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Properties

Label 4-3240e2-1.1-c1e2-0-36
Degree $4$
Conductor $10497600$
Sign $1$
Analytic cond. $669.336$
Root an. cond. $5.08640$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 2·7-s + 4·11-s + 2·13-s − 10·17-s − 10·19-s + 23-s − 2·29-s − 7·31-s − 2·35-s − 12·37-s − 4·43-s + 4·47-s + 7·49-s − 18·53-s + 4·55-s + 14·59-s + 11·61-s + 2·65-s − 14·67-s − 24·73-s − 8·77-s + 3·79-s − 83-s − 10·85-s − 4·91-s − 10·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s + 1.20·11-s + 0.554·13-s − 2.42·17-s − 2.29·19-s + 0.208·23-s − 0.371·29-s − 1.25·31-s − 0.338·35-s − 1.97·37-s − 0.609·43-s + 0.583·47-s + 49-s − 2.47·53-s + 0.539·55-s + 1.82·59-s + 1.40·61-s + 0.248·65-s − 1.71·67-s − 2.80·73-s − 0.911·77-s + 0.337·79-s − 0.109·83-s − 1.08·85-s − 0.419·91-s − 1.02·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10497600\)    =    \(2^{6} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(669.336\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 10497600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
good7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_ad
11$C_2^2$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_f
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.ac_aj
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.17.k_ch
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.19.k_cl
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) 2.23.ab_aw
29$C_2^2$ \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_az
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.h_s
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.41.a_abp
43$C_2^2$ \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_abb
47$C_2^2$ \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_abf
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.53.s_hf
59$C_2^2$ \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.59.ao_fh
61$C_2^2$ \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.61.al_ci
67$C_2^2$ \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.67.o_ez
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.73.y_le
79$C_2^2$ \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.79.ad_acs
83$C_2^2$ \( 1 + T - 82 T^{2} + p T^{3} + p^{2} T^{4} \) 2.83.b_ade
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2^2$ \( 1 + 16 T + 159 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.97.q_gd
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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

−8.652771486281700802270853949903, −8.481685264826619326907952318483, −7.53529207026854980876950084569, −7.26938505773011698913794415544, −6.84404953838356760133599024243, −6.51370929533518965043793197687, −6.29118350891873887849157861984, −6.09068094246640675306117823439, −5.42422349853287108314339035297, −5.05142703048046376322261375222, −4.31741149185476075733361302176, −4.31675408822436863934957868959, −3.65327248779224623622391518193, −3.55759338673608571972482624234, −2.58833484462354392581394954370, −2.37047146682838022570745201339, −1.70125352708718831020417605578, −1.45319610744050266347252752361, 0, 0, 1.45319610744050266347252752361, 1.70125352708718831020417605578, 2.37047146682838022570745201339, 2.58833484462354392581394954370, 3.55759338673608571972482624234, 3.65327248779224623622391518193, 4.31675408822436863934957868959, 4.31741149185476075733361302176, 5.05142703048046376322261375222, 5.42422349853287108314339035297, 6.09068094246640675306117823439, 6.29118350891873887849157861984, 6.51370929533518965043793197687, 6.84404953838356760133599024243, 7.26938505773011698913794415544, 7.53529207026854980876950084569, 8.481685264826619326907952318483, 8.652771486281700802270853949903

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