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Properties

Label 4-2070e2-1.1-c1e2-0-4
Degree $4$
Conductor $4284900$
Sign $1$
Analytic cond. $273.208$
Root an. cond. $4.06559$
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  + 2·2-s + 3·4-s + 2·5-s + 4·7-s + 4·8-s + 4·10-s + 4·11-s + 8·14-s + 5·16-s + 4·17-s + 6·20-s + 8·22-s + 2·23-s + 3·25-s + 12·28-s + 4·29-s − 8·31-s + 6·32-s + 8·34-s + 8·35-s − 8·37-s + 8·40-s + 12·44-s + 4·46-s + 6·49-s + 6·50-s + 4·53-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.894·5-s + 1.51·7-s + 1.41·8-s + 1.26·10-s + 1.20·11-s + 2.13·14-s + 5/4·16-s + 0.970·17-s + 1.34·20-s + 1.70·22-s + 0.417·23-s + 3/5·25-s + 2.26·28-s + 0.742·29-s − 1.43·31-s + 1.06·32-s + 1.37·34-s + 1.35·35-s − 1.31·37-s + 1.26·40-s + 1.80·44-s + 0.589·46-s + 6/7·49-s + 0.848·50-s + 0.549·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4284900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4284900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(273.208\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4284900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(12.30450242\)
\(L(\frac12)\) \(\approx\) \(12.30450242\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good7$C_4$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_k
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_be
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
29$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_cc
31$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_bu
37$D_{4}$ \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_de
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.41.a_by
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$D_{4}$ \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_as
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.59.au_ik
61$D_{4}$ \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.61.m_fu
67$D_{4}$ \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_fm
71$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \) 2.71.a_fe
73$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.73.e_w
79$C_2^2$ \( 1 + 150 T^{2} + p^{2} T^{4} \) 2.79.a_fu
83$D_{4}$ \( 1 + 8 T + 174 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.83.i_gs
89$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \) 2.89.a_fq
97$D_{4}$ \( 1 - 12 T + 222 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.97.am_io
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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

−9.099630985199850900613545259101, −9.049893486232341466448317370356, −8.447346626599010982624193152150, −8.184479842425552140327036894934, −7.52015209313273451244555518370, −7.33500505883364758123561611843, −6.79837747273482454366499359030, −6.54971099318643371860804315988, −6.02110080732991953757575825740, −5.56018838888448730508112358227, −5.22053570835672901738345440110, −5.11731069367088083522689459044, −4.37153796575677075830906830575, −4.19046474394123085095955836489, −3.38569082367677633454093907295, −3.37205974572949391128074036141, −2.36615202464516514783354411143, −2.10533781722412206466063352733, −1.39043206902009212889585974407, −1.16008839814667021142299015679, 1.16008839814667021142299015679, 1.39043206902009212889585974407, 2.10533781722412206466063352733, 2.36615202464516514783354411143, 3.37205974572949391128074036141, 3.38569082367677633454093907295, 4.19046474394123085095955836489, 4.37153796575677075830906830575, 5.11731069367088083522689459044, 5.22053570835672901738345440110, 5.56018838888448730508112358227, 6.02110080732991953757575825740, 6.54971099318643371860804315988, 6.79837747273482454366499359030, 7.33500505883364758123561611843, 7.52015209313273451244555518370, 8.184479842425552140327036894934, 8.447346626599010982624193152150, 9.049893486232341466448317370356, 9.099630985199850900613545259101

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