Properties

Label 4-320e2-1.1-c1e2-0-43
Degree $4$
Conductor $102400$
Sign $-1$
Analytic cond. $6.52911$
Root an. cond. $1.59850$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·9-s − 4·13-s + 25-s − 4·29-s − 16·37-s − 4·41-s + 2·49-s − 12·53-s − 5·81-s + 28·89-s − 16·97-s + 4·101-s + 24·109-s − 12·113-s + 8·117-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯
L(s)  = 1  − 2/3·9-s − 1.10·13-s + 1/5·25-s − 0.742·29-s − 2.63·37-s − 0.624·41-s + 2/7·49-s − 1.64·53-s − 5/9·81-s + 2.96·89-s − 1.62·97-s + 0.398·101-s + 2.29·109-s − 1.12·113-s + 0.739·117-s − 1.27·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-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: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 102400,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 10 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−9.161594690232246628047892723108, −8.928014109893746457580703256021, −8.366427849605726682718947495790, −7.78447817217622269241352141702, −7.34756400886185776811640070971, −6.82159304242439098453208036187, −6.29926616451589023950913375931, −5.62740191934179604340314048597, −5.07378980852191858617083112838, −4.75834455255754319948669802838, −3.76468828143529001580735881412, −3.26976264373048953038799305871, −2.47496385171731515848978398864, −1.68490315353034793653253970915, 0, 1.68490315353034793653253970915, 2.47496385171731515848978398864, 3.26976264373048953038799305871, 3.76468828143529001580735881412, 4.75834455255754319948669802838, 5.07378980852191858617083112838, 5.62740191934179604340314048597, 6.29926616451589023950913375931, 6.82159304242439098453208036187, 7.34756400886185776811640070971, 7.78447817217622269241352141702, 8.366427849605726682718947495790, 8.928014109893746457580703256021, 9.161594690232246628047892723108

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