Properties

Label 4-320e2-1.1-c1e2-0-29
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  + 4·3-s − 4·5-s + 6·7-s + 6·9-s + 2·11-s − 16·15-s + 2·17-s + 6·19-s + 24·21-s + 2·23-s + 11·25-s − 4·27-s − 14·29-s + 8·33-s − 24·35-s − 24·45-s + 14·47-s + 18·49-s + 8·51-s − 16·53-s − 8·55-s + 24·57-s − 6·59-s − 2·61-s + 36·63-s + 8·69-s + 6·73-s + ⋯
L(s)  = 1  + 2.30·3-s − 1.78·5-s + 2.26·7-s + 2·9-s + 0.603·11-s − 4.13·15-s + 0.485·17-s + 1.37·19-s + 5.23·21-s + 0.417·23-s + 11/5·25-s − 0.769·27-s − 2.59·29-s + 1.39·33-s − 4.05·35-s − 3.57·45-s + 2.04·47-s + 18/7·49-s + 1.12·51-s − 2.19·53-s − 1.07·55-s + 3.17·57-s − 0.781·59-s − 0.256·61-s + 4.53·63-s + 0.963·69-s + 0.702·73-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 102400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.151576536\)
\(L(\frac12)\) \(\approx\) \(3.151576536\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
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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

−11.50814938353155228541648729155, −11.49078568171744158644264814125, −11.18527061447747119851428558245, −10.68888766525442977472543815130, −9.528214016195289226326838627861, −9.478182554347819723251539097819, −8.742467991714931292882188491978, −8.545716628297638734549363976848, −8.085843928995543131546044172507, −7.69993671153038351534142081142, −7.35378707532596062831670289749, −7.27628014750412082123667030193, −5.77524046318002532213843835306, −5.29173233385468064927013894560, −4.42050212556708163117585892987, −4.15979097804442808367545409375, −3.30341074745633033539460164973, −3.22925393380734205009349706187, −2.11640262027913904639651854572, −1.40955617353133347827143458429, 1.40955617353133347827143458429, 2.11640262027913904639651854572, 3.22925393380734205009349706187, 3.30341074745633033539460164973, 4.15979097804442808367545409375, 4.42050212556708163117585892987, 5.29173233385468064927013894560, 5.77524046318002532213843835306, 7.27628014750412082123667030193, 7.35378707532596062831670289749, 7.69993671153038351534142081142, 8.085843928995543131546044172507, 8.545716628297638734549363976848, 8.742467991714931292882188491978, 9.478182554347819723251539097819, 9.528214016195289226326838627861, 10.68888766525442977472543815130, 11.18527061447747119851428558245, 11.49078568171744158644264814125, 11.50814938353155228541648729155

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