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Properties

Label 4-2560-1.1-c1e2-0-0
Degree $4$
Conductor $2560$
Sign $1$
Analytic cond. $0.163227$
Root an. cond. $0.635621$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 2·9-s − 4·13-s − 4·17-s − 2·25-s − 4·29-s + 4·37-s + 4·41-s − 2·45-s + 2·49-s − 4·53-s + 20·61-s + 4·65-s + 12·73-s − 5·81-s + 4·85-s + 4·89-s − 4·97-s − 4·101-s + 4·109-s − 28·113-s − 8·117-s + 10·121-s + 10·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.447·5-s + 2/3·9-s − 1.10·13-s − 0.970·17-s − 2/5·25-s − 0.742·29-s + 0.657·37-s + 0.624·41-s − 0.298·45-s + 2/7·49-s − 0.549·53-s + 2.56·61-s + 0.496·65-s + 1.40·73-s − 5/9·81-s + 0.433·85-s + 0.423·89-s − 0.406·97-s − 0.398·101-s + 0.383·109-s − 2.63·113-s − 0.739·117-s + 0.909·121-s + 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2560 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2560\)    =    \(2^{9} \cdot 5\)
Sign: $1$
Analytic conductor: \(0.163227\)
Root analytic conductor: \(0.635621\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2560,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6563117485\)
\(L(\frac12)\) \(\approx\) \(0.6563117485\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.e_w
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.23.a_abi
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ae_ck
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.ae_w
43$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.43.a_ck
47$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.47.a_ac
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.e_bu
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.59.a_bm
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.61.au_hy
67$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.67.a_as
71$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.71.a_bu
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.73.am_gk
79$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.79.a_abi
83$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \) 2.83.a_ew
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ae_eo
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.97.e_ha
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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

−12.95976264669693540025589192078, −12.48367628325372425340654340374, −11.80372322771740053448933326215, −11.23611624623193562674582428404, −10.65505975474798628359860403767, −9.774070737967069714239308188778, −9.480468180241242559684660262579, −8.555161022741242013168039523731, −7.81855471798383057814968612224, −7.20447908396518733838295689153, −6.57158847385268159714087019173, −5.49604786544735417724275437402, −4.59656659721016932035456370597, −3.83789795478194203908952192251, −2.35673748189235633398390138135, 2.35673748189235633398390138135, 3.83789795478194203908952192251, 4.59656659721016932035456370597, 5.49604786544735417724275437402, 6.57158847385268159714087019173, 7.20447908396518733838295689153, 7.81855471798383057814968612224, 8.555161022741242013168039523731, 9.480468180241242559684660262579, 9.774070737967069714239308188778, 10.65505975474798628359860403767, 11.23611624623193562674582428404, 11.80372322771740053448933326215, 12.48367628325372425340654340374, 12.95976264669693540025589192078

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