Properties

Label 4-320e2-1.1-c1e2-0-30
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  + 2·3-s + 2·5-s − 2·7-s + 2·9-s + 8·11-s + 6·13-s + 4·15-s − 6·17-s − 4·21-s − 6·23-s − 25-s + 6·27-s − 4·29-s + 16·33-s − 4·35-s + 6·37-s + 12·39-s + 12·41-s − 6·43-s + 4·45-s − 18·47-s + 2·49-s − 12·51-s − 10·53-s + 16·55-s − 4·63-s + 12·65-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s − 0.755·7-s + 2/3·9-s + 2.41·11-s + 1.66·13-s + 1.03·15-s − 1.45·17-s − 0.872·21-s − 1.25·23-s − 1/5·25-s + 1.15·27-s − 0.742·29-s + 2.78·33-s − 0.676·35-s + 0.986·37-s + 1.92·39-s + 1.87·41-s − 0.914·43-s + 0.596·45-s − 2.62·47-s + 2/7·49-s − 1.68·51-s − 1.37·53-s + 2.15·55-s − 0.503·63-s + 1.48·65-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: induced by $\chi_{320} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 102400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.848788027\)
\(L(\frac12)\) \(\approx\) \(2.848788027\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
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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

−11.68837005239998595916829500718, −11.42970449629801196089870221042, −11.02032927822570671658654264873, −10.33616522897688550485706945452, −9.717172452821239387655322594668, −9.462901133023270393262549780446, −9.071570556925653699035058463326, −8.801708660031791190087577861401, −8.203404459831289937590137128669, −7.76297179005460585182312075616, −6.70417239734258928387200378300, −6.56410494578771539097241739500, −6.21382746456286194991743353501, −5.71506287885804007776026299680, −4.43660479879381727218012065668, −4.16317567156087977885479179183, −3.52281069771939813450087708539, −2.96927320154170829308493493698, −1.93694329587974974996183938080, −1.44439507623810985111621831824, 1.44439507623810985111621831824, 1.93694329587974974996183938080, 2.96927320154170829308493493698, 3.52281069771939813450087708539, 4.16317567156087977885479179183, 4.43660479879381727218012065668, 5.71506287885804007776026299680, 6.21382746456286194991743353501, 6.56410494578771539097241739500, 6.70417239734258928387200378300, 7.76297179005460585182312075616, 8.203404459831289937590137128669, 8.801708660031791190087577861401, 9.071570556925653699035058463326, 9.462901133023270393262549780446, 9.717172452821239387655322594668, 10.33616522897688550485706945452, 11.02032927822570671658654264873, 11.42970449629801196089870221042, 11.68837005239998595916829500718

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