Properties

Label 4-1280e2-1.1-c1e2-0-47
Degree $4$
Conductor $1638400$
Sign $1$
Analytic cond. $104.465$
Root an. cond. $3.19700$
Motivic weight $1$
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  + 8·7-s + 6·9-s + 4·17-s − 8·23-s − 25-s − 16·31-s + 12·41-s + 8·47-s + 34·49-s + 48·63-s + 12·73-s + 27·81-s + 12·89-s − 28·97-s − 8·103-s + 36·113-s + 32·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s − 64·161-s + ⋯
L(s)  = 1  + 3.02·7-s + 2·9-s + 0.970·17-s − 1.66·23-s − 1/5·25-s − 2.87·31-s + 1.87·41-s + 1.16·47-s + 34/7·49-s + 6.04·63-s + 1.40·73-s + 3·81-s + 1.27·89-s − 2.84·97-s − 0.788·103-s + 3.38·113-s + 2.93·119-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.94·153-s + 0.0798·157-s − 5.04·161-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1638400\)    =    \(2^{16} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(104.465\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1280} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1638400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.406960782\)
\(L(\frac12)\) \(\approx\) \(4.406960782\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−9.783476976910268193793207618104, −9.556883724283425538484619981150, −9.049720005539771589039334819287, −8.601478980000924470268935778253, −7.934998029268371165969628099734, −7.920013100322538625068420066873, −7.45612804385445602217895831863, −7.35788545093417771082675510674, −6.72752498027776395137044191321, −5.98427364107726571581140300943, −5.41392899746999746386039428844, −5.38677145049316631031482791381, −4.60988225711371971150483130425, −4.40106414574905105150304509796, −3.91640263377770793539187093723, −3.60891605358348380574518074359, −2.27093942794656894904473638871, −2.00603431714875057107714298675, −1.50613779696024433616730292323, −1.01509157121117666870023522294, 1.01509157121117666870023522294, 1.50613779696024433616730292323, 2.00603431714875057107714298675, 2.27093942794656894904473638871, 3.60891605358348380574518074359, 3.91640263377770793539187093723, 4.40106414574905105150304509796, 4.60988225711371971150483130425, 5.38677145049316631031482791381, 5.41392899746999746386039428844, 5.98427364107726571581140300943, 6.72752498027776395137044191321, 7.35788545093417771082675510674, 7.45612804385445602217895831863, 7.920013100322538625068420066873, 7.934998029268371165969628099734, 8.601478980000924470268935778253, 9.049720005539771589039334819287, 9.556883724283425538484619981150, 9.783476976910268193793207618104

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