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Properties

Label 4-1512e2-1.1-c1e2-0-4
Degree $4$
Conductor $2286144$
Sign $1$
Analytic cond. $145.766$
Root an. cond. $3.47467$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 7-s − 3·11-s + 6·13-s − 14·17-s + 2·19-s − 4·23-s + 5·25-s − 4·29-s − 10·31-s − 2·35-s − 12·37-s + 7·41-s − 11·43-s + 8·47-s + 8·53-s − 6·55-s + 9·59-s − 4·61-s + 12·65-s − 9·67-s + 26·73-s + 3·77-s − 10·79-s − 28·85-s − 20·89-s − 6·91-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s − 0.904·11-s + 1.66·13-s − 3.39·17-s + 0.458·19-s − 0.834·23-s + 25-s − 0.742·29-s − 1.79·31-s − 0.338·35-s − 1.97·37-s + 1.09·41-s − 1.67·43-s + 1.16·47-s + 1.09·53-s − 0.809·55-s + 1.17·59-s − 0.512·61-s + 1.48·65-s − 1.09·67-s + 3.04·73-s + 0.341·77-s − 1.12·79-s − 3.03·85-s − 2.11·89-s − 0.628·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2286144\)    =    \(2^{6} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(145.766\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2286144,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.288195480\)
\(L(\frac12)\) \(\approx\) \(1.288195480\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_ab
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
13$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_x
17$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.17.o_df
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.19.ac_bn
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_ah
29$C_2^2$ \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_an
31$C_2^2$ \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_cr
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2^2$ \( 1 - 7 T + 8 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.41.ah_i
43$C_2^2$ \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.43.l_da
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_r
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.53.ai_es
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.59.aj_w
61$C_2^2$ \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_abt
67$C_2^2$ \( 1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.67.j_o
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.73.aba_md
79$C_2^2$ \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.79.k_v
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.83.a_adf
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2^2$ \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.97.h_abw
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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

−9.692852028292728143764759439233, −9.094513053636517639978277475789, −8.879187650211422732028014397875, −8.645828213102016659388457085066, −8.297007245003723769623724790716, −7.62716226424898798625868641622, −7.08625925731816898845934558883, −6.70192547022834597737070744180, −6.63477828194028078777092435341, −5.88838337266873988577668162725, −5.61926652235021639675547931560, −5.28323882528734955200197789927, −4.68025296421865968057462646792, −3.97883507794974064999297875240, −3.90451441694257815634677334175, −3.11636279146314831638953534525, −2.54151304709150034418287131407, −1.90381837922147008051796024635, −1.74540365061615371702421198917, −0.42075995596139965806164806089, 0.42075995596139965806164806089, 1.74540365061615371702421198917, 1.90381837922147008051796024635, 2.54151304709150034418287131407, 3.11636279146314831638953534525, 3.90451441694257815634677334175, 3.97883507794974064999297875240, 4.68025296421865968057462646792, 5.28323882528734955200197789927, 5.61926652235021639675547931560, 5.88838337266873988577668162725, 6.63477828194028078777092435341, 6.70192547022834597737070744180, 7.08625925731816898845934558883, 7.62716226424898798625868641622, 8.297007245003723769623724790716, 8.645828213102016659388457085066, 8.879187650211422732028014397875, 9.094513053636517639978277475789, 9.692852028292728143764759439233

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