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Properties

Label 4-2910e2-1.1-c1e2-0-8
Degree $4$
Conductor $8468100$
Sign $1$
Analytic cond. $539.933$
Root an. cond. $4.82042$
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 − 2·3-s + 3·4-s + 2·5-s − 4·6-s − 4·7-s + 4·8-s + 3·9-s + 4·10-s − 4·11-s − 6·12-s + 4·13-s − 8·14-s − 4·15-s + 5·16-s − 4·17-s + 6·18-s − 12·19-s + 6·20-s + 8·21-s − 8·22-s − 4·23-s − 8·24-s + 3·25-s + 8·26-s − 4·27-s − 12·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 1.51·7-s + 1.41·8-s + 9-s + 1.26·10-s − 1.20·11-s − 1.73·12-s + 1.10·13-s − 2.13·14-s − 1.03·15-s + 5/4·16-s − 0.970·17-s + 1.41·18-s − 2.75·19-s + 1.34·20-s + 1.74·21-s − 1.70·22-s − 0.834·23-s − 1.63·24-s + 3/5·25-s + 1.56·26-s − 0.769·27-s − 2.26·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8468100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8468100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8468100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 97^{2}\)
Sign: $1$
Analytic conductor: \(539.933\)
Root analytic conductor: \(4.82042\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 8468100,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
97$C_1$ \( ( 1 + T )^{2} \)
good7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$D_{4}$ \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_y
17$D_{4}$ \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_bg
19$D_{4}$ \( 1 + 12 T + 68 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.19.m_cq
23$D_{4}$ \( 1 + 4 T - 4 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_ae
29$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_bm
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$D_{4}$ \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_y
41$D_{4}$ \( 1 + 4 T + 80 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_dc
43$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_da
47$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.47.a_cs
53$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.53.a_k
59$D_{4}$ \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_fa
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$D_{4}$ \( 1 + 4 T + 132 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_fc
71$D_{4}$ \( 1 + 20 T + 236 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.71.u_jc
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.73.a_by
79$D_{4}$ \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_co
83$D_{4}$ \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_cw
89$D_{4}$ \( 1 - 12 T + 190 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_hi
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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

−8.387514582807915672683091317912, −8.343261729843645483978683095874, −7.45726448283998034758345206389, −7.36658840448366674776896360928, −6.42412884059089968485757139597, −6.39726087040750620133838787097, −6.33111904530028142937286194549, −6.17981405824955903208918252664, −5.46904086196127799753712223055, −5.14522999382221263336366431870, −4.72483653494599460499204992642, −4.36546076028463168090325177824, −3.80352769606223704818252881294, −3.57407673523949968491905169843, −2.73509757291269821587542591904, −2.63222500326033397852903827885, −1.70549035159687046531176410067, −1.67338330719789782119205927170, 0, 0, 1.67338330719789782119205927170, 1.70549035159687046531176410067, 2.63222500326033397852903827885, 2.73509757291269821587542591904, 3.57407673523949968491905169843, 3.80352769606223704818252881294, 4.36546076028463168090325177824, 4.72483653494599460499204992642, 5.14522999382221263336366431870, 5.46904086196127799753712223055, 6.17981405824955903208918252664, 6.33111904530028142937286194549, 6.39726087040750620133838787097, 6.42412884059089968485757139597, 7.36658840448366674776896360928, 7.45726448283998034758345206389, 8.343261729843645483978683095874, 8.387514582807915672683091317912

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