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Properties

Label 4-5202e2-1.1-c1e2-0-1
Degree $4$
Conductor $27060804$
Sign $1$
Analytic cond. $1725.42$
Root an. cond. $6.44501$
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  − 2·2-s + 3·4-s + 3·5-s + 7-s − 4·8-s − 6·10-s + 3·11-s + 13-s − 2·14-s + 5·16-s − 5·19-s + 9·20-s − 6·22-s + 6·23-s + 5·25-s − 2·26-s + 3·28-s − 3·29-s − 2·31-s − 6·32-s + 3·35-s − 17·37-s + 10·38-s − 12·40-s + 12·41-s − 5·43-s + 9·44-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 1.34·5-s + 0.377·7-s − 1.41·8-s − 1.89·10-s + 0.904·11-s + 0.277·13-s − 0.534·14-s + 5/4·16-s − 1.14·19-s + 2.01·20-s − 1.27·22-s + 1.25·23-s + 25-s − 0.392·26-s + 0.566·28-s − 0.557·29-s − 0.359·31-s − 1.06·32-s + 0.507·35-s − 2.79·37-s + 1.62·38-s − 1.89·40-s + 1.87·41-s − 0.762·43-s + 1.35·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27060804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27060804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27060804\)    =    \(2^{2} \cdot 3^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(1725.42\)
Root analytic conductor: \(6.44501\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 27060804,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.703257529\)
\(L(\frac12)\) \(\approx\) \(1.703257529\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
17 \( 1 \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_e
7$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) 2.7.ab_g
11$D_{4}$ \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_q
13$D_{4}$ \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_s
19$D_{4}$ \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.19.f_bk
23$D_{4}$ \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_w
29$D_{4}$ \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_ca
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.31.c_cl
37$D_{4}$ \( 1 + 17 T + 138 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.37.r_fi
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$D_{4}$ \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.43.f_s
47$D_{4}$ \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_cs
53$D_{4}$ \( 1 + 9 T + 118 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.53.j_eo
59$D_{4}$ \( 1 + 21 T + 220 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.59.v_im
61$D_{4}$ \( 1 + 11 T + 144 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.61.l_fo
67$D_{4}$ \( 1 + 5 T + 132 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.67.f_fc
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$D_{4}$ \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.73.ah_dg
79$D_{4}$ \( 1 - 13 T + 126 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.79.an_ew
83$D_{4}$ \( 1 + 6 T + 142 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.83.g_fm
89$D_{4}$ \( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.89.as_is
97$D_{4}$ \( 1 - 4 T + 165 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_gj
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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

−8.479853687771473896107488536321, −8.119065304785937716646001750003, −7.58400974578860888797903429393, −7.54655986724861487155907964612, −6.88602914109978761663627481993, −6.63750653665519392235415799516, −6.32560740079250889416913019117, −6.11927485014781324036501579754, −5.57707999541350759857834984871, −5.29637364683663610702982839269, −4.62795872382505905211505534239, −4.58629676217285924801429806014, −3.68384452879537065439259722649, −3.36596043182056447070147539617, −2.97896215943565841328410406391, −2.30244625202438665258377426555, −1.78359987574314078667668422644, −1.76357595907527198673602731125, −1.16945884042255214019079839356, −0.44836337438480035187298429336, 0.44836337438480035187298429336, 1.16945884042255214019079839356, 1.76357595907527198673602731125, 1.78359987574314078667668422644, 2.30244625202438665258377426555, 2.97896215943565841328410406391, 3.36596043182056447070147539617, 3.68384452879537065439259722649, 4.58629676217285924801429806014, 4.62795872382505905211505534239, 5.29637364683663610702982839269, 5.57707999541350759857834984871, 6.11927485014781324036501579754, 6.32560740079250889416913019117, 6.63750653665519392235415799516, 6.88602914109978761663627481993, 7.54655986724861487155907964612, 7.58400974578860888797903429393, 8.119065304785937716646001750003, 8.479853687771473896107488536321

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