Properties

Label 4-2880e2-1.1-c0e2-0-0
Degree $4$
Conductor $8294400$
Sign $1$
Analytic cond. $2.06585$
Root an. cond. $1.19887$
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  − 2·5-s − 2·13-s − 2·17-s + 3·25-s − 2·37-s − 4·41-s − 2·53-s + 4·65-s − 2·73-s + 4·85-s − 2·97-s + 2·113-s − 2·121-s − 4·125-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 + ⋯
L(s)  = 1  − 2·5-s − 2·13-s − 2·17-s + 3·25-s − 2·37-s − 4·41-s − 2·53-s + 4·65-s − 2·73-s + 4·85-s − 2·97-s + 2·113-s − 2·121-s − 4·125-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8294400\)    =    \(2^{12} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2.06585\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8294400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05943376493\)
\(L(\frac12)\) \(\approx\) \(0.05943376493\)
\(L(1)\) 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 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 + T^{4} \)
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$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
41$C_1$ \( ( 1 + T )^{4} \)
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$C_2$ \( ( 1 + T^{2} )^{2} \)
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

−9.028902312417474995090016858901, −8.623345293231622288704976911294, −8.483496274461369790474295068571, −7.87050787384386015494758656003, −7.80131188430928146097230568953, −7.04927263170452841074601809363, −7.04464824917153803179918194201, −6.70696618814236765793948525271, −6.41112449763050548042067858941, −5.42282872854658132784515926932, −5.16618443262300700895108142875, −4.71980958619243795448462942401, −4.57197749453620855803271023342, −4.05979962730519690993546485919, −3.59614965549997602557157618609, −2.98002060769753690289434480474, −2.92358627559292952382383428145, −1.98397832221700599696582231182, −1.62252415666222125032276672748, −0.14623722214497856810189978782, 0.14623722214497856810189978782, 1.62252415666222125032276672748, 1.98397832221700599696582231182, 2.92358627559292952382383428145, 2.98002060769753690289434480474, 3.59614965549997602557157618609, 4.05979962730519690993546485919, 4.57197749453620855803271023342, 4.71980958619243795448462942401, 5.16618443262300700895108142875, 5.42282872854658132784515926932, 6.41112449763050548042067858941, 6.70696618814236765793948525271, 7.04464824917153803179918194201, 7.04927263170452841074601809363, 7.80131188430928146097230568953, 7.87050787384386015494758656003, 8.483496274461369790474295068571, 8.623345293231622288704976911294, 9.028902312417474995090016858901

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