Properties

Label 4-320e2-1.1-c1e2-0-36
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  − 4·7-s + 4·17-s − 4·23-s − 25-s − 8·31-s − 8·41-s + 4·47-s − 8·71-s − 4·73-s − 9·81-s − 4·89-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 16·119-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 16·161-s + ⋯
L(s)  = 1  − 1.51·7-s + 0.970·17-s − 0.834·23-s − 1/5·25-s − 1.43·31-s − 1.24·41-s + 0.583·47-s − 0.949·71-s − 0.468·73-s − 81-s − 0.423·89-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.46·119-s + 2/11·121-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 + 0.0798·157-s + 1.26·161-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$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$D_{4}$ \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 64 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 80 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−14.2151067192, −13.8039112904, −13.2899048446, −12.9128056800, −12.6158677963, −12.0671157476, −11.7615203711, −11.1848198677, −10.5947354095, −10.0508482539, −9.94138507393, −9.35783223000, −8.90142020079, −8.40167910522, −7.67834979289, −7.39129337813, −6.69484144053, −6.31722754152, −5.70933179667, −5.32752819203, −4.45634499647, −3.68781758863, −3.36447939541, −2.60553505577, −1.57217239405, 0, 1.57217239405, 2.60553505577, 3.36447939541, 3.68781758863, 4.45634499647, 5.32752819203, 5.70933179667, 6.31722754152, 6.69484144053, 7.39129337813, 7.67834979289, 8.40167910522, 8.90142020079, 9.35783223000, 9.94138507393, 10.0508482539, 10.5947354095, 11.1848198677, 11.7615203711, 12.0671157476, 12.6158677963, 12.9128056800, 13.2899048446, 13.8039112904, 14.2151067192

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