-s + 3·121-s − 24·123-s + 127-s + 48·129-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s + 0.603·11-s − 0.970·17-s − 2·25-s + 0.769·27-s + 0.696·33-s − 1.87·41-s + 3.65·43-s − 1.42·49-s − 1.12·51-s − 0.977·67-s + 0.468·73-s − 2.30·75-s + 5/9·81-s + 0.878·83-s − 2.96·89-s − 0.406·97-s + 0.603·99-s − 0.773·107-s − 2.63·113-s + 3/11·121-s − 2.16·123-s + 0.0887·127-s + 4.22·129-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2230272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2230272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2230272\)    =    \(2^{11} \cdot 3^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(142.204\)
Root analytic conductor: \(3.45324\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 2230272,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.23.a_bq
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.43.ay_iw
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.71.a_bq
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.79.a_fy
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.83.ai_ha
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
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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

−7.59728616940349967649485030278, −7.18560206492637594737079166625, −6.56294199925233738325138182894, −6.49152714179339359324472370282, −5.65888596404534691311056509338, −5.54640234169864262168007165565, −4.52009791178801828235754419921, −4.48131796152143406195105869579, −3.72939112912468208779060606206, −3.67603058766281829826755122415, −2.76370782595700519478080690028, −2.45735474547779135132608451071, −1.80476785917260666627304613228, −1.28476892045959906041277637532, 0, 1.28476892045959906041277637532, 1.80476785917260666627304613228, 2.45735474547779135132608451071, 2.76370782595700519478080690028, 3.67603058766281829826755122415, 3.72939112912468208779060606206, 4.48131796152143406195105869579, 4.52009791178801828235754419921, 5.54640234169864262168007165565, 5.65888596404534691311056509338, 6.49152714179339359324472370282, 6.56294199925233738325138182894, 7.18560206492637594737079166625, 7.59728616940349967649485030278

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