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Properties

Label 4-1140e2-1.1-c1e2-0-2
Degree $4$
Conductor $1299600$
Sign $1$
Analytic cond. $82.8636$
Root an. cond. $3.01710$
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  + 3-s − 5-s − 4·7-s − 4·11-s − 15-s − 17-s − 19-s − 4·21-s + 8·23-s − 27-s + 6·29-s − 6·31-s − 4·33-s + 4·35-s + 8·37-s − 4·41-s − 12·43-s − 47-s − 2·49-s − 51-s − 5·53-s + 4·55-s − 57-s − 10·59-s − 14·61-s + 6·67-s + 8·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 1.51·7-s − 1.20·11-s − 0.258·15-s − 0.242·17-s − 0.229·19-s − 0.872·21-s + 1.66·23-s − 0.192·27-s + 1.11·29-s − 1.07·31-s − 0.696·33-s + 0.676·35-s + 1.31·37-s − 0.624·41-s − 1.82·43-s − 0.145·47-s − 2/7·49-s − 0.140·51-s − 0.686·53-s + 0.539·55-s − 0.132·57-s − 1.30·59-s − 1.79·61-s + 0.733·67-s + 0.963·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1299600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1299600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1299600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(82.8636\)
Root analytic conductor: \(3.01710\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1299600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8095973946\)
\(L(\frac12)\) \(\approx\) \(0.8095973946\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
19$C_2$ \( 1 + T + p T^{2} \)
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$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.13.a_an
17$C_2^2$ \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \) 2.17.b_aq
23$C_2^2$ \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_bp
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_h
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.31.g_ct
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.37.ai_dm
41$C_2^2$ \( 1 + 4 T - 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_az
43$C_2^2$ \( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_dx
47$C_2^2$ \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \) 2.47.b_abu
53$C_2^2$ \( 1 + 5 T - 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.53.f_abc
59$C_2^2$ \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.59.k_bp
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.o_ff
67$C_2^2$ \( 1 - 6 T - 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.67.ag_abf
71$C_2^2$ \( 1 + 8 T - 7 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.71.i_ah
73$C_2^2$ \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_acr
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.79.ai_ap
83$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.83.aw_lb
89$C_2^2$ \( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.89.ac_adh
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_adp
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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

−9.936633683336715192133654461454, −9.490569466185323981936903230696, −9.273543597525241943558225486854, −8.879933277381609031802718611186, −8.220413440744960637054566700392, −8.107650332960755336386650748805, −7.59494672793727622316772578214, −7.10764730579910796314033638024, −6.73094442762432761522906994012, −6.25729620937100098520054816486, −6.00564050687361173268780622435, −5.10247035472623644825125004760, −4.92770262612755435619333366564, −4.42771129142824449628815398124, −3.53830546618600769444687459202, −3.28166449189888195326264399220, −2.94951358973155154654716209911, −2.39866234635984487171182468256, −1.53749101803381352246038596213, −0.37173987528240481263761181968, 0.37173987528240481263761181968, 1.53749101803381352246038596213, 2.39866234635984487171182468256, 2.94951358973155154654716209911, 3.28166449189888195326264399220, 3.53830546618600769444687459202, 4.42771129142824449628815398124, 4.92770262612755435619333366564, 5.10247035472623644825125004760, 6.00564050687361173268780622435, 6.25729620937100098520054816486, 6.73094442762432761522906994012, 7.10764730579910796314033638024, 7.59494672793727622316772578214, 8.107650332960755336386650748805, 8.220413440744960637054566700392, 8.879933277381609031802718611186, 9.273543597525241943558225486854, 9.490569466185323981936903230696, 9.936633683336715192133654461454

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