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Properties

Label 4-6900e2-1.1-c1e2-0-8
Degree $4$
Conductor $47610000$
Sign $1$
Analytic cond. $3035.65$
Root an. cond. $7.42272$
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  + 2·3-s − 2·7-s + 3·9-s − 2·11-s + 10·13-s + 12·17-s − 2·19-s − 4·21-s + 2·23-s + 4·27-s + 4·29-s − 2·31-s − 4·33-s + 2·37-s + 20·39-s − 12·41-s + 2·47-s + 49-s + 24·51-s + 4·53-s − 4·57-s − 4·59-s − 18·61-s − 6·63-s − 10·67-s + 4·69-s − 4·71-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s + 9-s − 0.603·11-s + 2.77·13-s + 2.91·17-s − 0.458·19-s − 0.872·21-s + 0.417·23-s + 0.769·27-s + 0.742·29-s − 0.359·31-s − 0.696·33-s + 0.328·37-s + 3.20·39-s − 1.87·41-s + 0.291·47-s + 1/7·49-s + 3.36·51-s + 0.549·53-s − 0.529·57-s − 0.520·59-s − 2.30·61-s − 0.755·63-s − 1.22·67-s + 0.481·69-s − 0.474·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47610000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(47610000\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(3035.65\)
Root analytic conductor: \(7.42272\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 47610000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.187548314\)
\(L(\frac12)\) \(\approx\) \(6.187548314\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
23$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_d
11$D_{4}$ \( 1 + 2 T - 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_ae
13$D_{4}$ \( 1 - 10 T + 48 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.13.ak_bw
17$D_{4}$ \( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.17.am_cp
19$D_{4}$ \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_bk
29$D_{4}$ \( 1 - 4 T + 35 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_bj
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.31.c_cl
37$C_2^2$ \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_abh
41$D_{4}$ \( 1 + 12 T + 115 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_el
43$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.43.a_cw
47$D_{4}$ \( 1 - 2 T + 92 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.47.ac_do
53$D_{4}$ \( 1 - 4 T + 35 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_bj
59$D_{4}$ \( 1 + 4 T + 95 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_dr
61$D_{4}$ \( 1 + 18 T + 200 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.61.s_hs
67$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.67.k_gd
71$D_{4}$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.71.e_ab
73$D_{4}$ \( 1 - 26 T + 312 T^{2} - 26 p T^{3} + p^{2} T^{4} \) 2.73.aba_ma
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.79.ai_gs
83$D_{4}$ \( 1 + 4 T + 95 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_dr
89$D_{4}$ \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_ec
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.97.i_ic
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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

−8.104784856997803094661157477636, −8.064469453634941485872001694336, −7.44822699813276739718488383797, −7.23204890787818267989315165083, −6.63909905754951613797374590135, −6.51891153423254961118135236775, −5.96176092558733571222381817052, −5.76949299741449195740323007541, −5.38936856766440089731674279910, −4.99894245912163190656978098211, −4.32932100485992812867730832759, −4.06708873876119174207213017986, −3.50817757396754464785759795382, −3.39988712825126259588866035357, −2.96950662958396563576396039043, −2.89562950123847344779858423142, −1.96762590284761508250123543892, −1.49679278795648747522742976324, −1.20493256948798574455525916220, −0.60440772824373397582195787242, 0.60440772824373397582195787242, 1.20493256948798574455525916220, 1.49679278795648747522742976324, 1.96762590284761508250123543892, 2.89562950123847344779858423142, 2.96950662958396563576396039043, 3.39988712825126259588866035357, 3.50817757396754464785759795382, 4.06708873876119174207213017986, 4.32932100485992812867730832759, 4.99894245912163190656978098211, 5.38936856766440089731674279910, 5.76949299741449195740323007541, 5.96176092558733571222381817052, 6.51891153423254961118135236775, 6.63909905754951613797374590135, 7.23204890787818267989315165083, 7.44822699813276739718488383797, 8.064469453634941485872001694336, 8.104784856997803094661157477636

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