Properties

Label 4-800080-1.1-c1e2-0-1
Degree $4$
Conductor $800080$
Sign $1$
Analytic cond. $51.0137$
Root an. cond. $2.67252$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 3·5-s − 3·8-s + 2·9-s + 3·10-s − 6·13-s − 16-s + 2·17-s + 2·18-s − 3·20-s + 2·25-s − 6·26-s − 6·29-s + 5·32-s + 2·34-s − 2·36-s − 8·37-s − 9·40-s + 18·41-s + 6·45-s + 10·49-s + 2·50-s + 6·52-s + 10·53-s − 6·58-s + 12·61-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 1.34·5-s − 1.06·8-s + 2/3·9-s + 0.948·10-s − 1.66·13-s − 1/4·16-s + 0.485·17-s + 0.471·18-s − 0.670·20-s + 2/5·25-s − 1.17·26-s − 1.11·29-s + 0.883·32-s + 0.342·34-s − 1/3·36-s − 1.31·37-s − 1.42·40-s + 2.81·41-s + 0.894·45-s + 10/7·49-s + 0.282·50-s + 0.832·52-s + 1.37·53-s − 0.787·58-s + 1.53·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800080 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800080 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(800080\)    =    \(2^{4} \cdot 5 \cdot 73 \cdot 137\)
Sign: $1$
Analytic conductor: \(51.0137\)
Root analytic conductor: \(2.67252\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{800080} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 800080,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.749088267\)
\(L(\frac12)\) \(\approx\) \(2.749088267\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
73$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 6 T + p T^{2} ) \)
137$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 14 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 52 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−8.259953185648721717585060390559, −7.67831906356609756105753799076, −7.24918280352642515111724279751, −6.94023627621197936407396451010, −6.33705296122873803052343305263, −5.69333062404338169750038707160, −5.45716280393938487502005590404, −5.30447879159785385783089479861, −4.46893004561203578380230576655, −4.16240664100987404480577634716, −3.64585304810701299951511676538, −2.76993440073162879110471823217, −2.40272225978440983140143660772, −1.78645617020287955358854774712, −0.71345096422148734638967933716, 0.71345096422148734638967933716, 1.78645617020287955358854774712, 2.40272225978440983140143660772, 2.76993440073162879110471823217, 3.64585304810701299951511676538, 4.16240664100987404480577634716, 4.46893004561203578380230576655, 5.30447879159785385783089479861, 5.45716280393938487502005590404, 5.69333062404338169750038707160, 6.33705296122873803052343305263, 6.94023627621197936407396451010, 7.24918280352642515111724279751, 7.67831906356609756105753799076, 8.259953185648721717585060390559

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