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Properties

Label 4-2646e2-1.1-c1e2-0-12
Degree $4$
Conductor $7001316$
Sign $1$
Analytic cond. $446.409$
Root an. cond. $4.59656$
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 − 4·8-s − 3·11-s − 2·13-s + 5·16-s − 3·17-s + 19-s + 6·22-s − 6·23-s + 5·25-s + 4·26-s + 6·29-s − 8·31-s − 6·32-s + 6·34-s + 4·37-s − 2·38-s + 9·41-s + 43-s − 9·44-s + 12·46-s + 12·47-s − 10·50-s − 6·52-s + 12·53-s − 12·58-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s − 0.904·11-s − 0.554·13-s + 5/4·16-s − 0.727·17-s + 0.229·19-s + 1.27·22-s − 1.25·23-s + 25-s + 0.784·26-s + 1.11·29-s − 1.43·31-s − 1.06·32-s + 1.02·34-s + 0.657·37-s − 0.324·38-s + 1.40·41-s + 0.152·43-s − 1.35·44-s + 1.76·46-s + 1.75·47-s − 1.41·50-s − 0.832·52-s + 1.64·53-s − 1.57·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7001316\)    =    \(2^{2} \cdot 3^{6} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(446.409\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7001316,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.120542259\)
\(L(\frac12)\) \(\approx\) \(1.120542259\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.c_aj
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.ab_as
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_n
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_h
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2^2$ \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_av
41$C_2^2$ \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_bo
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) 2.43.ab_abq
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.47.am_fa
53$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.53.am_dn
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.59.g_ex
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.61.aq_he
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.67.ak_gd
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.73.l_bw
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_cj
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_acb
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) 2.97.f_acu
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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

−8.923651824034210541314266152522, −8.851120212131100275666077497982, −8.132841470099691173515657155274, −8.093500395163308513568710186676, −7.65015655580751846149006326545, −7.23089837584664638546175356192, −6.82712407970766879288589273264, −6.71158262676028955317412597818, −5.87058900706700462779698170274, −5.81270967541070715068806227330, −5.24573033346363685696718462661, −4.84857703102516482141371735729, −4.08403213183626612063990825431, −3.97438039560080224251254609982, −3.09595343212417031009774622149, −2.69939529476788285965806656157, −2.09315382817874986257842745361, −2.08747718039733568228513271301, −0.841499227779318127024314506398, −0.59159463084271176201386647019, 0.59159463084271176201386647019, 0.841499227779318127024314506398, 2.08747718039733568228513271301, 2.09315382817874986257842745361, 2.69939529476788285965806656157, 3.09595343212417031009774622149, 3.97438039560080224251254609982, 4.08403213183626612063990825431, 4.84857703102516482141371735729, 5.24573033346363685696718462661, 5.81270967541070715068806227330, 5.87058900706700462779698170274, 6.71158262676028955317412597818, 6.82712407970766879288589273264, 7.23089837584664638546175356192, 7.65015655580751846149006326545, 8.093500395163308513568710186676, 8.132841470099691173515657155274, 8.851120212131100275666077497982, 8.923651824034210541314266152522

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