Properties

Label 4-80000-1.1-c1e2-0-2
Degree $4$
Conductor $80000$
Sign $1$
Analytic cond. $5.10086$
Root an. cond. $1.50283$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s + 4-s − 2·6-s + 8-s − 3·9-s − 2·12-s + 8·13-s + 16-s − 3·18-s − 2·24-s + 8·26-s + 14·27-s + 4·31-s + 32-s − 3·36-s − 4·37-s − 16·39-s − 6·41-s + 8·43-s − 2·48-s − 10·49-s + 8·52-s − 12·53-s + 14·54-s + 4·62-s + 64-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s − 9-s − 0.577·12-s + 2.21·13-s + 1/4·16-s − 0.707·18-s − 0.408·24-s + 1.56·26-s + 2.69·27-s + 0.718·31-s + 0.176·32-s − 1/2·36-s − 0.657·37-s − 2.56·39-s − 0.937·41-s + 1.21·43-s − 0.288·48-s − 1.42·49-s + 1.10·52-s − 1.64·53-s + 1.90·54-s + 0.508·62-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(80000\)    =    \(2^{7} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(5.10086\)
Root analytic conductor: \(1.50283\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 80000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.447513187\)
\(L(\frac12)\) \(\approx\) \(1.447513187\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
5 \( 1 \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−9.930723730317650982781533521356, −9.108795455289480583011416986557, −8.746535765801052966997753191886, −8.071729218562685355881418106440, −7.945377299913127727532994829665, −6.69992826322664140620538681665, −6.35306147256574072432179047412, −6.29316757927509488643773547834, −5.34945783064988058695168525193, −5.29439943415338779506744522168, −4.48531193864795315694041050402, −3.53176922528875095704922238510, −3.29699975126484092788987476482, −2.18562128408643501857872580519, −0.919526999450248053192487619525, 0.919526999450248053192487619525, 2.18562128408643501857872580519, 3.29699975126484092788987476482, 3.53176922528875095704922238510, 4.48531193864795315694041050402, 5.29439943415338779506744522168, 5.34945783064988058695168525193, 6.29316757927509488643773547834, 6.35306147256574072432179047412, 6.69992826322664140620538681665, 7.945377299913127727532994829665, 8.071729218562685355881418106440, 8.746535765801052966997753191886, 9.108795455289480583011416986557, 9.930723730317650982781533521356

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