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Properties

Label 4-4600e2-1.1-c1e2-0-14
Degree $4$
Conductor $21160000$
Sign $1$
Analytic cond. $1349.17$
Root an. cond. $6.06062$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s − 9-s − 4·11-s + 3·13-s − 7·17-s − 4·19-s + 21-s + 2·23-s − 8·29-s − 8·31-s − 4·33-s + 3·37-s + 3·39-s + 4·43-s + 5·47-s − 9·49-s − 7·51-s − 13·53-s − 4·57-s − 7·59-s − 10·61-s − 63-s + 19·67-s + 2·69-s − 10·71-s − 11·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s − 1/3·9-s − 1.20·11-s + 0.832·13-s − 1.69·17-s − 0.917·19-s + 0.218·21-s + 0.417·23-s − 1.48·29-s − 1.43·31-s − 0.696·33-s + 0.493·37-s + 0.480·39-s + 0.609·43-s + 0.729·47-s − 9/7·49-s − 0.980·51-s − 1.78·53-s − 0.529·57-s − 0.911·59-s − 1.28·61-s − 0.125·63-s + 2.32·67-s + 0.240·69-s − 1.18·71-s − 1.28·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21160000\)    =    \(2^{6} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1349.17\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 21160000,\ (\ :1/2, 1/2),\ 1)\)

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$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_c
7$D_{4}$ \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) 2.7.ab_k
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$D_{4}$ \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.13.ad_y
17$D_{4}$ \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.17.h_bq
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
29$D_{4}$ \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_cf
31$D_{4}$ \( 1 + 8 T + 61 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_cj
37$D_{4}$ \( 1 - 3 T + 72 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.37.ad_cu
41$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \) 2.41.a_cn
43$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_w
47$D_{4}$ \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.47.af_ck
53$D_{4}$ \( 1 + 13 T + 144 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.53.n_fo
59$D_{4}$ \( 1 + 7 T + 126 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.59.h_ew
61$D_{4}$ \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.61.k_fa
67$D_{4}$ \( 1 - 19 T + 220 T^{2} - 19 p T^{3} + p^{2} T^{4} \) 2.67.at_im
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.71.k_gl
73$D_{4}$ \( 1 + 11 T + 172 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.73.l_gq
79$D_{4}$ \( 1 + 10 T + 166 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.79.k_gk
83$D_{4}$ \( 1 + 3 T + 130 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.83.d_fa
89$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.89.aq_ji
97$D_{4}$ \( 1 + 2 T + 178 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_gw
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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

−8.030416815781157959752584356548, −7.977217821520905099999363309142, −7.39354650327697566571191015463, −7.23667368674003826937941985311, −6.68976542427338450105956692599, −6.30802067948381416789183641287, −5.84967530213675521522322952175, −5.75368627639428514483384005596, −5.08264882773962702346377998581, −4.77077535909309278291272269609, −4.40203846995579791572239276957, −4.02077463706326672991960983930, −3.42245061045840514289501903833, −3.19472630821373086275988257918, −2.53398639010901402454692856359, −2.28413223652911904488728077064, −1.77644446224955025913741417599, −1.28608363067669696909857755640, 0, 0, 1.28608363067669696909857755640, 1.77644446224955025913741417599, 2.28413223652911904488728077064, 2.53398639010901402454692856359, 3.19472630821373086275988257918, 3.42245061045840514289501903833, 4.02077463706326672991960983930, 4.40203846995579791572239276957, 4.77077535909309278291272269609, 5.08264882773962702346377998581, 5.75368627639428514483384005596, 5.84967530213675521522322952175, 6.30802067948381416789183641287, 6.68976542427338450105956692599, 7.23667368674003826937941985311, 7.39354650327697566571191015463, 7.977217821520905099999363309142, 8.030416815781157959752584356548

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