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Properties

Label 4-1800e2-1.1-c1e2-0-29
Degree $4$
Conductor $3240000$
Sign $1$
Analytic cond. $206.585$
Root an. cond. $3.79118$
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 + 2·4-s − 8·13-s − 4·16-s + 16·26-s + 4·31-s + 8·32-s − 16·37-s − 4·41-s − 8·43-s + 10·49-s − 16·52-s − 12·53-s − 8·62-s − 8·64-s + 24·67-s − 24·71-s + 32·74-s − 20·79-s + 8·82-s − 32·83-s + 16·86-s − 20·89-s − 20·98-s + 24·106-s − 24·107-s + 22·121-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 2.21·13-s − 16-s + 3.13·26-s + 0.718·31-s + 1.41·32-s − 2.63·37-s − 0.624·41-s − 1.21·43-s + 10/7·49-s − 2.21·52-s − 1.64·53-s − 1.01·62-s − 64-s + 2.93·67-s − 2.84·71-s + 3.71·74-s − 2.25·79-s + 0.883·82-s − 3.51·83-s + 1.72·86-s − 2.11·89-s − 2.02·98-s + 2.33·106-s − 2.32·107-s + 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3240000\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(206.585\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3240000,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + p T + p T^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.29.a_aw
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.37.q_fi
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.59.a_ady
61$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.61.a_aes
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.83.bg_qg
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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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

−8.875998004626885302377960097467, −8.756825596544356717914668224294, −8.413234306678107424534958907818, −8.036091677772918509625916872702, −7.34566300328257519877336630627, −7.33253750549547717658542255343, −6.80117255286592095212583072175, −6.74204108544332128169277060569, −5.82640220486720773920543179375, −5.47089450309105897255216798680, −4.94223590153793988921772925873, −4.65203588439624544080891765752, −4.14305611505320979620291930505, −3.47535315893522131663500516509, −2.68565329769207784762041051970, −2.60309321654469820398165846484, −1.68115709682987794279755210826, −1.40173502057083032090356188986, 0, 0, 1.40173502057083032090356188986, 1.68115709682987794279755210826, 2.60309321654469820398165846484, 2.68565329769207784762041051970, 3.47535315893522131663500516509, 4.14305611505320979620291930505, 4.65203588439624544080891765752, 4.94223590153793988921772925873, 5.47089450309105897255216798680, 5.82640220486720773920543179375, 6.74204108544332128169277060569, 6.80117255286592095212583072175, 7.33253750549547717658542255343, 7.34566300328257519877336630627, 8.036091677772918509625916872702, 8.413234306678107424534958907818, 8.756825596544356717914668224294, 8.875998004626885302377960097467

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