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Properties

Label 4-1224e2-1.1-c1e2-0-47
Degree $4$
Conductor $1498176$
Sign $1$
Analytic cond. $95.5250$
Root an. cond. $3.12629$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 2·7-s − 2·11-s − 2·17-s − 4·19-s − 2·23-s + 2·25-s − 4·29-s − 2·31-s − 8·35-s − 4·37-s − 4·41-s − 12·43-s − 8·47-s − 6·49-s + 4·53-s + 8·55-s − 20·59-s − 4·61-s − 24·67-s − 14·71-s + 12·73-s − 4·77-s + 10·79-s − 12·83-s + 8·85-s + 24·89-s + ⋯
L(s)  = 1  − 1.78·5-s + 0.755·7-s − 0.603·11-s − 0.485·17-s − 0.917·19-s − 0.417·23-s + 2/5·25-s − 0.742·29-s − 0.359·31-s − 1.35·35-s − 0.657·37-s − 0.624·41-s − 1.82·43-s − 1.16·47-s − 6/7·49-s + 0.549·53-s + 1.07·55-s − 2.60·59-s − 0.512·61-s − 2.93·67-s − 1.66·71-s + 1.40·73-s − 0.455·77-s + 1.12·79-s − 1.31·83-s + 0.867·85-s + 2.54·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1498176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1498176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1498176\)    =    \(2^{6} \cdot 3^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(95.5250\)
Root analytic conductor: \(3.12629\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1498176,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
7$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_k
11$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_s
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
19$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_w
23$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_bq
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$D_{4}$ \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_cg
37$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_ac
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_dy
47$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_be
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.53.ae_eg
59$D_{4}$ \( 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.59.u_hq
61$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_bu
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$D_{4}$ \( 1 + 14 T + 186 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.71.o_he
73$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.73.am_dy
79$D_{4}$ \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.79.ak_fi
83$D_{4}$ \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_ha
89$D_{4}$ \( 1 - 24 T + 302 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.89.ay_lq
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−9.298828244529765316303193846723, −9.131772038426623968387761404142, −8.523708651450445713268596067504, −8.216077719992776177045841195155, −7.79289989260832537813691163972, −7.73944197492634853259667144352, −7.20563738270572016034536531760, −6.67938150915946395242158480830, −6.22207169980315580295092588391, −5.76737397285629147308665589277, −4.90779661389959046205516543029, −4.90708072138503181669603657872, −4.27479141814712282387756575600, −3.91523460680044068241241887717, −3.30350529781361205179862726188, −2.98951382154881400235982078814, −1.83639678084949254834894847007, −1.72835408891533948993926801453, 0, 0, 1.72835408891533948993926801453, 1.83639678084949254834894847007, 2.98951382154881400235982078814, 3.30350529781361205179862726188, 3.91523460680044068241241887717, 4.27479141814712282387756575600, 4.90708072138503181669603657872, 4.90779661389959046205516543029, 5.76737397285629147308665589277, 6.22207169980315580295092588391, 6.67938150915946395242158480830, 7.20563738270572016034536531760, 7.73944197492634853259667144352, 7.79289989260832537813691163972, 8.216077719992776177045841195155, 8.523708651450445713268596067504, 9.131772038426623968387761404142, 9.298828244529765316303193846723

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