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Properties

Label 4-2175e2-1.1-c1e2-0-10
Degree $4$
Conductor $4730625$
Sign $1$
Analytic cond. $301.628$
Root an. cond. $4.16742$
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 − 2·3-s + 2·4-s − 2·6-s − 2·7-s + 5·8-s + 3·9-s + 10·11-s − 4·12-s − 2·14-s + 5·16-s + 6·17-s + 3·18-s − 2·19-s + 4·21-s + 10·22-s + 8·23-s − 10·24-s − 4·27-s − 4·28-s + 2·29-s + 8·31-s + 10·32-s − 20·33-s + 6·34-s + 6·36-s + 8·37-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 4-s − 0.816·6-s − 0.755·7-s + 1.76·8-s + 9-s + 3.01·11-s − 1.15·12-s − 0.534·14-s + 5/4·16-s + 1.45·17-s + 0.707·18-s − 0.458·19-s + 0.872·21-s + 2.13·22-s + 1.66·23-s − 2.04·24-s − 0.769·27-s − 0.755·28-s + 0.371·29-s + 1.43·31-s + 1.76·32-s − 3.48·33-s + 1.02·34-s + 36-s + 1.31·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4730625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4730625\)    =    \(3^{2} \cdot 5^{4} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(301.628\)
Root analytic conductor: \(4.16742\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4730625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.060695040\)
\(L(\frac12)\) \(\approx\) \(5.060695040\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
29$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) 2.2.ab_ab
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.7.c_p
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.11.ak_bv
13$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.13.a_f
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_s
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.23.ai_ck
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.37.ai_dm
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.41.a_ac
43$D_{4}$ \( 1 - 10 T + 90 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.43.ak_dm
47$D_{4}$ \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_ef
53$D_{4}$ \( 1 + 10 T + 110 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.53.k_eg
59$D_{4}$ \( 1 + 6 T + 106 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_ec
61$D_{4}$ \( 1 + 2 T - 66 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_aco
67$D_{4}$ \( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.67.k_cx
71$D_{4}$ \( 1 + 10 T + 146 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.71.k_fq
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.73.i_gg
79$D_{4}$ \( 1 - 6 T + 146 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.79.ag_fq
83$D_{4}$ \( 1 - 14 T + 194 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.83.ao_hm
89$D_{4}$ \( 1 - 12 T + 193 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_hl
97$D_{4}$ \( 1 + 14 T + 222 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.97.o_io
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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

−9.251004163357870746073627954751, −9.162116754132302553154066006700, −8.442482924588290781486228239667, −7.78274481667225042864320658486, −7.65815231598995003054480680296, −7.10101946156118560670678427286, −6.71364743734635390796179695737, −6.30162114434693409334273495843, −6.28682244898965485177071784677, −6.03722914197282991917795322043, −5.17675596705504956908515818125, −4.77881315004347918853515289514, −4.48147268034813259344525036065, −4.13960371082335566514743268559, −3.47385968027462250930628983347, −3.23214326805480175550137597333, −2.58501583039268858435081274307, −1.49958259453774684915816459667, −1.39167544269547412145664125992, −0.881935906972457734636824661786, 0.881935906972457734636824661786, 1.39167544269547412145664125992, 1.49958259453774684915816459667, 2.58501583039268858435081274307, 3.23214326805480175550137597333, 3.47385968027462250930628983347, 4.13960371082335566514743268559, 4.48147268034813259344525036065, 4.77881315004347918853515289514, 5.17675596705504956908515818125, 6.03722914197282991917795322043, 6.28682244898965485177071784677, 6.30162114434693409334273495843, 6.71364743734635390796179695737, 7.10101946156118560670678427286, 7.65815231598995003054480680296, 7.78274481667225042864320658486, 8.442482924588290781486228239667, 9.162116754132302553154066006700, 9.251004163357870746073627954751

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