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Properties

Label 4-320e2-1.1-c1e2-0-39
Degree $4$
Conductor $102400$
Sign $-1$
Analytic cond. $6.52911$
Root an. cond. $1.59850$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·9-s − 4·17-s − 25-s − 12·49-s − 12·73-s + 7·81-s − 20·89-s − 28·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 16·153-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 4/3·9-s − 0.970·17-s − 1/5·25-s − 1.71·49-s − 1.40·73-s + 7/9·81-s − 2.11·89-s − 2.84·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.29·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(102400\)    =    \(2^{12} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(6.52911\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 102400,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.7.a_m
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.19.a_g
23$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.23.a_bs
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.31.a_ak
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \) 2.43.a_bk
47$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.47.a_ae
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.53.a_ady
59$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.59.a_di
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.a_aw
67$C_2^2$ \( 1 + 132 T^{2} + p^{2} T^{4} \) 2.67.a_fc
71$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.71.a_acg
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.79.a_be
83$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.83.a_e
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.97.bc_pa
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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

−14.2244947249, −13.8835788771, −13.4256287250, −12.9397621340, −12.5581414954, −12.0061877425, −11.4727193146, −11.2521344165, −10.8877080002, −10.2120909889, −9.80530603877, −9.19181875628, −8.83705824525, −8.33387907023, −7.97955559825, −7.31878569452, −6.68249073725, −6.31643406985, −5.64491259488, −5.26218660734, −4.51024177862, −3.95375223140, −3.06247334895, −2.63154732816, −1.63490624320, 0, 1.63490624320, 2.63154732816, 3.06247334895, 3.95375223140, 4.51024177862, 5.26218660734, 5.64491259488, 6.31643406985, 6.68249073725, 7.31878569452, 7.97955559825, 8.33387907023, 8.83705824525, 9.19181875628, 9.80530603877, 10.2120909889, 10.8877080002, 11.2521344165, 11.4727193146, 12.0061877425, 12.5581414954, 12.9397621340, 13.4256287250, 13.8835788771, 14.2244947249

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