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Properties

Label 4-4080e2-1.1-c1e2-0-9
Degree $4$
Conductor $16646400$
Sign $1$
Analytic cond. $1061.38$
Root an. cond. $5.70779$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·5-s + 7-s + 3·9-s + 11-s − 2·13-s + 4·15-s + 2·17-s + 19-s − 2·21-s − 8·23-s + 3·25-s − 4·27-s + 13·29-s − 8·31-s − 2·33-s − 2·35-s + 9·37-s + 4·39-s + 41-s − 8·43-s − 6·45-s − 3·47-s − 9·49-s − 4·51-s + 21·53-s − 2·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 0.377·7-s + 9-s + 0.301·11-s − 0.554·13-s + 1.03·15-s + 0.485·17-s + 0.229·19-s − 0.436·21-s − 1.66·23-s + 3/5·25-s − 0.769·27-s + 2.41·29-s − 1.43·31-s − 0.348·33-s − 0.338·35-s + 1.47·37-s + 0.640·39-s + 0.156·41-s − 1.21·43-s − 0.894·45-s − 0.437·47-s − 9/7·49-s − 0.560·51-s + 2.88·53-s − 0.269·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16646400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16646400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16646400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1061.38\)
Root analytic conductor: \(5.70779\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16646400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.218274271\)
\(L(\frac12)\) \(\approx\) \(1.218274271\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
17$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) 2.7.ab_k
11$D_{4}$ \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) 2.11.ab_s
13$C_4$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_k
19$D_{4}$ \( 1 - T + 34 T^{2} - p T^{3} + p^{2} T^{4} \) 2.19.ab_bi
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$D_{4}$ \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.29.an_ds
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$D_{4}$ \( 1 - 9 T + 56 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.37.aj_ce
41$D_{4}$ \( 1 - T - 24 T^{2} - p T^{3} + p^{2} T^{4} \) 2.41.ab_ay
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$D_{4}$ \( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.47.d_ak
53$D_{4}$ \( 1 - 21 T + 4 p T^{2} - 21 p T^{3} + p^{2} T^{4} \) 2.53.av_ie
59$D_{4}$ \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_eg
61$D_{4}$ \( 1 - 2 T + 106 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_ec
67$D_{4}$ \( 1 - 2 T - 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_as
71$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.71.ae_da
73$D_{4}$ \( 1 + 21 T + 252 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.73.v_js
79$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.79.ac_g
83$D_{4}$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_fe
89$D_{4}$ \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.89.aq_gs
97$D_{4}$ \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_fm
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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

−8.510203839435418480899377806906, −8.174391767498818491454105110195, −7.78354691531763440572749267764, −7.59853970542649093056392328168, −7.08330377345688089359487584648, −6.81709299104707576243394379703, −6.39348993850383040685470891157, −6.05943218701229777873808810423, −5.60163967980735655345191845708, −5.31831467017825097258819820361, −4.81783862462917972609698064619, −4.49798912628328329292787989743, −4.11258299473496025023532167824, −3.88631748839904468089155239947, −3.03595229064088046162316413225, −2.96965996218575630059966216571, −2.00298948191892838723692022532, −1.69364264963686892543734993444, −0.871100831039080794121158886632, −0.45129541017034157199197939395, 0.45129541017034157199197939395, 0.871100831039080794121158886632, 1.69364264963686892543734993444, 2.00298948191892838723692022532, 2.96965996218575630059966216571, 3.03595229064088046162316413225, 3.88631748839904468089155239947, 4.11258299473496025023532167824, 4.49798912628328329292787989743, 4.81783862462917972609698064619, 5.31831467017825097258819820361, 5.60163967980735655345191845708, 6.05943218701229777873808810423, 6.39348993850383040685470891157, 6.81709299104707576243394379703, 7.08330377345688089359487584648, 7.59853970542649093056392328168, 7.78354691531763440572749267764, 8.174391767498818491454105110195, 8.510203839435418480899377806906

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