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Properties

Label 4-4050e2-1.1-c1e2-0-35
Degree $4$
Conductor $16402500$
Sign $1$
Analytic cond. $1045.83$
Root an. cond. $5.68677$
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 + 2·7-s + 4·8-s − 6·11-s − 10·13-s + 4·14-s + 5·16-s − 2·19-s − 12·22-s − 20·26-s + 6·28-s − 6·29-s − 8·31-s + 6·32-s + 2·37-s − 4·38-s − 12·41-s − 4·43-s − 18·44-s − 12·47-s + 49-s − 30·52-s − 6·53-s + 8·56-s − 12·58-s − 18·59-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.755·7-s + 1.41·8-s − 1.80·11-s − 2.77·13-s + 1.06·14-s + 5/4·16-s − 0.458·19-s − 2.55·22-s − 3.92·26-s + 1.13·28-s − 1.11·29-s − 1.43·31-s + 1.06·32-s + 0.328·37-s − 0.648·38-s − 1.87·41-s − 0.609·43-s − 2.71·44-s − 1.75·47-s + 1/7·49-s − 4.16·52-s − 0.824·53-s + 1.06·56-s − 1.57·58-s − 2.34·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16402500\)    =    \(2^{2} \cdot 3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1045.83\)
Root analytic conductor: \(5.68677\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 16402500,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
good7$D_{4}$ \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_d
11$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_bc
13$D_{4}$ \( 1 + 10 T + 48 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.13.k_bw
17$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \) 2.17.a_bf
19$D_{4}$ \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_bk
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
29$D_{4}$ \( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_cm
31$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_co
37$D_{4}$ \( 1 - 2 T + 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_bw
41$D_{4}$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_dn
43$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_bq
47$D_{4}$ \( 1 + 12 T + 127 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_ex
53$D_{4}$ \( 1 + 6 T + 112 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_ei
59$D_{4}$ \( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.59.s_ho
61$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_be
67$D_{4}$ \( 1 - 2 T + 108 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_ee
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.71.a_bi
73$D_{4}$ \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.73.k_cl
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.79.ak_hb
83$D_{4}$ \( 1 - 18 T + 172 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.83.as_gq
89$D_{4}$ \( 1 + 12 T + 187 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_hf
97$D_{4}$ \( 1 - 2 T + 147 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_fr
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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

−8.061060542320696669377621734421, −7.65738815810577255045767667326, −7.41525961768152022699127735617, −7.40204160837163369580095951894, −6.62984932130217896910220541132, −6.49523578958426645075150725494, −5.80100682792637413061730186690, −5.48797342530150237349680401479, −5.07179135242880595502032433486, −4.93697459808850796344389738166, −4.64392000129370050832298859429, −4.32766533239264698960746386296, −3.47050978912644775388093663017, −3.22386322593479821925527512299, −2.85531147115277794387818491254, −2.18166200778727666782706340477, −2.01954065663644059984387200566, −1.59753729233013532347116786075, 0, 0, 1.59753729233013532347116786075, 2.01954065663644059984387200566, 2.18166200778727666782706340477, 2.85531147115277794387818491254, 3.22386322593479821925527512299, 3.47050978912644775388093663017, 4.32766533239264698960746386296, 4.64392000129370050832298859429, 4.93697459808850796344389738166, 5.07179135242880595502032433486, 5.48797342530150237349680401479, 5.80100682792637413061730186690, 6.49523578958426645075150725494, 6.62984932130217896910220541132, 7.40204160837163369580095951894, 7.41525961768152022699127735617, 7.65738815810577255045767667326, 8.061060542320696669377621734421

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