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Properties

Label 4-5202e2-1.1-c1e2-0-11
Degree $4$
Conductor $27060804$
Sign $1$
Analytic cond. $1725.42$
Root an. cond. $6.44501$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·7-s − 4·8-s + 4·13-s + 8·14-s + 5·16-s + 4·19-s − 12·23-s − 4·25-s − 8·26-s − 12·28-s − 4·31-s − 6·32-s + 8·37-s − 8·38-s − 12·41-s − 8·43-s + 24·46-s + 4·49-s + 8·50-s + 12·52-s − 12·53-s + 16·56-s + 12·59-s + 8·61-s + 8·62-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.51·7-s − 1.41·8-s + 1.10·13-s + 2.13·14-s + 5/4·16-s + 0.917·19-s − 2.50·23-s − 4/5·25-s − 1.56·26-s − 2.26·28-s − 0.718·31-s − 1.06·32-s + 1.31·37-s − 1.29·38-s − 1.87·41-s − 1.21·43-s + 3.53·46-s + 4/7·49-s + 1.13·50-s + 1.66·52-s − 1.64·53-s + 2.13·56-s + 1.56·59-s + 1.02·61-s + 1.01·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27060804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27060804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27060804\)    =    \(2^{2} \cdot 3^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(1725.42\)
Root analytic conductor: \(6.44501\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 27060804,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
17 \( 1 \)
good5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
7$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_m
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
13$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_g
19$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_s
23$D_{4}$ \( 1 + 12 T + 76 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.23.m_cy
29$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \) 2.29.a_ca
31$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_m
37$D_{4}$ \( 1 - 8 T + 84 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_dg
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.47.a_cs
53$D_{4}$ \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_eo
59$D_{4}$ \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_fa
61$D_{4}$ \( 1 - 8 T + 132 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_fc
67$D_{4}$ \( 1 - 4 T + 114 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_ek
71$D_{4}$ \( 1 + 12 T + 124 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.71.m_eu
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$D_{4}$ \( 1 + 4 T + 108 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.79.e_ee
83$D_{4}$ \( 1 + 12 T + 178 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_gw
89$D_{4}$ \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_eo
97$D_{4}$ \( 1 - 20 T + 270 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.97.au_kk
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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

−8.134224251733526988945850583399, −7.79899802204457089505195477502, −7.25184272051023650834158859061, −7.18024803657722560641057072321, −6.46546729201441883616077663616, −6.36783540662630640072246332774, −6.11092912932926989749835833976, −5.74644160020615702216216309142, −5.26643831948306429547065500409, −4.82321010434041451315326430526, −4.00309441200777132386434395017, −3.75745906649164815722220219942, −3.44808427229050337663380735980, −3.10271098527377872048487638693, −2.41704367384853537926625425602, −2.06148843032489325122553400852, −1.52531686293198501246355636871, −0.996523027521248524221400884381, 0, 0, 0.996523027521248524221400884381, 1.52531686293198501246355636871, 2.06148843032489325122553400852, 2.41704367384853537926625425602, 3.10271098527377872048487638693, 3.44808427229050337663380735980, 3.75745906649164815722220219942, 4.00309441200777132386434395017, 4.82321010434041451315326430526, 5.26643831948306429547065500409, 5.74644160020615702216216309142, 6.11092912932926989749835833976, 6.36783540662630640072246332774, 6.46546729201441883616077663616, 7.18024803657722560641057072321, 7.25184272051023650834158859061, 7.79899802204457089505195477502, 8.134224251733526988945850583399

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