Properties

Label 4-80e4-1.1-c1e2-0-32
Degree $4$
Conductor $40960000$
Sign $1$
Analytic cond. $2611.64$
Root an. cond. $7.14872$
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  − 6·9-s − 4·11-s + 12·19-s − 4·41-s + 6·49-s − 28·59-s + 27·81-s − 28·89-s + 24·99-s − 10·121-s + ⋯
L(s)  = 1  − 2·9-s − 1.20·11-s + 2.75·19-s − 0.624·41-s + 6/7·49-s − 3.64·59-s + 3·81-s − 2.96·89-s + 2.41·99-s − 0.909·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40960000\)    =    \(2^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2611.64\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 40960000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.19.am_cw
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.23.a_ba
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.37.a_cc
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.47.a_adi
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.53.a_acw
59$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.59.bc_mc
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.67.a_fe
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.73.a_fq
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.89.bc_ok
97$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.97.a_hm
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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

−7.75229516030121008740640202959, −7.65353314713179511761454547516, −7.26969093873281983373662725382, −6.87294975745001865655411901225, −6.19733434677069709919580948968, −6.10213792461213744027679911261, −5.70322339308376864081634271527, −5.26216740694083247746134469576, −5.06517884846806027744058186803, −4.94295306759318707148283033441, −4.15656857014323267417185756525, −3.69909903235276583604249618957, −3.18423756557940995554887083178, −3.02093624972826930217720233343, −2.64309608580125481927829059742, −2.32378814973219626397608918861, −1.40418623019367695513879974830, −1.12277764309779393628894605157, 0, 0, 1.12277764309779393628894605157, 1.40418623019367695513879974830, 2.32378814973219626397608918861, 2.64309608580125481927829059742, 3.02093624972826930217720233343, 3.18423756557940995554887083178, 3.69909903235276583604249618957, 4.15656857014323267417185756525, 4.94295306759318707148283033441, 5.06517884846806027744058186803, 5.26216740694083247746134469576, 5.70322339308376864081634271527, 6.10213792461213744027679911261, 6.19733434677069709919580948968, 6.87294975745001865655411901225, 7.26969093873281983373662725382, 7.65353314713179511761454547516, 7.75229516030121008740640202959

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