Properties

Label 4-320e2-1.1-c1e2-0-10
Degree $4$
Conductor $102400$
Sign $1$
Analytic cond. $6.52911$
Root an. cond. $1.59850$
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  + 2·3-s − 2·5-s − 2·7-s + 2·9-s + 2·13-s − 4·15-s + 2·17-s − 8·19-s − 4·21-s + 10·23-s − 25-s + 6·27-s + 4·35-s + 10·37-s + 4·39-s + 12·41-s − 6·43-s − 4·45-s + 14·47-s + 2·49-s + 4·51-s + 2·53-s − 16·57-s − 8·59-s − 4·61-s − 4·63-s − 4·65-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s − 0.755·7-s + 2/3·9-s + 0.554·13-s − 1.03·15-s + 0.485·17-s − 1.83·19-s − 0.872·21-s + 2.08·23-s − 1/5·25-s + 1.15·27-s + 0.676·35-s + 1.64·37-s + 0.640·39-s + 1.87·41-s − 0.914·43-s − 0.596·45-s + 2.04·47-s + 2/7·49-s + 0.560·51-s + 0.274·53-s − 2.11·57-s − 1.04·59-s − 0.512·61-s − 0.503·63-s − 0.496·65-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: induced by $\chi_{320} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 102400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.806256515\)
\(L(\frac12)\) \(\approx\) \(1.806256515\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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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

−11.92828920887095361776041972633, −11.28628376592839010459808379064, −10.78840241851468527504478962394, −10.69693638849651829150433791638, −9.897535205114377250911723937345, −9.310491780430194098450854394124, −9.086184361996998785091766103786, −8.578585476202832637249814237153, −8.083294707773213364670804230382, −7.76114321371533072991784338575, −7.11186202054915336867025013045, −6.63788300244926264859356528384, −6.15927039959388504878409697851, −5.41066169742392480345008753508, −4.53537320768920545153705018769, −4.08831241527517536506397399518, −3.56947203076070380168261943174, −2.83914565713673064494934908551, −2.42352590918712853466567242207, −0.965135808481614986405658292943, 0.965135808481614986405658292943, 2.42352590918712853466567242207, 2.83914565713673064494934908551, 3.56947203076070380168261943174, 4.08831241527517536506397399518, 4.53537320768920545153705018769, 5.41066169742392480345008753508, 6.15927039959388504878409697851, 6.63788300244926264859356528384, 7.11186202054915336867025013045, 7.76114321371533072991784338575, 8.083294707773213364670804230382, 8.578585476202832637249814237153, 9.086184361996998785091766103786, 9.310491780430194098450854394124, 9.897535205114377250911723937345, 10.69693638849651829150433791638, 10.78840241851468527504478962394, 11.28628376592839010459808379064, 11.92828920887095361776041972633

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