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Properties

Label 4-666e2-1.1-c1e2-0-20
Degree $4$
Conductor $443556$
Sign $1$
Analytic cond. $28.2815$
Root an. cond. $2.30608$
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-s + 4·5-s − 3·7-s − 8-s + 4·10-s + 4·11-s + 3·13-s − 3·14-s − 16-s + 6·17-s + 4·19-s + 4·22-s + 12·23-s + 5·25-s + 3·26-s + 14·31-s + 6·34-s − 12·35-s − 10·37-s + 4·38-s − 4·40-s + 6·41-s + 2·43-s + 12·46-s − 20·47-s + 7·49-s + 5·50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.78·5-s − 1.13·7-s − 0.353·8-s + 1.26·10-s + 1.20·11-s + 0.832·13-s − 0.801·14-s − 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.852·22-s + 2.50·23-s + 25-s + 0.588·26-s + 2.51·31-s + 1.02·34-s − 2.02·35-s − 1.64·37-s + 0.648·38-s − 0.632·40-s + 0.937·41-s + 0.304·43-s + 1.76·46-s − 2.91·47-s + 49-s + 0.707·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(443556\)    =    \(2^{2} \cdot 3^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(28.2815\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 443556,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.172649721\)
\(L(\frac12)\) \(\approx\) \(4.172649721\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
37$C_2$ \( 1 + 10 T + p T^{2} \)
good5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.5.ae_l
7$C_2^2$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_c
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2^2$ \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.13.ad_ae
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_t
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_ad
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.31.ao_eh
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_af
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.43.ac_dj
47$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.47.u_hm
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_ar
59$C_2^2$ \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.59.ak_bp
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.61.k_bn
67$C_2^2$ \( 1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.67.j_o
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_abj
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.73.w_kh
79$C_2^2$ \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.79.j_c
83$C_2^2$ \( 1 - 10 T + 17 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.83.ak_r
89$C_2^2$ \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.89.c_adh
97$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \) 2.97.be_qd
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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

−10.42790391111010164270195089840, −10.30637082436760405378015835831, −9.859606427278954681641562916471, −9.366099591444130747778960899739, −9.246103525652369731422918259512, −8.810764449079380003980270638965, −8.211607257244901956452984639615, −7.57131855657701854971926619254, −6.77462945688144403224894600111, −6.69439609353918609924924349637, −6.22331399321221923791957099472, −5.73018187751063312711422249233, −5.46115290745122647422760147866, −4.86575065065155141479311787074, −4.29416611402911422822415005020, −3.48899503573627805059940420583, −2.96052392194031261161816771197, −2.90681213586910936442331567722, −1.43282904485530448110077634790, −1.24625954342673677325471100967, 1.24625954342673677325471100967, 1.43282904485530448110077634790, 2.90681213586910936442331567722, 2.96052392194031261161816771197, 3.48899503573627805059940420583, 4.29416611402911422822415005020, 4.86575065065155141479311787074, 5.46115290745122647422760147866, 5.73018187751063312711422249233, 6.22331399321221923791957099472, 6.69439609353918609924924349637, 6.77462945688144403224894600111, 7.57131855657701854971926619254, 8.211607257244901956452984639615, 8.810764449079380003980270638965, 9.246103525652369731422918259512, 9.366099591444130747778960899739, 9.859606427278954681641562916471, 10.30637082436760405378015835831, 10.42790391111010164270195089840

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