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Properties

Label 4-9200e2-1.1-c1e2-0-6
Degree $4$
Conductor $84640000$
Sign $1$
Analytic cond. $5396.71$
Root an. cond. $8.57101$
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  + 2·3-s − 4·7-s + 2·9-s − 2·11-s − 4·17-s − 2·19-s − 8·21-s − 2·23-s + 6·27-s − 8·29-s + 8·31-s − 4·33-s − 18·37-s + 6·41-s + 14·43-s + 10·47-s + 3·49-s − 8·51-s + 2·53-s − 4·57-s − 2·59-s − 10·61-s − 8·63-s + 8·67-s − 4·69-s − 20·71-s + 10·73-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.51·7-s + 2/3·9-s − 0.603·11-s − 0.970·17-s − 0.458·19-s − 1.74·21-s − 0.417·23-s + 1.15·27-s − 1.48·29-s + 1.43·31-s − 0.696·33-s − 2.95·37-s + 0.937·41-s + 2.13·43-s + 1.45·47-s + 3/7·49-s − 1.12·51-s + 0.274·53-s − 0.529·57-s − 0.260·59-s − 1.28·61-s − 1.00·63-s + 0.977·67-s − 0.481·69-s − 2.37·71-s + 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(84640000\)    =    \(2^{8} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(5396.71\)
Root analytic conductor: \(8.57101\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 84640000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
23$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_c
7$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_n
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.11.c_x
13$C_2^2$ \( 1 + 21 T^{2} + p^{2} T^{4} \) 2.13.a_v
17$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_s
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
29$D_{4}$ \( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_cr
31$D_{4}$ \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.31.ai_cg
37$D_{4}$ \( 1 + 18 T + 150 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.37.s_fu
41$D_{4}$ \( 1 - 6 T + 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_l
43$D_{4}$ \( 1 - 14 T + 115 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.43.ao_el
47$D_{4}$ \( 1 - 10 T + 74 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.47.ak_cw
53$D_{4}$ \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.53.ac_dy
59$D_{4}$ \( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.59.c_ek
61$D_{4}$ \( 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.61.k_fm
67$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_cs
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.71.u_ji
73$D_{4}$ \( 1 - 10 T + 151 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.73.ak_fv
79$D_{4}$ \( 1 + 8 T + 49 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_bx
83$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.83.as_jn
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.89.ae_ha
97$D_{4}$ \( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.97.ak_dq
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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

−7.54753571409843942558060179842, −7.38041808620477699267997240543, −6.81066338923647229295331314839, −6.51407464472291515774231691940, −6.34800328553629890845211848359, −5.99565820418257625923549766905, −5.35088800782171616237558052392, −5.24497203391693027152205599549, −4.75362366306761134578013568305, −4.05893958557440277932928343158, −3.93299134823177919011048041874, −3.82258086861460767264678641233, −2.95325155079481733505107609739, −2.95225363993380610756723987635, −2.46384713756264172515887997472, −2.26421542895692290163360522509, −1.57225848276104265504312742088, −1.03628403156256222131554944701, 0, 0, 1.03628403156256222131554944701, 1.57225848276104265504312742088, 2.26421542895692290163360522509, 2.46384713756264172515887997472, 2.95225363993380610756723987635, 2.95325155079481733505107609739, 3.82258086861460767264678641233, 3.93299134823177919011048041874, 4.05893958557440277932928343158, 4.75362366306761134578013568305, 5.24497203391693027152205599549, 5.35088800782171616237558052392, 5.99565820418257625923549766905, 6.34800328553629890845211848359, 6.51407464472291515774231691940, 6.81066338923647229295331314839, 7.38041808620477699267997240543, 7.54753571409843942558060179842

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