Properties

Label 4-3200e2-1.1-c0e2-0-4
Degree $4$
Conductor $10240000$
Sign $1$
Analytic cond. $2.55043$
Root an. cond. $1.26372$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·13-s − 2·17-s + 4·29-s + 2·37-s − 2·53-s + 2·73-s − 81-s + 2·97-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 + 199-s + ⋯
L(s)  = 1  + 2·13-s − 2·17-s + 4·29-s + 2·37-s − 2·53-s + 2·73-s − 81-s + 2·97-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 + 199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10240000\)    =    \(2^{14} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2.55043\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10240000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.641697496\)
\(L(\frac12)\) \(\approx\) \(1.641697496\)
\(L(1)\) 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 \( 1 \)
good3$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_1$ \( ( 1 - T )^{4} \)
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$C_2$ \( ( 1 + T^{2} )^{2} \)
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

−8.898343517022490814201378567211, −8.630053899975606689695176076637, −8.253870150607420174793273787104, −8.186138585564438907729875065059, −7.61763891168823077212505296624, −7.06396529055362152126001760228, −6.58140971038435988410035407164, −6.45098444084630268463399899097, −6.06862884331546124309552301122, −5.95113047459660457500579286515, −4.98006158610815547455180929820, −4.78761345212855034210472940620, −4.38038051118067235606833245271, −4.13953403024065277636310202923, −3.29244737107063561412180363178, −3.24224194484881919221777398388, −2.41924396642072560160029415616, −2.22088826751870483828544639037, −1.25243281903463886346800228200, −0.914987420435682091305445612252, 0.914987420435682091305445612252, 1.25243281903463886346800228200, 2.22088826751870483828544639037, 2.41924396642072560160029415616, 3.24224194484881919221777398388, 3.29244737107063561412180363178, 4.13953403024065277636310202923, 4.38038051118067235606833245271, 4.78761345212855034210472940620, 4.98006158610815547455180929820, 5.95113047459660457500579286515, 6.06862884331546124309552301122, 6.45098444084630268463399899097, 6.58140971038435988410035407164, 7.06396529055362152126001760228, 7.61763891168823077212505296624, 8.186138585564438907729875065059, 8.253870150607420174793273787104, 8.630053899975606689695176076637, 8.898343517022490814201378567211

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