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Properties

Label 4-666e2-1.1-c1e2-0-1
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 5-s − 2·7-s + 8-s − 10-s − 4·11-s − 6·13-s + 2·14-s − 16-s + 3·17-s + 2·19-s + 4·22-s − 16·23-s + 5·25-s + 6·26-s − 2·29-s + 4·31-s − 3·34-s − 2·35-s − 11·37-s − 2·38-s + 40-s − 9·41-s + 20·43-s + 16·46-s − 4·47-s + 7·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s − 1.20·11-s − 1.66·13-s + 0.534·14-s − 1/4·16-s + 0.727·17-s + 0.458·19-s + 0.852·22-s − 3.33·23-s + 25-s + 1.17·26-s − 0.371·29-s + 0.718·31-s − 0.514·34-s − 0.338·35-s − 1.80·37-s − 0.324·38-s + 0.158·40-s − 1.40·41-s + 3.04·43-s + 2.35·46-s − 0.583·47-s + 49-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

\(L(1)\) \(\approx\) \(0.4806502072\)
\(L(\frac12)\) \(\approx\) \(0.4806502072\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 11 T + p T^{2} \)
good5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_ae
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_ad
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_x
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_ai
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_ap
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.23.q_eg
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.29.c_ch
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_bo
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.43.au_he
47$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.47.e_du
53$C_2^2$ \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.53.c_abx
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_dh
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.b_aci
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_acl
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_abj
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2^2$ \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.79.ak_v
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.83.a_adf
89$C_2^2$ \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) 2.89.b_adk
97$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \) 2.97.bi_sp
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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

−10.57558942196640710174124815119, −10.16356137322599288057736205662, −9.749414948887695676405478890478, −9.724100544094617858624987398541, −9.175968016865979394587002543034, −8.357015427386739172283851771363, −8.319472851834525247541570337237, −7.65950797607866890725037612800, −7.33572303700705219607863314739, −6.96233728797908714144420624898, −6.20885108567918048370892269039, −5.85917460319188376724296717262, −5.14422654700363670485629718103, −5.11325925422841372841486679924, −4.11469300728018895072364159528, −3.72187421079470915592075685526, −2.76565167002142138433502680168, −2.45697887136000946972439680267, −1.72851462967080437276638280968, −0.40129317819854791914808432228, 0.40129317819854791914808432228, 1.72851462967080437276638280968, 2.45697887136000946972439680267, 2.76565167002142138433502680168, 3.72187421079470915592075685526, 4.11469300728018895072364159528, 5.11325925422841372841486679924, 5.14422654700363670485629718103, 5.85917460319188376724296717262, 6.20885108567918048370892269039, 6.96233728797908714144420624898, 7.33572303700705219607863314739, 7.65950797607866890725037612800, 8.319472851834525247541570337237, 8.357015427386739172283851771363, 9.175968016865979394587002543034, 9.724100544094617858624987398541, 9.749414948887695676405478890478, 10.16356137322599288057736205662, 10.57558942196640710174124815119

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