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Properties

Label 4-320e2-1.1-c0e2-0-0
Degree $4$
Conductor $102400$
Sign $1$
Analytic cond. $0.0255043$
Root an. cond. $0.399625$
Motivic weight $0$
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  + 2·13-s − 2·17-s − 25-s − 2·37-s − 2·53-s + 2·73-s − 81-s + 2·97-s + 4·101-s − 2·113-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 2·13-s − 2·17-s − 25-s − 2·37-s − 2·53-s + 2·73-s − 81-s + 2·97-s + 4·101-s − 2·113-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(102400\)    =    \(2^{12} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.0255043\)
Root analytic conductor: \(0.399625\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 102400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6256554718\)
\(L(\frac12)\) \(\approx\) \(0.6256554718\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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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

−11.83342670478939600778000367090, −11.70730757930274210052070860578, −10.99823526752159599260567792653, −10.85084804999473395270961997714, −10.42385273045418601749797180161, −9.718829672758755078742899264270, −9.137370007529070708115527314309, −8.887964672811984972068635532586, −8.315241678892954349982762143012, −8.032732310269770728252854160536, −7.18830048997100790055883055519, −6.76267191062233945982580644459, −6.12915681482321798350687279061, −5.98658972706148741961646032438, −4.99971014154535995678329749748, −4.58466241709766521529310591437, −3.67342203931195572136873608633, −3.52007095924675564889294389541, −2.31263235827859086969458246725, −1.59942077213077209988873457183, 1.59942077213077209988873457183, 2.31263235827859086969458246725, 3.52007095924675564889294389541, 3.67342203931195572136873608633, 4.58466241709766521529310591437, 4.99971014154535995678329749748, 5.98658972706148741961646032438, 6.12915681482321798350687279061, 6.76267191062233945982580644459, 7.18830048997100790055883055519, 8.032732310269770728252854160536, 8.315241678892954349982762143012, 8.887964672811984972068635532586, 9.137370007529070708115527314309, 9.718829672758755078742899264270, 10.42385273045418601749797180161, 10.85084804999473395270961997714, 10.99823526752159599260567792653, 11.70730757930274210052070860578, 11.83342670478939600778000367090

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